text:
\begin{document}

\maketitle
\pagestyle{plain}
\footnote{2008 Mathematical Subject Classification 11F80 (primary), 11F85, 11S25 (secondary).
Keywords: $p$-adic Hodge theory, trianguline representations, B-pairs.}
\begin{abstract}
The aims of this article are to study deformation theory of
trianguline $B$-pairs and to construct a $p$-adic family of two dimensional trianguline
representations for any $p$-adic field. The deformation theory is the generalization
of Bella\"iche-Chenevier's and Chenevier's works in the $\mathbb{Q}_p$-case,
 where they used ($\varphi,\Gamma$)-modules over the Robba ring instead of using $B$-pairs.
 Generalizing and modifying Kisin's theory of $X_{fs}$ for any $p$-adic field, we construct
 a $p$-adic family of two dimensional trianguline representations.
 As an application of these theories, we prove a theorem concerning
 Zariski density of two dimensional
 crystalline representations for any $p$-adic field, which is a generalization of Colmez's and Kisin's theorem for $\mathbb{Q}_p$-case.
\end{abstract}
\setcounter{tocdepth}{1}
\tableofcontents

\section{Introduction.}
\subsection{Background.}
Let $p$ be a prime number and $K$ be a $p$-adic field, i.e. finite
extension of $\mathbb{Q}_p$. The theory of trianguline representations (which form a class of $p$-adic representations
of $G_K:=\mathrm{Gal}(\overline{K}/K)$), in particular the theory of their $p$-adic families turns out to be very important in the study of $p$-adic Galois representations parametrized by
$p$-adic families  of automorphic forms (in particular eigenvarieties).
Inspired by Kisin's $p$-adic Hodge theoretic study of Coleman-Mazur eigencurve (\cite{Ki03}),
 Colmez (\cite{Co08})
defined the notion of trianguline representations by using Fontaine's and Kedlaya's theory of ($\varphi,\Gamma$)-modules over the Robba ring in the study of $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$.
Based on their works, Bella\"iche-Chenevier  (\cite{Bel-Ch09}) and Chenevier (\cite{Ch09b}) studied deformation theory of trianguline representations
and $p$-adic families of trianguline representations. These theories are the most fundamental tools for their study
of unitary eigenvarieties.
Because all their studies are limited to the case $K=\mathbb{Q}_p$, we didn't have any results
concerning to $p$-adic Hodge theoretic properties of eigenvarieties  over a number field $F$
except when $F$ is $\mathbb{Q}$ or more generally is a number field in which $p$ splits completely.

On the other hands, in \cite{Na09}, the author of this article generalized
many results of  \cite{Co08} for any $p$-adic field $K$. The author proved some fundamental properties of trianguline representations and then classified two dimensional trianguline representations for any $p$-adic field, where we studied trianguline representations by using
$B$-pairs, which was defined by Berger in \cite{Be09}, instead of using ($\varphi,\Gamma$)-modules over the Robba ring.

The aim of this article is
to generalize Kisin's, Bella\"iche-Chenevier's  and Chenevier's works for any $p$-adic field $K$, more precisely,
to develop  deformation
theory of trianguline representations and to construct  a $p$-adic family of two dimensional
trianguline representations for any $p$-adic field $K$. The author thinks that these generalizations are fundamental for applications
to  $p$-adic Hodge theoretic study of eigenvarieties for more general number fields.
Moreover, as an application of these theories,
we prove some theorems (see Theorem \ref{000} and Theorem\ref{0000} in Introduction)
concerning Zariski density of two dimensional crystalline representations for any $p$-adic field.
These results are  generalizations of  a theorem of Colmez and Kisin when $K=\mathbb{Q}_p$, which played
some crucial roles in the proof of $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$ (\cite{Co10}, \cite{Ki10}, \cite{Pa10}).

In the next article (\cite{Na11}) which is based on many results of this article, we construct a $p$-adic family of $d$-dimensional trianguline representations
for any $d\in \mathbb{Z}_{\geqq 1}$  and for any $K$ and prove a theorem concerning to
Zariski density of $d$-dimensional crystalline representations for any $d$ and $K$.

In future works, the author wants to study eigenvarieties which parametrize finite
slope $p$-adic Hilbert modular forms by using the results of this article.

  \subsection{Overview.}

 Here, we explain the details of this article.
 \smallskip

 In $\S$ 2, we study the deformation theory of trianguline $B$-pairs, which is the
 generalization of the studies of \cite{Bel-Ch09}, \cite{Ch09b} for any $p$-adic field.

 In $\S$ 2.1, we recall the definition of $B$-pairs and some fundamental properties of trianguline
 $B$-pairs proved in \cite{Na09} and then we extend these to Artin local ring coefficients case. Let $E$ be a suitable finite extension of $\mathbb{Q}_p$ as in Notation below. We recall the definition of $E$-$B$-pairs of $G_K$, which is the $E$-coefficient version of $B$-pairs. We write $B_e:=B_{\mathrm{cris}}^{\varphi=1}$.
 An $E$-$B$-pair is a pair $W=(W_e,W^+_{\mathrm{dR}})$ where $W_e$ is a finite free $B_e\otimes_{\mathbb{Q}_p}E$-module
 with a continuous semi-linear $G_K$-action such that $W^+_{\mathrm{dR}}\subseteq W_{\mathrm{dR}}:=B_{\mathrm{dR}}\otimes_{B_e}W_e$
 is a $G_K$-stable $B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E$-lattice of $W_{\mathrm{dR}}$. The category of $E$-representations
 of $G_K$ is embedded in the category of $E$-$B$-pairs by $V\mapsto W(V):=(B_e\otimes_{\mathbb{Q}_p}V, B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}V)$.
 We say that an $E$-$B$-pair is split trianguline if $W$ is a successive extension of rank one $E$-$B$-pairs, i.e. $W$ has a filtration
 $0\subseteq W_1\subseteq W_2\subseteq \cdots \subseteq W_{n-1}\subseteq W_n=W$ such that $W_i$ is a saturated sub $E$-$B$-pair
 of $W$ and $W_i/W_{i-1}$ is a rank one $E$-$B$-pair for any $1\leqq i\leqq n$. We say that $W$ is trianguline
 if $W\otimes_{E}E'$ is a $E'$-split trianguline $E'$-$B$-pair for a finite extension $E'$ of $E$. We say that an $E$-representation $V$ is split trianguline
 (resp. trianguline) if $W(V)$ is split trianguline (resp. trianguline). By these definitions, to study trianguline $E$-$B$-pairs, we first need to classify
 rank one $E$-$B$-pairs and then we need to calculate extension class group of them, which were studied in \cite{Co08} for $K=\mathbb{Q}_p$ and in
 \cite{Na09} for general $K$. In $\S$ 2.1, we recall these results which we need to study the deformation theory of trianguline $E$-$B$-pairs.
We define the Artin local ring coefficient version of $B$-pairs. Let $\mathcal{C}_E$ be the category of Artin local $E$-algebras with the residue field isomorphic to $E$. For any $A\in \mathcal{C}_E$, we say that $W:=(W_e,W^+_{\mathrm{dR}})$ is an $A$-$B$-pair if $W_e$ is a finite free $B_e\otimes_{\mathbb{Q}_p}A$-module with a continuous semi-linear $G_K$-action and
 $W^+_{\mathrm{dR}}\subseteq W_{\mathrm{dR}}:=B_{\mathrm{dR}}\otimes_{B_e}W_e$ is a $G_K$-stable $B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A$-lattice. We generalize some results of \cite{Na09} for $A$-$B$-pairs.

 In $\S$ 2, we study two types of deformations of split trianguline $E$-$B$-pairs.
 In $\S$ 2.2, first we study the
 usual deformation for any $E$-$B$-pairs which is the generalization of Mazur's deformation theory
 of $p$-adic Galois representations.
 Let $W$ be an $E$-$B$-pair and $A\in \mathcal{C}_E$.
 We say that $(W_A,\iota)$ is a deformation of $W$ over $A$ if $W_A$ is an $A$-$B$-pair and $\iota:W_A\otimes_A E\isom W$ is an
 isomorphism. We define the deformation functor of $W$, $D_W:\mathcal{C}_E\rightarrow (\mathrm{Sets})$ by
 $D_W(A):=\{$equivalent classes of deformations $(W_A,\iota)$ of $W$ over $A$ $\}$. We prove the following proposition concerning to
 pro-representability and formally smoothness and dimension formula
 of $D_W$.

  \begin{prop} (Corollary \ref{13})
    Let $W$ be an $E$-$B$-pair of rank $n$. If $W$ satisfies the following conditions,
    \begin{itemize}
    \item[(1)]$\mathrm{End}_{G_K}(W)=E$,
    \item[(2)]$\mathrm{H}^2(G_K, \mathrm{ad}(W))=0$,
    \end{itemize}
    then the functor $D_W$ is pro-representable by $R_W$ such that
    $$R_W\isom E[[T_1,\cdots,T_d]]\,\,\text{ where } d:=[K:\mathbb{Q}_p]n^2 +1.$$
    \end{prop}

 In $\S$ 2.3, we study the other more important type of deformation, i.e. trianguline deformation. Let $W$ be a split trianguline $E$-$B$-pair of rank $n$ and
 $\mathcal{T}:0\subseteq W_1\subseteq W_2\subseteq \cdots \subseteq W_{n-1}\subseteq W_n=W$ be a fixed triangulation of $W$.  For any $A\in \mathcal{C}_E$,
 we say that $(W_A, \iota, \mathcal{T}_A)$ is a trianguline deformation of $(W,\mathcal{T})$ over $A$ if $(W_A,\iota)$ is a deformation of $W$
 over $A$ and $\mathcal{T}_A:0\subseteq W_{1,A}\subseteq \cdots \subseteq W_{n-1,A}\subseteq W_{n,A}=W_A$ is an $A$-triangulation of $W_A$
( i.e. $W_{i,A}$ is a saturated sub $A$-$B$-pair of $W_A$ such that $W_{i,A}/W_{i-1,A}$ is a rank one $A$-$B$-pair for any $i$) such that
$\iota(W_{i,A}\otimes_A E)=W_i$ for any $i$. We define the trianguline deformation functor $D_{W,\mathcal{T}}:
\mathcal{C}_E\rightarrow (\mathrm{Sets})$ of $(W,\mathcal{T})$ by $D_{W,\mathcal{T}}(A):=\{ $equivalent classes of trianguline deformations $(W_A,\iota,\mathcal{T}_A)$
of $(W,\mathcal{T})$ over $A$$\}$. We prove the following proposition concerning to the pro-representability and formally smoothness and dimension formula of this functor, which is a generalization of Proposition 3.6 of \cite{Ch09b} for any $p$-adic field.

\begin{prop}\label{000000}(Proposition \ref{19})
Let $W$ be a split trianguline $E$-$B$-pair of rank $n$ with a triangulation
$\mathcal{T}:0\subseteq W_1\subseteq \cdots\subseteq W_{n-1}\subseteq W_n=W$.
We assume that $(W,\mathcal{T})$ satisfies the following conditions,
\begin{itemize}
\item[(0)]$\mathrm{End}_{G_K}(W)=E$,
\item[(1)]For any $1\leqq i < j\leqq n$,  $\delta_j/\delta_i\not= \prod_{\sigma\in\mathcal{P}}\sigma^{k_{\sigma}}$
for any $\{k_{\sigma}\}_{\sigma\in\mathcal{P}}\in \prod_{\sigma\in\mathcal{P}}\mathbb{Z}_{\leqq 0}$
\item[(2)]For any $1\leqq i < j\leqq n$,
$\delta_i/\delta_j\not= |\mathrm{N}_{K/\mathbb{Q}_p}|\prod_{\sigma\in\mathcal{P}}\sigma^{k_{\sigma}}$
for any $\{k_{\sigma}\}_{\sigma\in\mathcal{P}}\in \prod_{\sigma\in\mathcal{P}}\mathbb{Z}_{\geqq 1}$,
\end{itemize}
then $D_{W,\mathcal{T}}$ is pro-represented by a quotient ring  $R_{W,\mathcal{T}}$ of $R_W$ such
that $$R_{W,\mathcal{T}}\isom E[[T_1,\cdots, T_{d_n}]]\text{ where }d_n:=\frac{n(n+1)}{2}[K:\mathbb{Q}_p]+1.$$
\end{prop}

In $\S$ 2.4, we define the notion of benign $E$-$B$-pairs, which forms a special good class of split trianguline and potentially crystalline
 $E$-$B$-pairs, and prove a theorem concerning to tangent spaces of the deformation rings of this class.
In $\S$ 2.4.1, we define the notion of benign $E$-$B$-pairs , in \cite{Ch09b} this class is called generic, in this article we follow the terminology of \cite{Ki10}. Let $W$ be a potentially crystalline $E$-$B$-pair of rank $n$ such that
$W|_{G_L}$ is crystalline for a finite totally ramified abel extension $L$ of $K$ (we call such a representation a crystabelline representation).
We assume that $D_{\mathrm{cris}}^{L}(W):=(B_{\mathrm{cris}}\otimes_{B_e}W_e)^{G_L}=K_0\otimes_{\mathbb{Q}_p}Ee_1\oplus
\cdots \oplus K_0\otimes_{\mathbb{Q}_p}Ee_n$ such that $K_0\otimes_{\mathbb{Q}_p}Ee_i$ are preserved by $(\varphi,\mathrm{Gal}(L/K))$-action
and that $\varphi^f(e_i)=\alpha_ie_i$ for some $\alpha_i\in E^{\times}$, here $f:=[K_0:\mathbb{Q}_p]$ and $K_0$ is the maximal unramified extension of $\mathbb{Q}_p$ in $K$.
We denote by $\{k_{1,\sigma}.k_{2,\sigma},\cdots,k_{n,\sigma}\}_{\sigma:K\hookrightarrow \overline{K}}$ the Hodge-Tate weight of $W$ such that
$k_{1,\sigma}\geqq k_{2,\sigma}\geqq \cdots \geqq k_{n,\sigma}$ for any $\sigma:K\hookrightarrow \overline{K}$, in this paper we define
the Hodge-Tate weight of $\mathbb{Q}_p(1)$ by $1$.
Let $\mathfrak{S}_n$ be the $n$-th permutation group. For any $\tau\in\mathfrak{S}_n$, we can define a filtration on $D^L_{\mathrm{cris}}(W)$
by $0\subseteq K_0\otimes_{\mathbb{Q}_p}Ee_{\tau(1)}\subseteq K_0\otimes_{\mathbb{Q}_p}Ee_{\tau(1)}\oplus K_0\otimes_{\mathbb{Q}_p}Ee_{\tau(2)}\subseteq
\cdots \subseteq K_0\otimes_{\mathbb{Q}_p}Ee_{\tau(1)}\oplus \cdots \oplus K_0\otimes_{\mathbb{Q}_p}Ee_{\tau(n-1)}\subseteq D^L_{\mathrm{cris}}(W)$ by
sub $E$-filtered $(\varphi, G_K)$-modules, where the filtration of $K_0\otimes_{\mathbb{Q}_p}Ee_{\tau(1)}\oplus \cdots \oplus K_0\otimes_{\mathbb{Q}_p}Ee_{\tau(i)}$ is
the one induced from that of $D^L_{\mathrm{dR}}(W)=L\otimes_{K_0}D^L_{\mathrm{cris}}(W)$. By the equivalence between
the category of potentially crystalline $B$-pairs and the category of filtered $(\varphi,G_K)$-modules, for any $\tau\in \mathfrak{S}_n$
we obtain the triangulation $\mathcal{T}_{\tau}:0\subseteq W_{\tau,1}\subseteq W_{\tau,2}\subseteq \cdots \subseteq W_{\tau,n}=W$
such that
$W_{\tau,i}$ is potentially crystalline and $D^L_{\mathrm{cris}}(W_{\tau,i})\isom K_0\otimes_{\mathbb{Q}_p}Ee_{\tau(1)}\oplus \cdots \oplus K_0\otimes_{\mathbb{Q}_p}E
e_{\tau(i)}$ for any $1\leqq i\leqq n$.

Under this situation, we define the notion of benign $E$-$B$-pairs as follows.
\begin{defn}\label{ben}
Let $W$ be a potentially crystalline $E$-$B$-pair of rank $n$ as above. Then we say that $W$ is benign if $W$ satisfies the following;
\begin{itemize}
\item[(1)] For any $i\not= j$, we have $\alpha_i/\alpha_j\not= 1, p^f, p^{-f}$,
\item[(2)] For any $\sigma:K\hookrightarrow \overline{K}$, we have $k_{1,\sigma}>k_{2,\sigma}>\cdots >k_{n-1,\sigma}> k_{n,\sigma}$,
\item[(3)] For any $\tau\in \mathfrak{S}_n$ and $1\leqq i \leqq n$, the Hodge-Tate weight of $W_{\tau,i}$ is $\{k_{1,\sigma},k_{2,\sigma},\cdots,k_{i,\sigma}\}_{\sigma :K\hookrightarrow \overline{K}}$.
\end{itemize}
\end{defn}

In $\S$ 2.4.2, we prove the main theorem of $\S$ 2. Let $W$ be a benign $E$-$B$-pair of rank $n$ as above. For any $\tau\in \mathfrak{S}_n$, we can define
the trianguline deformation functor $D_{W,\mathcal{T}_{\tau}}$. Let $R_W$ be the universal deformation ring of $D_W$ and
$R_{W,\mathcal{T}_{\tau}}$ be the universal deformation ring of $D_{W,\mathcal{T}_{\tau}}$ for any $\tau\in \mathfrak{S}_n$.

Let $t(R_{W})$ and $t(R_{W,\mathcal{T}_\tau})$  be the tangent spaces
of $R_W$ and $R_{W,\mathcal{T}_{\tau}}$. For any $\tau\in \mathfrak{S}_n$, $t(R_{W,\mathcal{T}_{\tau}})$ is a sub $E$-vector space of $t(R_W)$.
The main theorem of $\S$ 2 is the following, which is the generalization of Theorem 3.19 of  \cite{Ch09b} for any
$p$-adic field.
\begin{thm}$(\text{Theorem } \mathrm{\ref{33}} )$
Let $W$ be a benign $E$-$B$-pair of rank $n$, then we
 have an equality $$\sum_{\tau\in \mathfrak{S}_n}t(R_{W,\mathcal{T}_{\tau}})=t(R_W).$$
 \end{thm}

 This theorem is a crucial local result for applications to some Zariski density theorems of
 local or global $p$-adic Galois representations. In fact, using
 this theorem for $K=\mathbb{Q}_p$, Chenevier (\cite{Ch09b}) proved a theorem concerning to Zariski density of unitary automorphic Galois representations
 in universal deformation spaces of three dimensional self dual $p$-adic representations of $G_{F}$ for any CM field $F$
 in which $p$ splits completely. Moreover, this theorem is also a crucial result for the proof of
 Zariski density of crystalline representations in universal deformation spaces of $p$-adic Galois representations of $p$-adic fields.
 In the rest of this paper $\S$ 3 and $\S4$, we apply this theorem only for the two dimensional case. Using
 this theorem for $K=\mathbb{Q}_p$, Chenevier (\cite{Ch10}) recently
 proved Zariski density of crystalline representations for higher dimensional and $K=\mathbb{Q}_p$ case.
 In the next paper (\cite{Na11}), the author uses this theorem for proving Zariski density of crystalline representations for higher dimensional and
 any $p$-adic field case.

 \smallskip

 In $\S$ 3, we construct $p$-adic families of two dimensional trianguline representations for any $p$-adic field
 by generalizing Kisin's theory of finite slope sub space $X_{fs}$ in \cite{Ki03} for any $p$-adic field.
 As in $\mathbb{Q}_p$ case of \cite{Ki03}, \cite{Ki10}, this family is essential for the proof of Zariski density of two dimensional crystalline
 representations in $\S4$ and for $p$-adic Hodge theoretic studies of Hilbert modular eigenvarieties, which
 will be a subject  in author's subsequent works.

 In $\S$ 3.1, we prove some propositions concerning Banach $G_K$-modules which we need for constructing
 $p$-adic families of trianguline representations. In particular, we show that these Banach $G_K$-modules can be
 obtained naturally from some almost $\mathbb{C}_p$-representations (\cite{Fo03}). For us, one of the important properties of these
 Banach $G_K$-modules is orthonormalizability as Banach modules over some Banach algebras. All these properties follow from some general facts of almost
 $\mathbb{C}_p$-representations.

 In $\S$ 3.2, for any separated rigid analytic space $X$ over $E$ and for any
 finite free $\mathcal{O}_X$-module $M$ with a continuous $G_K$-action and for any invertible function
 $Y$ on $X$, we construct a Zariski closed sub rigid analytic space $X_{fs}$ of $X$, which is ``roughly" defined as
 the subspace of $X$ consisting of points $x\in X$ such that $D_{\mathrm{cris}}^+(M(x))^{\varphi^f=Y(x)}\not= 0$,
 where  $M(x)$ is the fiber of $M$ at $x$ and $f:=[K_0:\mathbb{Q}_p]$. For the precise characterization
 of $X_{fs}$, see Theorem \ref{5.8}. This construction is the generalization of
 Kisin's $X_{fs}$ in $\S5$ of \cite{Ki03} for any $p$-adic field. After obtaining the results in $\S$ 3.1, the construction
 and the proof is almost all the same as that of  \cite{Ki03}, a difference is that we need to consider all the
 embeddings $\tau:K\hookrightarrow \overline{K}$. But, for convenience of readers or the author, we choose to re-prove this
 construction in full detail.

 In $\S$ 3.3, we apply this construction to the rigid analytic spaces associated to
 the universal deformation ring of two dimensional mod $p$-representations of $G_K$.
 Let $\bar{\rho}:G_K\rightarrow \mathrm{GL}_2(\mathbb{F})$ be
 a two dimensional mod $p$ representation of $G_K$, where $\mathbb{F}$ is the residue field of $E$.
 For simplicity, in this paper, we assume that $\mathrm{End}_{\mathbb{F}[G_K]}(\bar{\rho})=\mathbb{F}$, then there exists the universal deformation ring $R_{\bar{\rho}}$
 of $\bar{\rho}$, which is a local complete noetherian $\mathcal{O}$-algebra, where $\mathcal{O}$ is the integer ring of $E$. Let $\mathfrak{X}(\bar{\rho})$ be the rigid
 analytic space over $E$ associated to $R_{\bar{\rho}}$. The universal deformation $V^{\mathrm{univ}}$ of $\bar{\rho}$ over $R_{\bar{\rho}}$ defines a rank two free $\mathcal{O}_{\mathfrak{X}(\bar{\rho})}$-module $\tilde{V}^{\mathrm{univ}}$ with a continuous $\mathcal{O}_{\mathfrak{X}(\bar{\rho})}$-linear $G_K$-action.
 $\mathfrak{X}(\bar{\rho})$ parametrizes $p$-adic representations $V$ of $G_K$ whose reductions are isomorphic to
 $\bar{\rho}$ for some $G_K$-stable lattices of $V$.
  Let $\mathcal{W}$ be the rigid analytic space over $E$ which represents the functor $D_{\mathcal{W}}:\{$ rigid analytic spaces over $E\}
  \rightarrow (\mathrm{Sets})$ defined by $D_{\mathcal{W}}(X):=\{\delta:\mathcal{O}_{K}^{\times}\rightarrow \mathcal{O}_X^{\times}:$ continuous homomorphisms $\}$ for any rigid analytic space $X$ over $E$.
  Let $\delta^{\mathrm{univ}}:\mathcal{O}_K^{\times}\rightarrow \mathcal{O}_{\mathcal{W}}^{\times}$ be the
  universal homomorphism. If we fix a uniformizer $\pi_K\in K$, there exists unique character $\tilde{\delta}^{\mathrm{univ}}
  :G_K^{\mathrm{ab}}\rightarrow \mathcal{O}_{\mathcal{W}}^{\times}$ such that $\tilde{\delta}^{\mathrm{univ}}\circ \mathrm{rec}_K|_{\mathcal{O}_K^{\times}}=\delta^{\mathrm{univ}}$ and $\tilde{\delta}^{\mathrm{univ}}\circ \mathrm{rec}_K(\pi_K)=1$, where
  $\mathrm{rec}_K:K\hookrightarrow G_K^{\mathrm{ab}}$ is the reciprocity map of local
  class field theory.
  We denote by $X(\bar{\rho}):=\mathfrak{X}(\bar{\rho})\times_E\mathcal{W}\times_E \mathbb{G}_{m,E}^{\mathrm{an}}$ and
  denote by $p_1:X(\bar{\rho})\rightarrow \mathfrak{X}(\bar{\rho})$, $p_2:X(\bar{\rho})\rightarrow \mathcal{W}$ and
  $p_3:X(\bar{\rho})\rightarrow \mathbb{G}_{m,E}^{\mathrm{an}}$ the canonical projections.
  For any $x\in X(\bar{\rho})$, we denote by $E(x)$ the residue field at $x$  which is a finite extension of $E$.

  For any two dimensional trianguline representation $V\in \mathfrak{X}(\bar{\rho})$ with a triangulation $\mathcal{T}:0\subseteq W(\delta_1)
  \subseteq W(V\otimes_E E')$ for some $E'$, we define a point
  $x_{(V,\mathcal{T})}:=(V,\delta_1|_{\mathcal{O}_K}^{\times},\delta_1(\pi_K))\in X(\bar{\rho})$. We define
  $M:=p_1^*(\tilde{V}^{\mathrm{univ}})((p_2^*\tilde{\delta}^{\mathrm{univ}})^{-1})$ a rank two $\mathcal{O}_{X(\bar{\rho})}$-module
  with a continuous $\mathcal{O}_{X(\bar{\rho})}$-linear $G_K$-action. Let $Y:=p_3^*(T)\in \mathcal{O}_{X(\bar{\rho})}^{\times}$ be the pullback
  of the canonical coordinate of $\mathbb{G}_{m,E}^{\mathrm{an}}$. If we apply the construction of $X_{fs}$ to
  the triple $(X(\bar{\rho}), M, Y)$, we obtain a Zariski closed sub rigid analytic space $X(\bar{\rho})_{fs}$ of $X(\bar{\rho})$, which we denote by $\mathcal{E}(\bar{\rho}):=X(\bar{\rho})_{fs}$. The main result of $\S$ 3 is the following theorem (see Theorem \ref{5.14} and
  Theorem \ref{5.18} for more precise statements), which is a generalization of Proposition 10.4 and 10.6 of \cite{Ki03}.

  \begin{thm}
  $\mathcal{E}(\bar{\rho})$ satisfies the following properties.
  \begin{itemize}
  \item[(1)] For any point $x:=([V_x],\delta_x,\lambda_x)\in \mathcal{E}(\bar{\rho})$, $V_x$ is a trianguline representation.
  \item[(2)] Conversely, if $x=[V_x]\in \mathfrak{X}(\bar{\rho})$ is a point such that $V_x$ is a split trianguline $E(x)$-representation with a triangulation
  $\mathcal{T}:0\subseteq W(\delta_1)\subseteq W(V_x)$ satisfying all the conditions in
  Proposition \ref{000000}, then the point $x_{(V_x,\mathcal{T})}\in X(\bar{\rho})$ defined above is contained in $\mathcal{E}(\bar{\rho})$.
  \item[(3)] For any point $x_{(V_x,\mathcal{T})}$ as in $(2)$, there exists an isomorphism $\hat{\mathcal{O}}_{\mathcal{E}(\bar{\rho}),x_{(V_x,\mathcal{T})}}\isom
  R_{V_x,\mathcal{T}}$, in particular $\mathcal{E}(\bar{\rho})$ is smooth of dimension $3[K:\mathbb{Q}_p]+1$ at such points.
  \end{itemize}

  \end{thm}

  In the next paper (\cite{Na11}), the author will generalize all these results for higher dimensional case.

  \smallskip

  In the final section $\S$ 4, as an application of $\S$ 2 (in the two dimensional case) and of $\S$ 3, we prove
  the following theorems concerning Zariski density of two
  dimensional crystalline representations.
  We denote by $\mathfrak{X}(\bar{\rho})_{\mathrm{reg}-\mathrm{cris}}:=\{x\in \mathfrak{X}(\bar{\rho})|
  V_x $ (the $p$-adic representation corresponding to $x$) is crystalline with Hodge-Tate weight
  $\{k_{1,\sigma},k_{2,\sigma}\}_{\sigma:K\hookrightarrow \overline{K}}$ such that $k_{1,\sigma}\not= k_{2,\sigma}$ for
  any $\sigma \}$, $\mathfrak{X}(\bar{\rho})_{\mathrm{b}}:=\{x\in \mathfrak{X}(\bar{\rho})| V_x $ is crystalline
  and benign $\}$. We denote by $\overline{\mathfrak{X}(\bar{\rho})}_{\mathrm{b}}$ the Zariski closure of $\mathfrak{X}(\bar{\rho})_{\mathrm{b}}$ in $\mathfrak{X}(\bar{\rho})$.
  \begin{thm}$(\text{Theorem } \mathrm{\ref{6.14}} )$\label{000}
  If $\mathfrak{X}(\bar{\rho})_{\mathrm{reg}-\mathrm{cris}}$ is non empty,
  then $\overline{\mathfrak{X}(\bar{\rho})}_{\mathrm{b}}$ is non empty and a union of irreducible components of $\mathfrak{X}(\bar{\rho})$.

  \end{thm}

   We note that we can prove all the above theorems even if $\mathrm{End}_{G_K}(\bar{\rho})\not=\mathbb{F}$ without any additional
  difficulties using the universal framed deformations instead of using usual deformations.

  Moreover, under the following assumptions, we prove the following stronger theorem.

  \begin{thm}$(\text{Theorem } \mathrm{\ref{4.17}})$\label{0000}
We assume the following conditions,
\begin{itemize}
\item[(0)]$\mathrm{End}_{G_K}(\bar{\rho})=\mathbb{F}$,
\item[(1)]$\mathfrak{X}(\bar{\rho})_{\mathrm{reg}-\mathrm{cris}}$ is not empty,
\item[(2)]$\bar{\rho}\not\sim \begin{pmatrix} 1 & \ast \\ 0 & \omega \end{pmatrix}\otimes \chi$ for
any $\chi:G_K\rightarrow \mathbb{F}^{\times}$, where $\omega$ is the mod $p$ cyclotomic character,

\item[(3)]$[K(\zeta_p):K]\not=2$ or, for any $\chi$, $\bar{\rho}|_{I_K}\not\sim \begin{pmatrix} \chi_2^i & 0 \\ 0 & \chi_2^{ip^f}\end{pmatrix}\otimes \chi$ such that $i=\frac{p^f+1}{2}$, where $\zeta_p$ is a primitive $p$-th root of $1$ and $\chi_2:I_K\rightarrow
\mathbb{F}^{\times}$ is a fundamental character of the second kind,
\end{itemize}
then we have an equality $\overline{\mathfrak{X}(\bar{\rho})}_{\mathrm{b}}=\mathfrak{X}(\bar{\rho})$.

\end{thm}

  This theorem generalizes the results of Colmez (\cite{Co08}) and Kisin (\cite{Ki10}) for  any $p$-adic field case.
  As in stated in the above paragraph, Chenevier recently proved similar results for higher dimensional
  and for $K=\mathbb{Q}_p$ and the author will prove these theorems in full generality (i.e. for higher dimensional
  and for any $p$-adic field) in the next paper.

\subsection*{Notation.}
Let $p$ be a prime number. $K$ be a finite extension of $\mathbb{Q}_p$, $\overline{K}$  a
fixed algebraic closure of $K$, $K_0$ the maximal unramified extension of $\mathbb{Q}_p$ in $K$,
$K^{\mathrm{nor}}$ the Galois closure of $K$ in $\overline{K}$. Let $G_K:=\mathrm{Gal}(\overline{K}/K)$ be the
absolute Galois group of $K$ equipped with pro-finite topology. $\mathcal{O}_K$ is the ring of integers of $K$, $\pi_K\in\mathcal{O}_K$ is
a uniformizer of $K$, $k:=\mathcal{O}_K/\pi_K\mathcal{O}_K$ is the residue field
of $K$, $q=p^f:=\sharp k$ is the order of $k$, $\chi_p:G_K\rightarrow \mathbb{Z}_p^{\times}$ is the $p$-adic cyclotomic character
(i.e. $g(\zeta_{p^n})=\zeta_{p^n}^{\chi(g)}$ for any $p^n$-th roots of unity and for any $g\in G_K$). Let $\mathbb{C}_p:=\hat{\overline{K}}$
be the $p$-adic completion of $\overline{K}$, which is an algebraically closed $p$-adically completed field, and $\mathcal{O}_{\mathbb{C}_p}$ be its ring of integers. We denote by $v_p$ the normalized valuation on $\mathbb{C}_p^{\times}$ such that
$v_p(p)=1$. We denote by $|-|_p:\mathbb{C}_p\rightarrow \mathbb{R}_{\geqq 0}$ the absolute value such that
$|p|_p=\frac{1}{p}$. Let $E$ be a finite extension of $\mathbb{Q}_p$ in $\overline{K}$ such that
$K^{\mathrm{nor}}\subseteq E$. In this paper, we use the notation $E$ as a coefficient field of representations.
We denote by $\mathcal{P}:=\{\sigma:K\hookrightarrow \overline{K}\}=\{\sigma:K\hookrightarrow E\}$ the set of
$\mathbb{Q}_p$-algebra homomorphisms from $K$ to $\overline{K}$ ( or $E$).
Let $\chi_{\mathrm{LT}}:G_K\rightarrow \mathcal{O}_K^{\times}\hookrightarrow \mathcal{O}_E^{\times}$ be
the Lubin-Tate character associated with the fixed uniformizer $\pi_K$. Let $\mathrm{rec}_K:K^{\times}
\rightarrow G_K^{\mathrm{ab}}$ be the reciprocity map of local class field theory such that
$\mathrm{rec}_K(\pi_K)$ is a lifting of the inverse of $q$-th power Frobenius on $k$, then
$\chi_{\mathrm{LT}}\circ \mathrm{rec}_K:K^{\times}\rightarrow \mathcal{O}_K^{\times}$ satisfies
$\chi_{\mathrm{LT}}\circ\mathrm{rec}_K(\pi_K)=1$ and $\chi_{\mathrm{LT}}\circ\mathrm{rec}_K|_{\mathcal{O}_K^{\times}}
=\mathrm{id}_{\mathcal{O}_K^{\times}}$.
\subsection*{Acknowledgement.}
The author would like to thank Kenichi Bannai for reading the first version of this article and for helpful comments.
He also would like to thank Tadashi Ochiai for many valuable and interesting discussions and also
thank Yoichi Mieda, Atsushi Shiho and Takahiro Tsushima for discussing
about rigid geometry which we use in $\S$ 4 and  also thank Ga\"etan Chenevier for discussing
about generalization to higher dimensional case and also thank Go Yamashita and Seidai Yasuda for
pointing out some mistakes in the previous version of this article and for many valuable discussions.

The author is supported by KAKENHI (21674001).  This research was also supported in part by
the JSPS Core-to-Core program 28005 (Represented by Makoto Matsumoto).
Part of this paper was written at l'Institute Henri Poincar\'e during Galois Trimester.  The author would like to thank his host
Pierre Colmez and all of the people at Keio University related to the JSPS ITP program which enabled my stay.

\section{Deformation theory of trianguline $B$-pairs.}
\subsection{Review of $B$-pairs.}
\subsubsection{$E$-$B$-pairs}
We start by recalling the definition of $E$-$B$-pairs (\cite{Be09},  \cite{Na09}) and then recall fundamental
properties of them established in \cite{Na09}.
First, we recall some $p$-adic period rings ([Fo94]) which we need for defining $B$-pairs.
Let $\widetilde{\mathbb{E}}^+:=\varprojlim_n\mathcal{O}_{\mathbb{C}_p}\isom
\varprojlim_n\mathcal{O}_{\mathbb{C}_p}/p\mathcal{O}_{\mathbb{C}_p}$, where the limits are taken with respect to $p$-th power maps. It is known that $\widetilde{\mathbb{E}}^+$ is a complete valuation ring of characteristic  $p$ whose valuation
is defined by $v(x):=v_{p}(x^{(0)})$  (here $x=(x^{(n)})_{n\geqq 0}\in \varprojlim_n\mathcal{O}_{\mathbb{C}_p}$).
We fix a system of $p^n$-th roots of unity $\{\varepsilon^{(n)}\}_{n\ge 0}$
such that $\varepsilon^{(0)}=1$, $(\varepsilon^{(n+1) })^p=\varepsilon^{(n)}$,
$\varepsilon^{(1)}\not= 1$. Then $\varepsilon:=(\varepsilon^{(n)})$ is an element of $\widetilde{\mathbb{E}}^+$ such that $v(\varepsilon -1)=p/(p-1)$. $G_K$ acts on this ring continuously in natural way. We put  $\widetilde{\mathbb{A}}^+:=W(\widetilde{\mathbb{E}}^+)$, where, for a perfect ring $R$,
$W(R)$ is the Witt ring of $R$. We put $\widetilde{\mathbb{B}}^+:=\widetilde{\mathbb{A}}^+[\frac{1}{p}]$. These rings are equipped with the weak topology and also have natural continuous $G_K$-actions and Frobenius
actions $\varphi$. Then we have a $G_K$-equivariant surjection $\theta:\widetilde{\mathbb{A}}^+
\rightarrow \mathcal{O}_{\mathbb{C}_p}:\sum_{k=0}^{\infty}p^k[x_k]\mapsto \sum_{k=0}^{\infty} p^k x_k^{(0)}$, where $[\,\, ]:\widetilde{\mathbb{E}}^+\rightarrow \widetilde{\mathbb{A}}^+$ is the
Teichm\"uller lifting.
By inverting $p$, we obtain a surjection $\widetilde{\mathbb{B}}^+\rightarrow \mathbb{C}_p$.
We put $B_{\mathrm{dR}}^+:=\varprojlim_n\widetilde{\mathbb{B}}^+/(\mathrm{Ker}(\theta))^n$, which is
a complete discrete valuation ring with residue field $\mathbb{C}_p$ and
is equipped with the projective limit topology of the $\mathbb{Q}_p$-Banach spaces
$\widetilde{\mathbb{B}}^+/(\mathrm{Ker}(\theta))^n$ whose $\mathbb{Z}_p$-lattice is the image of the natural map
$\tilde{\mathbb{A}}^+\rightarrow
\tilde{\mathbb{B}}^+/(\mathrm{Ker}(\theta))^n$.
Let $A_{\mathrm{max}}$ be the $p$-adic completion of $\widetilde{\mathbb{A}}^+[\frac{[\tilde{p}]]}{p}]$, where $\tilde{p}:=(p^{(n)})$ is an element in $\widetilde{\mathbb{E}}^+$ such that $p^{(0)}=p, (p^{(n+1) })^p=p^{(n)}$. We put $B_{\mathrm{max}}^+:=A_{\mathrm{max}}[\frac{1}{p}]$. $A_{\mathrm{max}}$ and
$B^+_{\mathrm{max}}$ have continuous $G_K$-actions and Frobenius actions $\varphi$. We have a natural
$G_K$-equivariant embedding $K\otimes_{K_0}B^+_{\mathrm{max}}\hookrightarrow B^+_{\mathrm{dR}}$.
If we put $t:=\mathrm{log}([\varepsilon])=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{([\varepsilon]-1)^n}{n}$, then we can see that $t\in A_{\mathrm{max}}$,
$\varphi(t)=pt, g(t)=\chi_p(g)t$ for any $g\in G_K$ and $\mathrm{Ker}(\theta)=tB_{\mathrm{dR}}^+
\subset B_{\mathrm{dR}}^+$ is the maximal ideal of $B^+_{\mathrm{dR}}$. If we put $B_{\mathrm{max}}:=B_{\mathrm{max}}^+
[\frac{1}{t}], B_{\mathrm{dR}}:=B_{\mathrm{dR}}^+[\frac{1}{t}]$, we have a natural embedding
$K\otimes_{K_0}B_{\mathrm{max}}\hookrightarrow B_{\mathrm{dR}}$.
We put $B_e:=B_{\mathrm{max}}^{\varphi =1}$ on which we equipped the locally convex inductive limit topology of
$B_e=\cup_{n}(\frac{1}{t^n}B^{+}_{\mathrm{max}})^{\varphi=1}$, where the topology on each $(\frac{1}{t^n}B^{+}_{\mathrm{max}})^{\varphi=1}
=\frac{1}{t^n}B^{+ \varphi=p^n}_{\mathrm{max}}$ is induced that of $B^+_{\mathrm{max}}$. We put $\mathrm{Fil}^i B_{\mathrm{dR}}:=t^i B_{\mathrm{dR}}^+$ for any $i\in\mathbb{Z}$.  On $B_{\mathrm{dR}}$, we also equipped with the locally convex inductive limit
topology of $B_{\mathrm{dR}}=\varinjlim_n\frac{1}{t^n}B^+_{\mathrm{dR}}$.

In this paper, we fix a coefficient field of $p$-adic representations or $B$-pairs of $G_K$.
Hence we start by recalling the definition of $E$-coefficient versions of $p$-adic representations and $B$-pairs.
\begin{defn}\label{a}
An $E$-representation of $G_K$ is a finite dimensional $E$-vector space $V$ with a continuous $E$-linear action of $G_K$. We call $E$-representation for simplicity when there is no risk
of confusion about $K$.
\end{defn}
\begin{defn} \label{b}
A pair $W:=(W_e,W^+_{\mathrm{dR}})$ is an $E$-$B$-pair if
\begin{itemize}
\item[(1)] $W_e$ is a finite $B_e\otimes_{\mathbb{Q}_p}E$-module which is free over $B_e$ with
a continuous semi-linear $G_K$-action.
\item[(2)] $W^+_{\mathrm{dR}}$ is a $G_K$-stable finite sub $B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E$-module
of $B_{\mathrm{dR}}\otimes_{B_e}W_e$ which generates $B_{\mathrm{dR}}\otimes_{B_e}W_e$ as a $B_{\mathrm{dR}}$-module.
\end{itemize}
\end{defn}
We have an exact  fully faithful functor $W(-)$ from the category of $E$-representations to the
category of $E$-$B$-pairs defined by
$$W(V):=(B_e\otimes_{\mathbb{Q}_p}V, B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}V)$$
for any $E$-representation $V$,
where the fully faithfulness follows from Bloch-Kato's fundamental short exact sequence,
$$0\rightarrow \mathbb{Q}_p\rightarrow B_e\oplus B^+_{\mathrm{dR}}\rightarrow B_{\mathrm{dR}}\rightarrow 0.$$
$W_e$ is a free $B_e\otimes_{\mathbb{Q}_p}E$-module and
$W^+_{\mathrm{dR}}$ is a free $B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E$-module by Lemma 1.7, 1.8 of \cite{Na09}. We define the rank of $W$ by
$$\mathrm{rank}(W):=\mathrm{rank}_{B_e\otimes_{\mathbb{Q}_p}E}(W_e).$$
For $E$-$B$-pairs $W_1:=(W_{1,e},W^+_{1,\mathrm{dR}})$ and $W_2:=(W_{2,e}, W^+_{2,\mathrm{dR}})$,
we define the tensor product of $W_1$ and $W_2$ by
$$W_1\otimes W_2:=(W_{1,e}\otimes_{B_e\otimes_{\mathbb{Q}_p}E}W_{2,e}, W^+_{1,\mathrm{dR}}\otimes_{B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E}W^+_{2,\mathrm{dR}})$$
 and define the dual of $W_1$
by
$$W_1^{\vee}:=(\mathrm{Hom}_{B_e\otimes_{\mathbb{Q}_p}E}(W_{1,e}, B_{e}\otimes_{\mathbb{Q}_p}E), W^{+,\vee}_{1,\mathrm{dR}})$$
where we define
$$W_{1,\mathrm{dR}}^{+,\vee}:=\{f\in \mathrm{Hom}_{B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E}(B_{\mathrm{dR}}
\otimes_{B_e}W_{1,e}, B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E)| f(W^+_{1,\mathrm{dR}})\subseteq B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E\})$$
  (remark: there is a mistake in
definition 1.9 of \cite{Na09}).
The category of $E$-$B$-pairs is not an abelian category. In particular, an inclusion $W_1\hookrightarrow W_2$
does not have a quotient in the category of $E$-$B$-pairs in general. However we can always take the saturation
$$W^{\mathrm{sat}}_{1}:=(W^{\mathrm{sat}}_{1.e}, W^{+,\mathrm{sat}}_{1,\mathrm{dR}})$$
such that $W^{\mathrm{sat}}_1$ sits in
$W_1\hookrightarrow W^{\mathrm{sat}}_1\hookrightarrow W_2$ and $W_{1,e}=W^{\mathrm{sat}}_{1,e}$ and $W_2/W^{\mathrm{sat}}_{1}$ is an $E$-$B$-pair (Lemma 1.14 of \cite{Na09}). We say that an inclusion $W_1\hookrightarrow W_2$ is saturated if $W_2/W_1$ is an $E$-$B$-pair.

Next, we recall $p$-adic Hodge theory for $B$-pairs. Let $W=(W_e,W^+_{\mathrm{dR}})$ be an $E$-$B$-pair.
We define
$$D_{\mathrm{cris}}(W):=(B_{\mathrm{cris}}\otimes_{B_e}W_e)^{G_K}, \,D^L_{\mathrm{cris}}(W):=
(B_{\mathrm{cris}}\otimes_{B_e}W_e)^{G_L}$$ for any finite extension $L$ of $K$ and define
$$D_{\mathrm{dR}}(W):=(B_{\mathrm{dR}}\otimes_{B_e}
W_e)^{G_K},\, D_{\mathrm{HT}}(W):=(B_{\mathrm{HT}}\otimes_{\mathbb{C}_p}(W^+_{\mathrm{dR}}/tW^+_{\mathrm{dR}}))^{G_K},$$
 here
$B_{\mathrm{HT}}:=\mathbb{C}_p[T, T^{-1}]$ on which $G_K$ acts by $g(aT^i):=\chi_p(g)^ig(a)T^i$ for any $g\in G_K, a\in \mathbb{C}_p, i\in \mathbb{Z}$.
\begin{defn}\label{c}
We say that $W$ is crystalline (resp. de Rham, resp. Hodge-Tate) if
$\mathrm{dim}_{K_{\ast}}D_{\mathrm{\ast}}(W)=[E:\mathbb{Q}_p]\mathrm{rank}(W)$ for $\ast= \mathrm{cris}$ (resp. $\ast=\mathrm{dR}$, resp.
$\ast=\mathrm{HT}$), where $K_{\ast}=K_0$ when $\ast=\mathrm{cris}$ and $K_{\ast}=K$ when $\ast=\mathrm{dR},\mathrm{HT}$.
We say that $W$ is potentially crystalline if $\mathrm{dim}_{L_0}(D^L_{\mathrm{cris}}(W))=[E:\mathbb{Q}_p]\mathrm{rank}(W)$ for
a finite extension $L$ of $K$, where $L_0$ is the maximal unramified extension of $\mathbb{Q}_p$ in $L$.
\end{defn}

\begin{defn}\label{d}
Let $L$ be a finite Galois extension of $K$ and $G_{L/K}:=\mathrm{Gal}(L/K)$.
We say that $D$ is an $E$-filtered ($\varphi,G_{L/K}$)-module over $K$ if
\begin{itemize}
\item[(1)] $D$ is a finite free $L_0\otimes_{\mathbb{Q}_p} E$-module with a $\varphi$-semi-linear
action $\varphi_D$ and a semi-linear action of $G_{L/K}$ such that
$\varphi_D:D\isom D$ is an isomorphism and that $\varphi_D$ and  $G_{L/K}$-action commute, where ($\varphi$-)semi-linear means that
$\varphi_D(a\otimes b\cdot x)=\varphi(a)\otimes b\cdot \varphi_D(x)$, $g(a\otimes b\cdot x)=g(a)\otimes b\cdot g(x)$ for
any $a\in L_0, b\in E, x\in D,  g\in G_{L/K}$.
\item[(2)] $D_L:=L\otimes_{L_0}D$ has a separated and exhausted decreasing filtration $\mathrm{Fil}^iD_L$ by
sub $L\otimes_{\mathbb{Q}_p}E$-modules such that the $G_{L/K}$-action on $D_L$ preserves this filtration.
\end{itemize}
\end{defn}

Let $W$ be a potentially crystalline $E$-$B$-pair such that $W|_{G_L}$ is crystalline
for a finite Galois extension $L$ of $K$, then we define an $E$-filtered ($\varphi, G_{L/K}$)-module's
structure on $D^L_{\mathrm{cris}}(W)$ as follows. First, $D^L_{\mathrm{cris}}(W)$ has a Frobenius action
induced from that on $B_{\mathrm{cris}}$ and has a $G_{L/K}$-action induced from those on $B_{\mathrm{cris}}$ and
$W_e$.
We define a filtration on $L\otimes_{L_0}D^L_{\mathrm{cris}}(W)=L\otimes_{K}D_{\mathrm{dR}}(W)$ by
$$\mathrm{Fil}^i(L\otimes_{L_0}D^{L}_{\mathrm{cris}}(W)):=(L\otimes_{K}D_{\mathrm{dR}}(W))\cap t^iW^+_{\mathrm{dR}}.$$

Let $D:=L_0e$ be a rank one $\mathbb{Q}_p$-filtered ($\varphi, G_{L/K}$)-module with a base $e$, then
we define $t_N(D):=v_p(\alpha)$ where $\varphi_D(e)=\alpha\cdot e$ and define $t_H(D):=i$ such that
$\mathrm{Fil}^iD_L/\mathrm{Fil}^{i+1}D_L\not= 0$. For general $D$ of rank $d$, we define $t_N(D):=t_N(\wedge^dD)$,
$t_H(D):=t_H(\wedge^dD)$. Then we say that $D$ is weakly admissible if $t_N(D)=t_H(D)$ and $t_N(D')\geqq t_H(D')$ for any
sub $\mathbb{Q}_p$-filtered ($\varphi,G_{L/K}$)-module $D'$ of $D$.
\begin{thm}\label{e}
Let $L$ be a finite Galois extension of $K$, then we have the following results.
\begin{itemize}
\item[(1)] The functor $W\mapsto D^L_{\mathrm{cris}}(W)$ gives an equivalence of categories between
the category of potentially crystalline $E$-$B$-pairs which are crystalline if restricted to $G_L$ and the
category of $E$-filtered $(\varphi,G_{L/K})$-modules over $K$.
\item[(2)] Restricting the above functor to $E$-representations, the functor $V\mapsto D^{L}_{\mathrm{cris}}(V)$
gives an equivalence of categories between the category of potentially crystalline $E$-representations which are
crystalline if restricted to $G_L$  and the category of weakly admissible $E$-filtered $(\varphi,G_{L/K})$-modules over $K$.
\end{itemize}
\end{thm}
\begin{proof}
See Proposition 2.3.4 and Theorem 2.3.5 of \cite{Be09} or Theorem 1.18 of \cite{Na09}.
\end{proof}

Next, we recall the definition of trianguline $E$-$B$-pairs, whose deformation theory we study in detail in this chapter.
\begin{defn}\label{f}
Let $W$ be an $E$-$B$-pair of rank $n$, then we say that $W$ is split trianguline if
there exists a filtration
$$\mathcal{T}:0\subseteq W_1\subseteq W_2\subseteq \cdots \subseteq W_n=W$$
 by sub $E$-$B$-pairs such that $W_{i}$ is saturated in $W_{i+1}$ and $W_{i+1}/W_{i}$ is a rank one
$E$-$B$-pair for any $0\leqq i\leqq n-1$. We say that $W$ is trianguline if $W\otimes_{E}E'$, the base change of $W$  to $E'$,
is a split trianguline $E'$-$B$-pair for a finite extension $E'$ of $E$.
\end{defn}

By this definition, to study split trianguline $E$-$B$-pairs, it is important to classify rank one $E$-$B$-pairs
and calculate extension classes of rank one $E$-$B$ pairs, which were studied in \cite{Na09}.
We recall some  results concerning these.

\begin{thm}\label{g}
There exists a canonical one to one correspondence $\delta\mapsto W(\delta)$ between the set of
continuous homomorphisms $\delta:K^{\times}\rightarrow E^{\times}$ and  the set of isomorphism classes
of rank one $E$-$B$-pairs.
\end{thm}
\begin{proof}
See  Proposition 3.1 of \cite{Co08} for $K=\mathbb{Q}_p$ and Theorem 1.45 of \cite{Na09} for general $K$.
For the construction of $W(\delta)$, see $\S$1.4 of \cite{Na09}.
\end{proof}

This correspondence is compatible with local class field theory, i.e. for a unitary homomorphism
$\delta:K^{\times}\rightarrow \mathcal{O}_E^{\times}$ if we take  the character $\tilde{\delta}:G^{\mathrm{ab}}_K\rightarrow \mathcal{O}_E^{\times}$
satisfying $\tilde{\delta}\circ \mathrm{rec}_K=\delta$, then we have an isomorphism $$W(\delta)\isom W(E(\tilde{\delta})).$$
This correspondence is also compatible with tensor products and with duals, i.e for continuous homomorphisms $\delta_1, \delta_2:K^{\times}
\rightarrow E^{\times}$, we have  $$W(\delta_1)\otimes W(\delta_2)\isom W(\delta_1\delta_2)\text{ and } W(\delta_1)^{\vee}
\isom W(\delta_1^{-1}).$$

There are some important examples of rank one $E$-$B$-pairs which we recall now.
For any $\{k_{\sigma}\}_{\sigma\in\mathcal{P}}\in \prod_{\sigma\in \mathcal{P}}\mathbb{Z}$, we define a homomorphism
$$\prod_{\sigma\in\mathcal{P}}\sigma^{k_{\sigma}}:K^{\times}\rightarrow E^{\times}:y\mapsto \prod_{\sigma\in \mathcal{P}}
\sigma(y)^{k_{\sigma}},$$  then we have an isomorphism
$$W(\prod_{\sigma\in\mathcal{P}}\sigma^{k_{\sigma}})\isom(B_e\otimes_{\mathbb{Q}_p}E,
\oplus_{\sigma\in\mathcal{P}}t^{k_{\sigma}}B^+_{\mathrm{dR}}\otimes_{K,\sigma}E)$$ ( Lemma 2.12 of \cite{Na09}).
 Let  $N_{K/\mathbb{Q}_p}:K^{\times}\rightarrow \mathbb{Q}_p^{\times}$ be the norm and
 $\allowbreak|-|:\mathbb{Q}_p^{\times}\rightarrow \mathbb{Q}^{\times}\hookrightarrow E^{\times}$ be the $p$-adic absolute value
 such that $|p|=\frac{1}{p}$, and we define $|N_{K/\mathbb{Q}_p}|:K^{\times}\rightarrow E^{\times}$  the composite
 of $N_{K/\mathbb{Q}_p}$ and $|-|$, then we have an isomorphism
 $$W(|N_{K/\mathbb{Q}_p}|\prod_{\sigma\in \mathcal{P}}\sigma)\isom W(E(\chi_p)),$$ which is the $E$-$B$-pair
 associated to $p$-adic cyclotomic character.

 Next, we recall the definition and some properties of Galois cohomology of $E$-$B$-pairs.
 For an $E$-$B$-pair $W:=(W_e, W^+_{\mathrm{dR}})$, we put $W_{\mathrm{dR}}:=B_{\mathrm{dR}}\otimes_{B_e}W_e$.
 We have natural inclusions, $W_e\hookrightarrow W_{\mathrm{dR}}$, $W^+_{\mathrm{dR}}
 \hookrightarrow W_{\mathrm{dR}}$. We define the Galois cohomology $\mathrm{H}^i(G_K, W)$ of $W$ as
 the continuous cochains' cohomology of the complex
 $$W_e\oplus W^+_{\mathrm{dR}}\rightarrow W_{\mathrm{dR}}:(x,y)\mapsto
 x-y$$
  (see the appendix of this article or $\S$2.1 of \cite{Na09} for the precise definition). As in the usual $p$-adic representation case, we have the following isomorphisms
  of $E$-vector spaces
  $$\mathrm{H}^0(G_K, W)
 \isom\mathrm{Hom}_{G_K}(B_E, W),\, \mathrm{H}^1(G_K, W)\isom\mathrm{Ext}^1(B_E, W),$$
  where $B_E:=(B_e\otimes_{\mathbb{Q}_p}E,
 B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E)$ is the trivial $E$-$B$-pair and $\mathrm{Hom}_{G_K}(-,-)$ is the group of homomorphisms
 of $E$-$B$-pairs and $\mathrm{Ext}^1(-,-)$ is the extension class group in the category of $E$-$B$-pairs. If
 $V$ is an $E$-representation of $G_K$, we have a canonical isomorphism
 $$\mathrm{H}^i(G_K, V)\isom \mathrm{H}^i(G_K, W(V)),$$ which follows from
 Bloch-Kato's fundamental short exact sequence.
 Moreover, we have the following theorem, Euler-Poincar\'e characteristic  formula and Tate duality theorem for
 $B$-pairs.
 \begin{thm}\label{h}
 Let $W$ be an $E$-$B$-pair.
 \begin{itemize}
 \item[(1)] For $i=0,1,2$, $\mathrm{H}^i(G_K, W)$ is finite dimensional over $E$ and $\mathrm{H}^i(G_K, W)=0$ for
 $i\not= 0,1,2$.
 \item[(2)] $\sum_{i=0}^2(-1)^{i-1} \mathrm{dim}_E\mathrm{H}^i(G_K, W)=[K:\mathbb{Q}_p] \mathrm{rank}(W)$.
 \item[(3)] Let $W$ be a $\mathbb{Q}_p$-$B$-pair. For any $i=0,1,2$, there is a natural perfect pairing defined by cup
 product,
 \item[] \[
              \begin{array}{ll}
              \mathrm{H}^i(G_K, W)\times\mathrm{H}^{2-i}(G_K, W^{\vee}(\chi_p))&\rightarrow \mathrm{H}^2(G_K, W\otimes W^{\vee}(
 \chi_p)) \\
  &\rightarrow \mathrm{H}^2(G_K, W(\mathbb{Q}_p(\chi_p))\allowbreak \isom \mathbb{Q}_p,
  \end{array}
  \]
 \item[] where the last isomorphism is Tate's trace map.
 \end{itemize}
 \end{thm}
 \begin{proof}
 See Theorem \ref{5-6} and and Theorem \ref{5-7} in the appendix.
 \end{proof}
 Using these formulae, we obtain the following dimension formulae of Galois cohomologies of rank one $E$-$B$-pairs.
 \begin{prop}\label{i}
 Let $\delta:K^{\times}\rightarrow E^{\times}$ be a continuous homomorphism, then we have$:$
 \begin{itemize}
 \item[(1)] $\mathrm{H}^0(G_K, W(\delta))\isom E $ if $\delta=\prod_{\sigma\in \mathcal{P}}\sigma^{k_{\sigma}}$
 such that $\{k_{\sigma}\}_{\sigma\in \mathcal{P}}\in \prod_{\sigma\in \mathcal{P}}\mathbb{Z}_{\leqq 0}$
 , and $\mathrm{H}^0(G_K, W(\delta))=0$ otherwise.
 \item[(2)] $\mathrm{H}^2(G_K, W(\delta))\isom E$ if $\delta=|N_{K/\mathbb{Q}_p}|\prod_{\sigma\in\mathcal{P}}\sigma^{k_{\sigma}}$
 such that  $\{k_{\sigma}\}_{\sigma\in \mathcal{P}}\in \prod_{\sigma\in \mathcal{P}}\mathbb{Z}_{\geqq 1}$, and $\mathrm{H}^2(G_K, W(\delta))=0$ otherwise.
 \item[(3)] $\mathrm{dim}_E\mathrm{H}^1(G_K, W(\delta))=[K:\mathbb{Q}_p] +1$ if $\delta=\prod_{\sigma\in \mathcal{P}}\sigma
 ^{k_{\sigma}}$ such that $\{k_{\sigma}\}_{\sigma\in \mathcal{P}}\in \prod_{\sigma\in \mathcal{P}}\mathbb{Z}_{\leqq 0}$ or $\delta=
 |N_{K/\mathbb{Q}_p}|\prod_{\sigma\in \mathcal{P}}\sigma^{k_{\sigma}}$ such that $\{k_{\sigma}\}_{\sigma\in \mathcal{P}}\in \prod_{\sigma\in \mathcal{P}}\mathbb{Z}_{\geqq 1}$, and $\mathrm{dim}_E\mathrm{H}^1(G_K, W(\delta))=[K:\mathbb{Q}_p]$ otherwise.
 \end{itemize}
 \end{prop}
 \begin{proof}
 See Theorem 2.9 and Theorem 2.22 of \cite{Co08} for $K=\mathbb{Q}_p$. For general $K$, the results can be proved by using
 Proposition 2.14 and Proposition 2.15 of \cite{Na09} and Tate duality for $B$-pairs.
 \end{proof}

\subsubsection{$B$-pairs over Artin local rings}
Here, we define $B$-pairs with Artin ring coefficients, which we need to define the notion of deformations of $E$-$B$-pairs.
Let $\mathcal{C}_E$ be the category of Artin local $E$-algebra $A$ with the residue field $E$. The morphisms in $\mathcal{C}_E$ are given by
local $E$-algebra homomorphisms. For any $A\in \mathcal{C}_E$, we denote by  $\mathfrak{m}_A$ the maximal ideal of $A$. We define the $A$-coefficient version of $B$-pairs as follows.
\begin{defn}\label{1}
We call a pair $W:=(W_e, W^+_{\mathrm{dR}})$ an $A$-$B$-pair of $G_K$ if
\begin{itemize}
\item[(1)] $W_e$ is a finite $B_e\otimes_{\mathbb{Q}_p}A$-module which is flat over $A$
and is free over $B_e$, with a continuous semi-linear $G_K$-action.
\item[(2)] $W^+_{\mathrm{dR}}$ is a finite generated sub $B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A$-module of $B_{\mathrm{dR}}\otimes_{B_e}W_e$ which is stable by the $G_K$-action and which generates
$B_{\mathrm{dR}}\otimes_{B_e}W_e$ as a $B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A$-module such that
$W^+_{\mathrm{dR}}/tW^+_{\mathrm{dR}}$ is flat over $A$.
\end{itemize}
For an $A$-$B$-pair $W:=(W_e,W^+_{\mathrm{dR}})$, we put $W_{\mathrm{dR}}:=B_{\mathrm{dR}}
\otimes_{B_e}W_e$.
\end{defn}
We simply call an $A$-$B$-pair if there is no risk of confusing about $K$.

\begin{lemma}\label{2}
Let $W:=(W_e,W^+_{\mathrm{dR}})$ be an $A$-$B$-pair. Then
$W_e$ is a finite free $B_e\otimes_{\mathbb{Q}_p}A$-module,
$W^+_{\mathrm{dR}}$ is a finite free $B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A$-module
and $W^+_{\mathrm{dR}}/tW^+_{\mathrm{dR}}$ is a finite free $\mathbb{C}_p\otimes_{\mathbb{Q}_p}A$-module.
\end{lemma}
\begin{proof}
First, we prove for $W_e$. Because the sub module $\mathfrak{m}_A W_e\subseteq W_e$ is a $G_K$-stable finite generated
torsion free $B_e$-module and because $B_e$ is a B\'ezout domain by Proposition 1.1.9 of \cite{Be08}, $\mathfrak{m}_AW_e$ is a finite free $B_e$-module by Lemma 2.4 of \cite{Ke04}.
By Lemma 2.1.4 of \cite{Be08}, the cokernel $W_e\otimes_A E$ is also a finite free $B_e$-module
(with an $E$-action). By Lemma 1.7 of \cite{Na09}, $W_e\otimes_AE$ is a finite free $B_e\otimes_{\mathbb{Q}_p}E$-module of
some rank $n$.
We take a $B_e\otimes_{\mathbb{Q}_p}A$-linear morphism $f:(B_e\otimes_{\mathbb{Q}_p}A)^n\rightarrow W_e$ which
is a lift of a $B_e\otimes_{\mathbb{Q}_p}E$-linear isomorphism $(B_e\otimes_{\mathbb{Q}_p}E)^n\isom W_e\otimes_A E$.
Because $A$ is Artinian, then $f$ is surjective. Because $W_e$ is $A$-flat, we have $\mathrm{Ker}(f)\otimes_AE=0$,
hence $\mathrm{Ker}(f)=0$. Hence $W_e$ is a free $B_e\otimes_{\mathbb{Q}_p}A$-module.

Next, we prove that $W^+_{\mathrm{dR}}$ is a free $B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A$-module.
Because $W_e$ is a free $B_e\otimes_{\mathbb{Q}_p}A$-module, $W_{\mathrm{dR}}$ is a free $B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A$-module, in particular this is flat over $A$. Because $W^+_{\mathrm{dR}}/tW^+_{\mathrm{dR}}$ is a flat $A$-module, $W_{\mathrm{dR}}/W^+_{\mathrm{dR}}$ is also  a flat $A$-module. Hence $W^+_{\mathrm{dR}}$ is also flat over $A$.
By the $A$-flatness of $W_{\mathrm{dR}}/W^+_{\mathrm{dR}}$, we have an inclusion
$W^+_{\mathrm{dR}}\otimes_A E\hookrightarrow W_{\mathrm{dR}}\otimes_{A} E$, hence
$W^{+}_{\mathrm{dR}}\otimes_A E$ is a finite generated torsion free $B^+_{\mathrm{dR}}$-module, hence
free $B^+_{\mathrm{dR}}$-module. By Lemma 1.8 of \cite{Na09}, we can show that
$W^+_{\mathrm{dR}}$ is a free $B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A$-module in the same way as in the case of $W_e$.
$\mathbb{C}_p\otimes_{\mathbb{Q}_p}A$-freeness of $W^+_{\mathrm{dR}}/tW^+_{\mathrm{dR}}$ follows from
$B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A$-freeness of $W^+_{\mathrm{dR}}$.

\end{proof}

\begin{defn}
Let $W=(W_e, W^+_{\mathrm{dR}})$ be an $A$-$B$-pair. We define the rank of $W$ by
$\mathrm{rank}(W):=\mathrm{rank}_{B_e\otimes_{\mathbb{Q}_p}A}(W_e)$.
\end{defn}

\begin{defn}\label{3}
Let $f:A\rightarrow A'$ be a morphism in $\mathcal{C}_E$ and $W=(W_e,W_{\mathrm{dR}})$ be an $A$-$B$-pair.
We define the base change of $W$ to $A'$ by
$$W\otimes_A A':=(W_e\otimes_A A',W^+_{\mathrm{dR}}\otimes_A A').$$
By Lemma $\ref{2}$, we can easily see that this is an $A'$-$B$-pair.
\end{defn}

\begin{defn}\label{4}
Let $W_1=(W_{e,1},W^+_{\mathrm{dR},1})$, $W_2=(W_{e,2}, W^+_{\mathrm{dR},2})$ be $A$-$B$-pairs.
We define the tensor product of $W_1$ and $W_2$ by $W_1\otimes W_2:=(W_{e,1}\otimes_{B_e\otimes_{\mathbb{Q}_p}A}W_{e,2},
W^+_{\mathrm{dR},1}\otimes_{B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A}W^+_{\mathrm{dR},2})$ ,
and define the dual of $W_1$ by $W_1^{\vee}:=(\mathrm{Hom}_{B_e\otimes_{\mathbb{Q}_p}A}(W_{e,1},\allowbreak B_e\otimes_{\mathbb{Q}_p}A),
W^{+,\vee}_{\mathrm{dR},1})$. Here, $W^{+,\vee}_{\mathrm{dR},1}:=\{f\in\mathrm{Hom}_{B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A}(
W_{\mathrm{dR},1}, B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A)|f(W^{+}_{\mathrm{dR},1})\subseteq B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A\}$. By Lemma $\ref{2}$, we can easily see that these are $A$-$B$-pairs.
\end{defn}

Next, we classify rank one $A$-$B$-pairs.
Let $\delta:K^{\times}\rightarrow A^{\times}$ be a continuous homomorphism, then we define a rank one $A$-$B$-pair $W(\delta)$ as follows.
Let $\bar{u}\in E^{\times}$ be the reduction of $u:=\delta(\pi_K)$.
We define a homomorphism $\delta_0:K^{\times}\rightarrow A^{\times}$ such that $\delta_0|_{\mathcal{O}_K^{\times}}
=\delta|_{\mathcal{O}_K^{\times}}$, $\delta_0(\pi_K)=u/\bar{u}$. Because $u/\bar{u}\in 1+\mathfrak{m}_A$ ,  $(u/\bar{u})^{p^n}$
converges to $1\in A^{\times}$. If we fix an isomorphism $K^{\times}\isom \mathcal{O}_K^{\times}\times \mathbb{Z}
:v\pi_K^n\mapsto (v,n)$ ($v\in\mathcal{O}_K^{\times}$), then $\delta_0$ uniquely extends to a continuous homomorphism $\delta_0':\mathcal{O}_K^{\times}\times
\hat{\mathbb{Z}}\rightarrow \mathcal{O}_K^{\times}\times \mathbb{Z}_p\rightarrow A^{\times}$. By local class field theory, then there exists unique character $\tilde{\delta}_0:G_K^{\mathrm{ab}}\rightarrow A^{\times}$ such
that $\delta_0=\tilde{\delta}_0\circ\mathrm{rec}_K$, where $\mathrm{rec}_K:K^{\times}\rightarrow G_K^{\mathrm{ab}}$ is the reciprocity map as in Notation. Using $\tilde{\delta}_0$, we define an
\'etale rank one $A$-$B$-pair $W(A(\tilde{\delta}_0))$, which is the $A$-$B$-pair associated to the rank one $A$-representation $A(\tilde{\delta}_0)$.
Next, we define a non-\'etale rank one $A$-$B$-pair by using $\bar{u}\in E^{\times}$.
For this, first we define a rank one $E$-filtered $\varphi$-module $D_{\bar{u}}:=K_0\otimes_{\mathbb{Q}_p}Ee_{\bar{u}}$ such that
$\varphi^f(e_{\bar{u}}):=\bar{u}e_{\bar{u}}$ and $\mathrm{Fil}^0(K\otimes_{K_0}D_{\bar{u}}):=K\otimes_{K_0}D_{\bar{u}}$, $\mathrm{Fil}^1(K\otimes_{K_0}D_{\bar{u}}):=0$.
From this, we obtain the rank one crystalline $E$-$B$-pair $W(D_{\bar{u}})$ such that $D_{\mathrm{cris}}(W(D_{\bar{u}}))\isom D_{\bar{u}}$ which is
pure of slope $\frac{v_p(\bar{u})}{f}$.
By tensoring these, we define a rank one $A$-$B$-pair $W(\delta)$ by $W(\delta):=(W(D_{\bar{u}})\otimes_EA)\otimes W(A(\tilde{\delta}_0))$,
 which is pure of slope $\frac{v_p(\bar{u})}{f}$.

 The following proposition is the $A$-coefficient version of Theorem 1.45 of \cite{Na09}.

 \begin{prop}\label{5}
 This construction $\delta\mapsto W(\delta)$ does not depend on the
 choice of uniformizer $\pi_K$ and gives a bijection between
 the set of continuous homomorphisms $\delta:K^{\times}\rightarrow A^{\times}$ and
 the set of isomorphism classes of rank one $A$-$B$-pairs.
 \end{prop}
 \begin{proof}
 The independence of the choice of uniformizer and the injection is proved in the same
 way as in the proof of Theorem 1.45 of \cite{Na09}. We prove the surjection. Let $W$ be a rank one $A$-$B$-pair, then
 as an $E$-$B$-pair, $W$ is a successive extension of rank one $E$-$B$-pair $W\otimes_A E$.
 $W\otimes_AE$ is pure of slope $\frac{n}{fe_E}$ for some $n\in \mathbb{Z}$ by Lemma 1.42 of \cite{Na09}.
 Hence, by Theorem 1.6.6 of \cite{Ke08}, $W$ is also pure of slope $\frac{n}{fe_E}$.
 We define a rank one $E$-filtered $\varphi$-module $D_{\pi_E^n}:=K_0\otimes_{\mathbb{Q}_p}Ee_{\pi_E^n}$
 in the same way as in $D_{\bar{u}}$, where $\pi_E$ is a uniformizer of $E$, then $W\otimes (W(D_{\pi_E^n}) \otimes_E A)^{\vee}$ is pure of slope zero by Lemma
 1.34 of \cite{Na09}. Hence, there exists $\tilde{\delta}':G_K^{\mathrm{ab}}\rightarrow A^{\times}$ such that
 $W\otimes (W(D_{\pi_E^n})\otimes_E A)^{\vee}\isom W(A(\tilde{\delta}'))$. We put $\delta':=\tilde{\delta}'\circ\mathrm{rec}_K:
 K^{\times}\rightarrow A^{\times}$ and define $\delta:K^{\times}\rightarrow A^{\times}$ such that $\delta|_{\mathcal{O}_K^{\times}}
 :=\delta'|_{\mathcal{O}_K^{\times}}$ and $\delta(\pi_K):=\delta'(\pi_K)\pi_E^n$, then we have an isomorphism
 $W\isom W(\delta)$, which can be easily seen from the construction of $W(\delta)$.

 \end{proof}
 By local class field theory, we have a canonical bijection $\delta\mapsto A(\tilde{\delta})$ from  the set of unitary continuous homomorphisms
 from $K^{\times}$ to $A^{\times}$ (where unitary means that the reduction of the image of $\delta$ is contained in $\mathcal{O}_E^{\times}$)
  to the set of isomorphism class of rank one $A$-representations of $G_K$, where $\tilde{\delta}:G_K^{ab}\rightarrow
  A^{\times}$ is the continuous homomorphism such that $\delta=\tilde{\delta}\circ\mathrm{rec}_K$.
  By the definition of $W(\delta)$  and by the above proof, it is easy to see that there exists an isomorphism
  $W(\delta)\isom W(A(\tilde{\delta}))$ for any unitary homomorphism $\delta:K^{\times}\rightarrow A^{\times}$.
 Moreover, it is easy to see that for any continuous homomorphisms $\delta_1,\delta_2: K^{\times}
 \rightarrow A^{\times}$ we have isomorphisms $W(\delta_1)\otimes W(\delta_2)\isom W(\delta_1\delta_2)$ and $W(\delta_1)^{\vee}
 \isom W(\delta_1^{-1})$.

Next, we generalize the functor $D_{\mathrm{cris}}$ to potentially crystalline $A$-$B$-pairs.
First, we define the $A$-coefficient version of filtered $(\varphi,G_K)$-modules.
Let $L$ be a finite Galois extension of $K$, we denote by $G_{L/K}:=\mathrm{Gal}(L/K)$.
\begin{defn}
Let $A$ be an object of $\mathcal{C}_E$.
We say that $D$ is an $A$-filtered $(\varphi,G_{L/K})$-module of $K$ if $D$ satisfies
the following conditions.
\begin{itemize}
\item[(1)]$D$ is a finite $L_0\otimes_{\mathbb{Q}_p}A$-module which is free as an
$A$-module with
a $\varphi$-semi-linear action $\varphi:D\isom D$.
\item[(2)]$D_L:=L\otimes_{L_0}D$ has a decreasing filtration $\mathrm{Fil}^iD_L$ by
sub $L\otimes_{\mathbb{Q}_p}A$-modules such that $\mathrm{Fil}^kD_L=0$ and
$\mathrm{Fil}^{-k}D_L=D_L$ for sufficiently large $k$ and that
$\mathrm{Fil}^kD_L/\mathrm{Fil}^{k+1}D_L$ are free $A$-modules for any $k$.
\item[(3)]$G_{L/K}$ acts on $D$ by $L_0\otimes_{\mathbb{Q}_p}A$-semi-linear automorphism
which commutes with the action of $\varphi$ and preserves the filtration.
\end{itemize}
\end{defn}
\begin{rem}
Using  the $\varphi$-structure on $D$, we can see that $D$ is a free $L_0\otimes_{\mathbb{Q}_p}A$-module.
\end{rem}

Let $W_A:=(W_{A,e},W^+_{A,\mathrm{dR}})$ be an $A$-$B$-pair such that $W_A|_{G_L}$ is crystalline (as an $E$-$B$-pair)
for a finite
Galois extension $L$ of $K$. As in the case of $E$-$B$-pairs, we define
$D^L_{\mathrm{cris}}(W_A):=(B_{\mathrm{cris}}\otimes_{B_e}W_e)^{G_L}$ with a $\varphi$-action induced from
that on $B_{\mathrm{cris}}$, then the natural map $L\otimes_{L_0}D^L_{\mathrm{cris}}(W_A)\rightarrow
D^L_{\mathrm{dR}}(W_A):=(B_{\mathrm{dR}}\otimes_{B_e}W_e)^{G_L}$ is isomorphism.
We define $\mathrm{Fil}^kD^L_{\mathrm{dR}}(W_A):=D^L_{\mathrm{dR}}(W_A)\cap t^kW^+_{\mathrm{dR}}$ for
any $k\in \mathbb{Z}$. These are equipped with a $G_{L/K}$-action naturally.
\begin{lemma}\label{s1}
In the above situation, $D^L_{\mathrm{cris}}(W_A)$ is an $A$-filtered $(\varphi,G_{L/K})$-module of $K$.
\end{lemma}
\begin{proof}
It suffices only to show the $A$-freeness of $D^L_{\mathrm{cris}}(W_A)$, $\mathrm{Fil}^kD^L_{\mathrm{dR}}(W_A)$,
$\mathrm{Fil}^kD^L_{\mathrm{dR}}\allowbreak (W_A)/\mathrm{Fil}^{k+1}D^L_{\mathrm{dR}}(W_A)$.
Here, we only prove the $A$-freeness of $D^L_{\mathrm{cris}}(W_A)$, other cases can be proved in a similar way.
By the exactness of $D^L_{\mathrm{cris}}$ for $E$-$B$-pairs which are crystalline when restricted to $G_L$, we have
a natural isomorphism $D^L_{\mathrm{cris}}(W_A)\otimes_A N\isom D^L_{\mathrm{cris}}(W_A\otimes_A N)$  for any finite $A$-module $N$.
From this, for any $A$-linear injection $N_1\hookrightarrow N_2$ of finite $A$-modules, we have
an inclusion $D^L_{\mathrm{cris}}(W_A)\otimes_A N_1=D^L_{\mathrm{cris}}(W_A\otimes_A N_1)\hookrightarrow
D^L_{\mathrm{cris}}(W_A\otimes_A N_2)=D^L_{\mathrm{cris}}(W_A)\otimes_A N_2$ because
$W_{A,e}$ is $A$-flat. Hence, $D^L_{\mathrm{cris}}(W_A)$ is $A$-flat.
\end{proof}

Conversely, let $D$ be  an $A$-filtered $(\varphi,G_{L/K})$-module of $K$, then we define
$W_e(D):=(B_{\mathrm{cris}}\otimes_{L_0}D)^{\varphi=1}$. We have
a natural isomorphism $B_{\mathrm{dR}}\otimes_{B_e}W_e(D)\isom B_{\mathrm{dR}}\otimes_{L}D_L$.
We define $W^+_{\mathrm{dR}}(D):=\mathrm{Fil}^0(B_{\mathrm{dR}}\otimes_LD_L)\subseteq
B_{\mathrm{dR}}\otimes_{B_e}W_e(D)$. We write $W(D):=(W_e(D), W^+_{\mathrm{dR}}(D))$ which is
a potentially crystalline $E$-$B$-pair with an $A$-action.
\begin{lemma}\label{s2}
In the above situation, $W(D)$ is a potentially crystalline
$A$-$B$-pair.
\end{lemma}
\begin{proof}
It suffices to show the $A$-flatness of $W_e(D)$ and $W^+_{\mathrm{dR}}(D)/tW^+_{\mathrm{dR}}(D)$.
We can prove these in the same way as in Lemma $\ref{s1}$ by using the
exactness of the functor $W(D)$ and the $A$-flatness of $D$ and $\mathrm{Fil}^kD_L/\mathrm{Fil}^{k+1}D_L$ for any
$k$.
\end{proof}

\begin{corollary}\label{s3}
Let $A$ be an object in $\mathcal{C}_E$.
The functor $D^L_{\mathrm{cris}}$ gives an equivalence of categories between
the category of potentially crystalline $A$-$B$-pairs which are crystalline if restricted to $G_L$ and
the category of $A$-filtered $(\varphi,G_{L/K})$-modules of $K$.
\end{corollary}
\begin{proof}
This follows from Lemma $\ref{s1}$ and Lemma $\ref{s2}$ and Theorem $\ref{e}$.
\end{proof}

Next, we prove some lemmas which will be used in later sections.

Let $W_A$ be an $A$-$B$-pair which is not potentially crystalline in general and $L$ be a finite Galois extension
of $K$, then we can define $D^L_{\mathrm{cris}}(W_A)$ in the same way as in the case where $W_A$ is potentially crystalline.
This is an $E$-filtered $(\varphi,G_{L/K})$-module, but in general this may not be an
$A$-filtered $(\varphi,G_{L/K})$-module.

\begin{lemma}\label{s3-5}
Let $W$ be an $E$-$B$-pair. Let $D\subseteq D_{\mathrm{cris}}(W)$ be
a rank one sub $E$-filtered $\varphi$-module whose filtration is induced
from that of $D_{\mathrm{cris}}(W)$, then there exists a natural saturated inclusion
$W(D)\hookrightarrow W$.

\end{lemma}
\begin{proof}
Twisting $W$ by a suitable crystalline character of the form $\prod_{\sigma\in\mathcal{P}}\sigma(\chi_{\mathrm{LT}})^{k_{\sigma}}$,
we may assume that $\mathrm{Fil}^0(D_{K})=D_{K}$ and $\mathrm{Fil}^1(D_{K})=0$, where we put
$D_{K}:=K\otimes_{K_0}D$.
We have natural inclusions $W(D)_e=(B_{\mathrm{cris}}\otimes_{K_0}D)^{\varphi=1}\subseteq
(B_{\mathrm{cris}}\otimes_{K_0}D_{\mathrm{cris}}(W))^{\varphi=1}\subseteq (B_{\mathrm{cris}}\otimes_{B_e}W_e)^{\varphi=1}
=W_e$ and, under the above assumption, $W^+_{\mathrm{dR}}(D)\allowbreak =\mathrm{Fil}^0(B_{\mathrm{dR}}\otimes_{K}D_{K})=
B^+_{\mathrm{dR}}\otimes_KD_{K}\subseteq \mathrm{Fil}^0(B_{\mathrm{dR}}\otimes_{K}D_{\mathrm{dR}}(W))\subseteq
W^+_{\mathrm{dR}}$, which define an inclusion
$W(D)\hookrightarrow W$. Hence, it suffices to show that this inclusion is saturated, i.e. $B^+_{\mathrm{dR}}\otimes_{K}D_{K}
=(B_{\mathrm{dR}}\otimes_KD_{K})\cap W^+_{\mathrm{dR}}$. We can write $(B_{\mathrm{dR}}\otimes_KD_{K})\cap W^+_{\mathrm{dR}}=
\oplus_{\sigma\in \mathcal{P}}\frac{1}{t^{k_{\sigma}}}B^+_{\mathrm{dR}}\otimes_{K,\sigma}D_{K,\sigma}$ for some $k_{\sigma}\in \mathbb{Z}_{\geqq 0}$, where we decompose $D_{K}$ by
$D_{K}\isom \oplus_{\sigma\in \mathcal{P}}D_{K}\otimes_{K\otimes_{\mathbb{Q}_p}E, \sigma\otimes id_E}E=:\oplus_{\sigma\in \mathcal{P}}D_{K,\sigma}$.
If $k_{\sigma}\geqq 1$ for a $\sigma\in \mathcal{P}$, then $D_{K,\sigma}\subseteq t^{k_{\sigma}}W^+_{\mathrm{dR}}$. Because the filtration on $D$ is induced from
$D_{\mathrm{cris}}(W)$, this implies that $\mathrm{Fil}^{k_{\sigma}}D_{K,\sigma}=D_{K,\sigma}$, this contradicts
to $\mathrm{Fil}^1D_{K,\sigma}=0$. Hence $k_{\sigma}=0$ for any $\sigma\in \mathcal{P}$, this implies
that $B^+_{\mathrm{dR}}\otimes_{K}D_{K}
=(B_{\mathrm{dR}}\otimes_KD_{K})\cap W^+_{\mathrm{dR}}$.

\end{proof}
\begin{lemma}\label{s4}
Let $W_A$ be an $A$-$B$-pair.

Let $D\subseteq D_{\mathrm{cris}}(W_A)$ be a sub $E$-filtered $\varphi$-module which is
an $A$-filtered $\varphi$-module
of rank one, where the filtration on $D$ is the one induced from that of
$D_{\mathrm{cris}}(W_A)$. We assume that the natural map
$D\otimes_A E\rightarrow D_{\mathrm{cris}}(W_A\otimes_A E)$ remains injective.

Then, we have a natural injection of $A$-$B$-pairs $W(D)\hookrightarrow W_A$ such that
the cokernel $W_A/W(D)$ is also an $A$-$B$-pair.
\end{lemma}
\begin{proof}
In the same way as in the above proof,

we have a natural injection $W(D)\hookrightarrow W_A$. Because the natural map
$D\otimes_A E\rightarrow D_{\mathrm{cris}}(W_A\otimes_A E)$ is injection,
we obtain an injection $W(D)\otimes_AE \isom W(D\otimes_A E)\hookrightarrow W_A\otimes_A E$ and this injection is saturated by the above lemma.
Hence, it suffices to show that if $W_1\hookrightarrow W_2$ is an inclusion of $A$-$B$-pairs such that
$W_1\otimes_{A}E\rightarrow W_{2}\otimes_AE$ remains injection and saturated, then the cokernel $W_2/W_1$ exists and
is an $A$-$B$-pair.
We put $W_{3,e}$, $W^+_{3,\mathrm{dR}}$ the cokernels of
$W_{1,e}\hookrightarrow W_{2,e}$, $W^+_{1,\mathrm{dR}}\hookrightarrow W^+_{2,\mathrm{dR}}$ respectively.
By Lemma 2.2.3 (i) of \cite{Bel-Ch09}, these are $A$-flat. Hence, it suffices to show that these are
free over $B_e$, $B^+_{\mathrm{dR}}$ respectively. We can prove this claim in the same way as in Lemma 2.2.3 (iii) of \cite{Bel-Ch09}.

\end{proof}

\subsection{Deformations of $B$-pairs}

In this subsection, we develop the deformation theory of $B$-pairs, which
are the generalization of Mazur's (characteristic zero) deformation theory of $p$-adic Galois
representations.

\begin{defn}\label{6}
Let $A$ be an object in $\mathcal{C}_E$, $W$ be an $E$-$B$-pair.
We say that a pair $(W_A,\iota)$ is a deformation of $W$ over $A$
if $W_A$ is an $A$-$B$-pair and $\iota: W_A\otimes_AE\isom W$
is an isomorphism of $E$-$B$-pairs. Let $(W_A,\iota)$, $(W'_A,\iota')$ be
two deformations of $W$ over $A$. Then we say that $(W_A,\iota)$ and $(W'_A,\iota')$ are
equivalent if there exists an isomorphism $f:W_A\isom W'_A$ of $A$-$B$-pairs which
satisfies $\iota=\iota'\circ\bar{f}$, where $\bar{f}:W_A\otimes_A E\isom W'_A\otimes_AE$
is the reduction of $f$.
\end{defn}
\begin{defn} \label{7}
Let $W$ be an $E$-$B$-pair, then we define the deformation functor
$D_W$ from the category $\mathcal{C}_E$ to the category of sets by
$$D_W(A):=\{ \text{ equivalent classes }  (W_A,\iota) \text{ of deformations of }  W \text{ over } A \}$$
for any $A\in \mathcal{C}_E$.
\end{defn}
We simply denote by $W_A$ if there is no risk of confusing about $\iota$.

Next, we prove the pro-representability
of the functor $D_W$ under suitable conditions.
For this, we recall Schlessinger's criterion for pro-representability of functors from
$\mathcal{C}_E$ to the category of sets.
We call a morphism $f:A'\rightarrow A$ in $\mathcal{C}_E$ a small extension if it is surjective and
the kernel $\mathrm{Ker}(f)=(t)$ is generated by a nonzero single element $t\in A'$
and $\mathrm{Ker}(f)\cdot \mathfrak{m}_{A'}=0$. $E[\varepsilon]$ is the ring defined by $E[\varepsilon]:=E[X]/(X^2)$.

\begin{thm}\label{8}
Let $F$ be a functor from $\mathcal{C}_E$ to the category of sets such that
$F(E)$ is a single point. For morphisms $A'\rightarrow A$, $A"\rightarrow A$ in
$\mathcal{C}_E$, consider the natural map
\begin{itemize}
\item[(1)] $F(A'\times_{A}A")\rightarrow F(A')\times_{F(A)}F(A")$,
\end{itemize}
then $F$ is pro-representable if and only if $F$ satisfies properties $(H_1)$,
$(H_2)$, $(H_3)$, $(H_4)$ below:
\begin{itemize}
\item[$(H_1)$] $(1)$ is surjective if $A"\rightarrow A$ is surjective.
\item[$(H_2)$] $(1)$ is bijective when $A=E$ and $A"=E[\varepsilon]$.
\item[$(H_3)$] $\mathrm{dim}_E(t_F)<\infty$ $($ where $t_F:=F(E[\varepsilon])$ and, under
the condition $(H_2)$, it is known that $t_F$ has a natural $E$-vector space structure$)$.
\item[$(H_4)$] $(1)$ is  bijective if $A'=A"$ and $A'\rightarrow A$ is a small extension.
\end{itemize}
\end{thm}
\begin{proof}
See \cite{Schl68} or $\S 18$ of \cite{Ma97}.
\end{proof}

Using this criterion, we prove the pro-representability of
$D_W$.

\begin{prop}\label{9}
Let $W$ be an $E$-$B$-pair.
If $\mathrm{End}_{G_K}(W)=E$, then $D_W$ is pro-representable by a complete
noetherian local $E$-algebra $R_W$ with residue field $E$.
\end{prop}

For proving this proposition, we first prove some lemmas.

\begin{lemma}\label{10}
Let $\mathrm{ad}(W):=\mathrm{Hom}(W,W)(\isom W\otimes W^{\vee})$ be the internal endomorphism of $W$,
then there exists an isomorphism of $E$-vector spaces
$D_W(E[\varepsilon])\isom \mathrm{H}^1(G_K,\mathrm{ad}(W))$.
\end{lemma}
\begin{proof}
Let $W_{E[\varepsilon]}:=(W_{E[\varepsilon],e}, W^+_{E[\varepsilon],\mathrm{dR}})$
be a deformation of $W$ over $E[\varepsilon]$. From this, we define an element
in $\mathrm{H}^1(G_K,\mathrm{ad}(W))$ as follows. Because $\varepsilon W_{E[\varepsilon],e}\isom
W_e$ and $W_{E[\varepsilon],e}/\varepsilon W_{E[\varepsilon],e} \isom W_e$ (where we put $W:=(W_e,W^+_{\mathrm{dR}})$),
we have a natural exact sequence of $B_e\otimes_{\mathbb{Q}_p}E[G_K]$-modules
\begin{equation*}
0\rightarrow W_e\rightarrow W_{E[\varepsilon],e}\rightarrow W_e\rightarrow 0
\end{equation*}
We fix an isomorphism of $B_e\otimes_{\mathbb{Q}_p}E$-modules
$W_{E[\varepsilon],e}\isom W_e e_1\oplus W_ee_2$ such that first factor $W_e e_1$is equal to $\varepsilon W_{E[\varepsilon]}$
as $B_e\otimes_{\mathbb{Q}_p}E[G_K]$-module and that the above natural projection maps the second factor $W_e e_2$ to $W_e$ by $xe_2\mapsto x$ for any $x\in W_e$.  We define a continuous one cocycle by
$$c_e: G_K\rightarrow \mathrm{Hom}_{B_e\otimes_{\mathbb{Q}_p}E}(W_e,W_e)\text{ by }
g(ye_2):=c_e(g)(gy)e_1+gye_2$$
 for  any $g\in G_K$ and $y\in W_e$.
 For $W^+_{\mathrm{dR}}$,  we fix an isomorphism $W^+_{E[\varepsilon],\mathrm{dR}}\isom W^+_{\mathrm{dR}}e_1\oplus
 W^+_{\mathrm{dR}}e'_2$ as in the case of $W_e$, then we define a one cocycle by
 $$c_{\mathrm{dR}}:G_K\rightarrow \mathrm{Hom}_{B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E}(W^+_{\mathrm{dR}},W^+_{\mathrm{dR}})\text{ by  }g(ye'_2):=c_{\mathrm{dR}}(g)(gy)e_1+gye'_2$$
  for
 any $g\in G_K$ and for any $y\in W^+_{\mathrm{dR}}$. Next, we define $c\in\mathrm{Hom}_{B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E}
 (W_{\mathrm{dR}},W_{\mathrm{dR}})$ as follows. Tensoring $W_{E[\varepsilon],e}$ and $W^+_{E[\varepsilon],\mathrm{dR}}$ with
 $B_{\mathrm{dR}}$ over $B_e$ or $B^+_{\mathrm{dR}}$, we have
 an isomorphism $f:W_{\mathrm{dR}}e_1\oplus W_{\mathrm{dR}}e_2\isom W_{E[\varepsilon],\mathrm{dR}}\isom
 W_{\mathrm{dR}}e_1\oplus W_{\mathrm{dR}}e'_2$ of $B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E$-modules.
 We define by
 $$c:W_{\mathrm{dR}}\rightarrow W_{\mathrm{dR}}\text{ by } f(ye_2):=c(y)e_1+ye'_2$$
  for any $y\in W_{\mathrm{dR}}$.
 By the definition, the triple ($c_e$, $c_{\mathrm{dR}}$, $c$) satisfies
 $$c_e(g)-c_{\mathrm{dR}}(g)=gc-c\text{ in }\mathrm{Hom}_{B_{\mathrm{dR}}
 \otimes_{\mathbb{Q}_p}E}(W_{\mathrm{dR}},W_{\mathrm{dR}})$$ for any $g\in G_K$, i.e. the triple $(c_e,c_{\mathrm{dR}},c)$ defines an
 element of $ \mathrm{H}^1(G_K,\mathrm{ad}(W))$ by the definition of Galois cohomology of $B$-pairs ($\S$ 2.1 of \cite{Na09}), then it is standard to check that this definition is independent of
 the choice of fixed isomorphism $W_{E[\varepsilon],e}\isom W_ee_1\oplus W_ee_2$, etc, and it is easy to check that
 this map defines an isomorphism $D_W(E[\varepsilon])\isom \mathrm{H}^1(G_K,\mathrm{ad}(W))$.

\end{proof}

\begin{lemma}\label{11}
Let $W_A$ be a deformation of $W$ over $A$.
If $\mathrm{End}_{G_K}(W)=E$, then $\mathrm{End}_{G_K}(W_A)=A$.
\end{lemma}

\begin{proof}
We prove this lemma by induction on the length of $A$.
When $A=E$, this is trivial. We assume that this lemma is proved for the rings of length $n$ and assume that
$A$ is of length $n+1$. We take a small extension
$f:A\rightarrow A'$. Because $\mathrm{End}_{G_K}(W)=\mathrm{H}^0(G_K,
W^{\vee}\otimes W)$, then we have the following short exact sequence,
\begin{equation*}
0\rightarrow \mathrm{Ker}(f)\otimes_E\mathrm{End}_{G_K}(W)
\rightarrow \mathrm{End}_{G_K}(W_A)\rightarrow
\mathrm{End}_{G_K}(W_A\otimes_A A').
\end{equation*}
From this and the induction hypothesis, we have
\[
\begin{array}{ll}
\mathrm{length}(\mathrm{End}_{G_K}(W_A))&\leqq
\mathrm{length}(\mathrm{End}_{G_K}(W_A\otimes_A A'))+
\mathrm{length}(\mathrm{Ker}(f)\otimes_E \mathrm{End}_{G_K}(W))\\
&=\mathrm{length}(A')+1=\mathrm{length}(A).
\end{array}
\]
 On the other hand,
we have a natural inclusion $A\subseteq \mathrm{End}_{G_K}(W_A)$.
Comparing length, we have an equality $A=\mathrm{End}_{G_K}(W_A)$.
\end{proof}

\begin{proof} (of proposition)
Let $W$ be a rank $n$ $E$-$B$-pair satisfying $\mathrm{End}_{G_K}(W)=E$.
For this $W$, we check the conditions $(H_i)$ of Schlessinger's criterion.
First, by Lemma $\ref{10}$, we have
$$\mathrm{dim}_E(D_{W}(E[\varepsilon]))=
\mathrm{dim}_E(\mathrm{H}^1(G_K,\mathrm{ad}(W)))<\infty,$$ hence $(H_3)$ is satisfied.
Next we check $(H_1)$. Let  $f:A'\rightarrow A$, $g:A"\rightarrow A$ be morphisms in
$\mathcal{C}_E$ such that $g$ is surjective. Let $([W_{A'}],[W_{A"}])$ be an element in
$D_W(A')\times_{D_W(A)}D_W(A")$. We take deformations $W_{A'}:=(W_{A', e},W^{ +}_{A',  \mathrm{dR}})
$, $W_{A"}:=(W_{A", e},W^{+}_{A" ,\mathrm{dR}})$ over $A'$ and $A"$ which are contained in  equivalent classes
$[W_{A'}]$ and $[W_{A"}]$ respectively, then we have an isomorphism $h:W_{A'}\otimes_{A'}A\isom W_{A"}\otimes_{A"}A=:W_A
:=(W_{A, e}, W^+_{A,\mathrm{dR}})$
which defines an equivalent class in $D_W(A)$. We fix a basis $e_1,\cdots,e_n$ of $W_{A' ,e}$ as a $B_e\otimes_{\mathbb{Q}_p}A'$
-module and denote by $\overline{e}_1,\cdots,\overline{e}_n$ the basis of $W_{A' ,e}\otimes_{A'}A$ induced from $e_1,\cdots,e_n$.
By the surjection of $g:A"\rightarrow A$ and by the $A"$-flatness of $W_{A", e}$, we can take a basis
$\tilde{e}_1,\cdots,\tilde{e}_n$ of $W_{A", e}$ such that the basis
$\bar{\tilde{e}}_1,\cdots,\bar{\tilde{e}}_n$ of $W_{A", e}\otimes_{A"}A$ induced from $\tilde{e}_1,\cdots,\tilde{e}_n$
satisfies $h(\overline{e}_i)=\bar{\tilde{e}}_i$ for any $i$. If we define by
$$W^{'"}_e:=W_{A', e}\times_{W_{A ,e}}W_{A" ,e}
:=\{(x,y)\in W_{A', e}\times W_{A" ,e}| h(\bar{x})=\bar{y}\}, $$ then
$W^{'"}_e$ is a free $B_e\otimes_{\mathbb{Q}_p}(A'\times_A A")$-module
 with a basis $(e_1,\tilde{e}_1), \cdots, (e_n,\tilde{e}_n)$. In the same way, we define by
 $W^{'"+}_{\mathrm{dR}}:=W^{+}_{A' ,\mathrm{dR}}\times_{W^+_{A, \mathrm{dR}}} W^{+}_{A", \mathrm{dR}}$,
 which is a free $B^{+}_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}(A'\times_A A")$-module. If we put
 $W_{A'}\times_{W_A} W_{A"}:=(W^{'"}_e,W^{'" +}_{\mathrm{dR}})$, then this is a $(A'\times_A A")$-$B$-pair which is a deformation
 of $W$ over $A'\times_A A"$ such that the equivalent class $[W_{A'}\times_{W_A}W_{A"}]\in D_W(A'\times_A A")$ maps
 $([W_{A'}], [W_{A"}])\in D_W(A')\times_{D_W(A)}D_W(A")$. Hence, we have proved $(H_1)$.

 Finally, we prove that if $g:A"\rightarrow A$ is surjective, then the natural map
 $D_W(A'\times_AA")\rightarrow D_W(A')\times_{D_W(A)}D_W(A")$ is bijective, which
 proves the condition $(H_2)$ and $(H_4)$, hence we can prove the pro-representability of $D_W$.
 Let $W^{'"}_1$, $W^{'"}_2$ be deformations of $W$ over $A'\times_A A"$ such
 that $[W^{'"}_1\otimes_{A'\times_AA"}A']=[W^{'"}_2\otimes_{A'\times_{A}A"}A']$ in $D_W(A')$ and
 $[W^{'"}_1\otimes_{A'\times_AA"}A"]=[W^{'"}_2\otimes_{A'\times_{A}A"}A"]$ in $D_W(A")$.
 Under this situation, we want to show $[W^{'"}_1]=[W^{'"}_2]$ in $D_W(A'\times_A A")$.
 We put $W_{1 A'}:=W^{'"}_1\otimes_{A'\times_A A"}A'$, $W_{1 A"}:=W_1^{'"}\otimes_{A'\times_A A"}A"$
 , $W_{1 A}:=W_1^{'"}\otimes_{A'\times_A A"} A$, and same for $W_{2 A'}$, $W_{2 A"}$, $W_{2 A}$,
 then we have natural isomorphisms $W^{'"}_1\isom W_{1 A'}\times_{W_{1 A}} W_{1 A"}$ and $W_2^{'"}
 \isom W_{2 A'}\times_{W_{2 A}} W_{2 A"}$ defined as in the last paragraph. Because
 $[W_{1 A'}]=[W_{2 A'}]$ and $[W_{1 A"}]=[W_{2 A"}]$, we have isomorphisms
 $h':W_{1 A'}\isom W_{2 A'}$ and $h":W_{1 A"}\isom W_{2 A"}$. By reduction of these isomorphisms,
 we obtain an automorphism $\bar{h}'\circ \bar{h}^{" -1}:W_{2 A}\isom W_{1 A}\isom W_{2 A}$.
 By Lemma $\ref{11}$ and by the surjection $g:A^{" \times}\rightarrow A^{\times}$, we can find an automorphism
 $\tilde{h}:W_{2 A"}\isom W_{2 A"}$ such that $\bar{\tilde{h}}=\bar{h}'\circ \bar{h}^{" -1}$.
 If we define a morphism
 $$h^{'"}:W_{1 A'}\times_{W_{1 A}} W_{1 A"}\rightarrow
 W_{2 A"}\times_{W_{2 A}} W_{2 A'}: (x,y)\mapsto (h_1(x), \tilde{h}\circ h_2(y)),$$ then we can see that this is
 well-defined and is isomorphism. Hence we finish to prove this proposition.
 \end{proof}

 \begin{prop}\label{12}
 Let $W:=(W_e, W^+_{\mathrm{dR}})$ be an $E$-$B$-pair of rank $n$.
  If $\mathrm{H}^2(G_K, \mathrm{ad}(W))\allowbreak =0$, then the functor
 $D_W$ is formally smooth.
 \end{prop}
 \begin{proof}
 Let $A'\rightarrow A$ be a small extension in $\mathcal{C}_E$,
 we denote the kernel by $I\subseteq A'$.
 Let $W_A:=(W_{e,A}, W^+_{\mathrm{dR},A})$ be a deformation of $W$
 over $A$, then it suffices to show that there exists an $A'$-$B$-pair $W_{A'}$
 such that $W_{A'}\otimes_{A'}A\isom W_A$.
 We fix a basis of $W_{e,A}$ as a $B_e\otimes_{\mathbb{Q}_p}A$-module.
 Using this basis and the $G_K$-action on $W_{e,A}$,
 we obtain a continuous one cocycle $$\rho_e:G_K\rightarrow \mathrm{GL}_n(B_e\otimes_{\mathbb{Q}_p}A).$$
  In the same way, if we fix a basis of $W^+_{\mathrm{dR},A}$ as a $B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A$-module,
  we obtain a continuous cocycle $$\rho_{\mathrm{dR}}:G_K\rightarrow \mathrm{GL}_n(B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A).$$
   From the canonical isomorphism $W_{e,A}\otimes_{B_e}B_{\mathrm{dR}}\isom W^+_{\mathrm{dR},A}
   \otimes_{B^+_{\mathrm{dR}}}
   B_{\mathrm{dR}}$, we obtain a matrix $P\in\mathrm{GL}_n(B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A)$ such that
   $$P\rho_e(g)g(P)^{-1}=\rho_{\mathrm{dR}}(g)\text{ for any }g\in G_K.$$
   We fix an $E$-linear section $s:A\rightarrow A'$ of $A'\rightarrow A$ and fix a lifting
   $\tilde{P}\in \mathrm{GL}_n(B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A')$ of $P$. Using this section,
   we obtain  continuous liftings
   $$\tilde{\rho}_e:=s\circ \rho_e:G_K\rightarrow \mathrm{GL}_n(B_e\otimes_{\mathbb{Q}_p}A')$$ of $\rho_e$
   and $$\tilde{\rho}_{\mathrm{dR}}:=s\circ \rho_{\mathrm{dR}}:G_K\rightarrow \mathrm{GL}_n(B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A')$$ of
   $\rho_{\mathrm{dR}}$.
   Using these liftings, we define
   $$c_e:G_K\times G_K\rightarrow I\otimes_E \mathrm{Hom}_{B_e\otimes_{\mathbb{Q}_p}E}(W_e, W_e)$$
   by

   \[
\begin{array}{ll}
\tilde{\rho}_e(g_1g_2)g_1(\tilde{\rho}_e(g_2))^{-1}\tilde{\rho}_e(g_1)^{-1}=1+c_{e}(g_1,g_2)&\in
   1+I \otimes_{A'}\mathrm{M}_n(B_e\otimes_{\mathbb{Q}_p}A')\\
   &=1+I\otimes_{E}\mathrm{Hom}_{B_e\otimes_{\mathbb{Q}_p}E}
   (W_e,W_e)
   \end{array}
   \]

    for any $g_1,g_2\in G_K$. In the same way, we define
    $$c_{\mathrm{dR}}:G_K\times G_K \rightarrow I\otimes_E\mathrm{Hom}_{B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E}
   (W^+_{\mathrm{dR}}, W^+_{\mathrm{dR}})$$ by
   $$\tilde{\rho}_{\mathrm{dR}}(g_1g_2)g_1(\tilde{\rho}_{\mathrm{dR}}(g_2))^{-1}\tilde{\rho}_{\mathrm{dR}}(g_1)^{-1}
   =1+c_{\mathrm{dR}}(g_1,g_2).$$  We define
   $$c:G_K\rightarrow I\otimes_E\mathrm{Hom}_{B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E}(W_{\mathrm{dR}}, W_{\mathrm{dR}})$$
   by
   $$\tilde{P}\tilde{\rho}_e(g)g(\tilde{P})^{-1}\tilde{\rho}_{\mathrm{dR}}(g)^{-1}=1 + c(g)\in 1+I\otimes_E\mathrm{Hom}_{B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E}(W_{\mathrm{dR}}, W_{\mathrm{dR}}).$$  These $c_e$ and $c_{\mathrm{dR}}$ are continuous two cocycles, i.e. satisfy
   $$g_1c_{\ast}(g_2, g_3)-c_{\ast}(g_1g_2,g_3)+c_{\ast}(g_1, g_2g_3)
   -c_{\ast}(g_1,g_2)=0$$ for any $g_1,g_2,g_3\in G_K$ ($\ast=e, \mathrm{dR}$). Moreover, we can check that $c_e$ and $c_{\mathrm{dR}}$ and $c$ satisfy
   $$c_e(g_1,g_2)-c_{\mathrm{dR}}(g_1,g_2)=g_1(c(g_2))-c(g_1g_2)+c(g_1)$$ for any
   $g_1,g_2,g_3\in G_K$, here we note that the isomorphism $\mathrm{Hom}_{B_e\otimes_{\mathbb{Q}_p}E}(W_e, W_e)
   \otimes_{B_e}B_{\mathrm{dR}}\isom \mathrm{Hom}_{B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E}(W^+_{\mathrm{dR}},W^+_{\mathrm{dR}})\otimes_{B^+_{\mathrm{dR}}}B_{\mathrm{dR}}$ is given by $c\mapsto \bar{P}^{-1}c\bar{P}$, where $\bar{P}\in \mathrm{GL}_n(B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E)$
    is the reduction of $P\in \mathrm{GL}_n(B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A)$. By the definition of Galois cohomology
    of $B$-pairs, these mean that $(c_e,c_{\mathrm{dR}}, c)$ defines an element $[(c_e,c_{\mathrm{dR}},c)]$ in $I\otimes_E\mathrm{H}^2(G_K, \mathrm{ad}(W))$ . Until here, we  don't use the condition $\mathrm{H}^2(G_K,\mathrm{ad}(W))=0$ and we can show that $[(c_e,c_{\mathrm{dR}},c)]$ doesn't depend on the choice of $s$ or $\tilde{P}$, i.e. depends only on $W_A$.
    Under the assumption that $\mathrm{H}^2(G_K,\mathrm{ad}(W))=0$, there exists a triple $(f_e,f_{\mathrm{dR}}, f)
    $ such that $f_e:G_K\rightarrow I\otimes_{E}\mathrm{Hom}_{B_e\otimes_{\mathbb{Q}_p}E}(W_e, W_e)$ and
    $f_{\mathrm{dR}}:G_K\rightarrow I\otimes_E\mathrm{Hom}_{B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E}(W^+_{\mathrm{dR}}, W^+_{\mathrm{dR}})$ are continuous maps and $f\in I\otimes_E\mathrm{Hom}_{B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E}(W_{\mathrm{dR}}, W_{\mathrm{dR}})$
    and these satisfy $$c_e(g_1,g_2)=g_1f_e(g_2)-f_e(g_1g_2)+f_e(g_1)$$ and $$c_{\mathrm{dR}}(g_1,g_2)=g_1f_{\mathrm{dR}}(g_2)
    -f_{\mathrm{dR}}(g_1g_2)+f_{\mathrm{dR}}(g_1)$$ and $$c(g_1)=f_{\mathrm{dR}}(g_1)-\bar{P}^{-1}f_{e}(g_1)\bar{P}+(g_1f-f)$$
    for any $g_1,g_2\in G_K$.
    Using these, we define new liftings $\rho'_{e}:G_K\rightarrow \mathrm{GL}_n(B_e\otimes_{\mathbb{Q}_p}A')$ by
    $$\rho'_e(g):=(1+f_e(g))\tilde{\rho}_e(g),$$
    $\rho'_{\mathrm{dR}}(g):G_K\rightarrow \mathrm{GL}_n(B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A')$ by $$\rho'_{\mathrm{dR}}(g):=(1+f_{\mathrm{dR}}(g))\tilde{\rho}_{\mathrm{dR}}(g)$$ and define
    $$P':=(1+f)\tilde{P}\in \mathrm{GL}_n(B_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A'),$$ then we can check
    that these satisfy $$\rho'_e(g_1g_2)=\rho'_e(g_1)g_1(\rho'_{e}(g_2))\text{ and }\rho'_{\mathrm{dR}}(g_1g_2)
    =\rho'_{\mathrm{dR}}(g_1)g_1(\rho'_{\mathrm{dR}}(g_2))$$ and $$P'\rho'_e(g_1)g(P')^{-1}=\rho'_{\mathrm{dR}}(g_1)$$ for
    any $g_1,g_2\in G_K$.  By the definition of $A'$-$B$-pair, this means that $(\rho'_e,\rho'_{\mathrm{dR}},P')$ defines
    an $A'$-$B$-pair and this is a lift of $W_A$ by the definition. We finish the proof of this proposition.
    \end{proof}

    \begin{corollary}\label{13}
    Let $W$ be an $E$-$B$-pair of rank $n$.
    If $\mathrm{End}_{G_K}(W)=E$ and $\mathrm{H}^2(G_K, \mathrm{ad}(W))=0$,
    then the functor $D_W$ is pro-representable by $R_W$ such that
    $$R_W\isom E[[T_1,\cdots,T_d]]\,\,\text{ where } d:=[K:\mathbb{Q}_p]n^2 +1.$$
    \end{corollary}
    \begin{proof}
    The existence and formally smoothness of $R_W$ follows from Proposition $\ref{9}$ and Proposition $\ref{12}$.
    For the dimension, we have
     \[
\begin{array}{ll}
    \mathrm{dim}_ED_W(E[\varepsilon])&=\mathrm{dim}_E\mathrm{H}^1(G_K, \mathrm{ad}(W))\\
                                                    &=[K:\mathbb{Q}_p]n^2
    +\mathrm{dim}_E\mathrm{H}^0(G_K,\mathrm{ad}(W))+\mathrm{dim}_E\mathrm{H}^2(G_K,\mathrm{ad}(W)) \\
    &=[K:\mathbb{Q}_p]n^2
    +1
    \end{array}
    \]
     by Theorem $\ref{h}$ and Lemma $\ref{10}$.
    \end{proof}

\subsection{Trianguline  deformations of trianguline $B$-pairs}

In this subsection, we define the trianguline deformation functor for
split trianguline $E$-$B$-pairs and prove the pro-representability and the formally smoothness
under some conditions and calculate the dimension of the universal
deformation ring of  this functor. These are the generalizations of Bella\"iche-Chenevier's works
in the $\mathbb{Q}_p$ case. In $\S$ 2 of \cite{Bel-Ch09}, Bella\"iche-Chenevier
proved all of these in the case
of $K=\mathbb{Q}_p$ by using $(\varphi,\Gamma)$-modules over the Robba ring and by using Colmez's
theory of trianguline representations for $K=\mathbb{Q}_p$ (\cite{Co08}).
We generalize their theory by using
$B$-pairs and the theory of trianguline representations for any $p$-adic field (\cite{Na09} or $\S$ 2.1).

We first define the notion of split  trianguline $A$-$B$-pairs  as follows.
\begin{defn}
Let $W$ be a rank $n$ $A$-$B$-pairs. We say that $W$ is a split trianguline
$A$-$B$-pair if there exists a sequence of sub $A$-$B$-pairs
$\mathcal{T}: 0=W_0\subseteq W_1\subseteq W_2\subseteq\cdots \subseteq W_{n-1}
\subseteq W_n=W$ such that  $W_i$ is saturated in $W_{i+1}$ and the
quotient $W_{i+1}/W_i$ is a rank one $A$-$B$-pair for any $0\leqq i\leqq
n-1$.

By Proposition $\ref{5}$, there exist continuous homomorphisms $\delta_i:K^{\times}\rightarrow A^{\times}$
such that $W_i/W_{i-1}\isom W(\delta_i)$ for any $1\leqq i\leqq n$.
We say that $\{\delta_i\}_{i=1}^n$ is the parameter of the triangulation
$\mathcal{T}$.
\end{defn}

Next, we define the trianguline deformation functor.
Let $W$ be a rank $n$ split trianguline $E$-$B$-pair. We fix a triangulation
$\mathcal{T}: 0\subseteq W_1\subseteq\cdots\subseteq W_{n-1}\subseteq
W_n=W$ of $W$. Under this situation, we define the trianguline deformation
as follows.

\begin{defn}
Let $A$ be an object in $\mathcal{C}_E$.
We say that $(W_A,\iota, \mathcal{T}_A)$ is a trianguline deformation of $(W,\mathcal{T})$ over $A$
if $(W_A,\iota)$ is a deformation of $W$ over $A$ and $W_A$ is a split trianguline $A$-$B$-pair with
a triangulation $\mathcal{T}_A:0\subseteq W_{1,A}\subseteq \cdots \subseteq W_{n,A}=W_A$
 such that  $\iota(W_{i,A}\otimes_A E)=W_i$ for any $1\leqq i\leqq n$.
Let $(W_A,\iota,\mathcal{T}_A)$ and $(W'_A,\iota',\mathcal{T}'_A)$ be two trianguline
deformations of $(W,\mathcal{T})$ over $A$. We say that $(W_A,\iota,\mathcal{T}_A)$ and
$(W'_A,\iota',\mathcal{T}'_A)$ are equivalent if there exists an isomorphism of $A$-$B$-pairs
$f:W_A\isom W'_A$ satisfying $\iota=\iota'\circ \bar{f}$ and $f(W_{i,A})=W'_{i,A}$ for
any $1\leqq i\leqq n$.
\end{defn}
\begin{defn}
Let $W$ be a split trianguline $E$-$B$-pair with a triangulation $\mathcal{T}$.
We define the trianguline deformation functor $D_{W,\mathcal{T}}$
from the category $\mathcal{C}_E$ to the category of sets by
\[
\begin{array}{ll}
D_{W,\mathcal{T}}(A):=\{&\text{equivalent classes } (W_A,\iota,\mathcal{T}_A) \text{ of }\\
&\text{trianguline deformations of } (W,\mathcal{T})\text{ over }A  \}.
\end{array}
\]
for any $A\in\mathcal{C}_E$.
\end{defn}

By definition, we have a natural map of functors from $D_{W,\mathcal{T}}$ to $D_{W}$ by forgetting
the triangulation, i.e. defined by
$$D_{W,\mathcal{T}}(A)\rightarrow D_{W}(A): [(W_A,\iota,\mathcal{T}_A)]
\mapsto [(W_A,\iota)].$$
 In general, $D_{W,\mathcal{T}}$ is not a sub functor of
$D_W$ by this map, i.e. a deformation $W_A$ can have many liftings of the triangulation $\mathcal{T}$.
Here, we give a sufficient condition for $D_{W,\mathcal{T}}$ to be a sub functor of $D_W$.
Let $\{\delta_i\}_{i=1}^n$ be the parameter of triangulation $\mathcal{T}$.
\begin{lemma}\label{14}
If we have $\delta_j/\delta_i\not= \prod_{\sigma\in\mathcal{P}}\sigma^{k_{\sigma}}$  for any $1\leqq i<j\leqq n$
and for
any $\{k_{\sigma}\}_{\sigma\in\mathcal{P}}\in \prod_{\sigma\in\mathcal{P}} \mathbb{Z}_{\leqq 0}$, then
the functor $D_{W,\mathcal{T}}$ is a sub functor of $D_W$.
\end{lemma}
\begin{proof}
Let $W_A$ be a deformation of $W$ over $A$, let $0\subseteq W_{A ,1}\subseteq \cdots \subseteq W_{A ,n-1}\subseteq W_{A}$
and $0\subseteq W'_{A ,1}\subseteq \cdots \subseteq W'_{A, n-1}\subseteq W_{A}$ be two triangulations which are
lifts of $\mathcal{T}$. It suffices to show that $W_{A, i}=W'_{A, i}$ for any $i$. By induction, it suffices
to show $W_{A ,1}= W'_{A ,1}$. For proving this, first we consider $\mathrm{Hom}_{G_K}(W_{1,A}, W_A)$.
This is equal to  $\mathrm{H}^0(G_K, W^{\vee}_{1, A}\otimes W_A)$. Because $\mathrm{H}^0(G_K, -)$ is left exact
and $\mathrm{H}^0(G_K, W(\delta))=0$ for any $\delta:K^{\times}\rightarrow E^{\times}$ such that
$\delta\not= \prod_{\sigma\in \mathcal{P}}\sigma^{k_{\sigma}}$ for any $\{k_{\sigma}\}_{\sigma\in\mathcal{P}}
\in \prod_{\sigma\in\mathcal{P}}\mathbb{Z}_{\leqq 0}$ by Proposition $\ref{i}$, we have
$$\mathrm{H}^0(G_K, W_{1, A}^{\vee}\otimes
(W_{i+1, A}/W_{i, A}))\allowbreak =\mathrm{H}^0(G_K, W_{1, A}^{\vee}\otimes( W'_{i+1, A}/W'_{i, A}))=0$$
 for any $i\geqq 1$.
Hence, we obtain equalities  $$\mathrm{Hom}_{G_K}(W_{1, A}, W_{1,A})\allowbreak =\mathrm{Hom}_{G_K}(W_{1,A}, \allowbreak W_A)=\mathrm{Hom}_{G_K}(W_{1, A},
W'_{1,A}).$$ This means that the given inclusion $W_{1, A}\hookrightarrow W_A$ factors through $W'_{1, A}\hookrightarrow W_A$.
By the same reason, the inclusion $W'_{1,A}\hookrightarrow W_A$ also factors through $W_{1, A}\hookrightarrow W_{A}$.
Hence we obtain an equality $W_{1,A}=W'_{1, A'}$.
\end{proof}

Next, we prove relative representability of $D_{W,\mathcal{T}}$. But, before doing this, we need to define the following
functor which is a $B$-pair version of Lemma 2.3.8 of \cite{Bel-Ch09}.
Let $W=(W_e,W^+_{\mathrm{dR}})$ be an $E$-$B$-pair. Then we define by
$$F(W):=\{x\in W_e\cap W^+_{\mathrm{dR}}| \exists n\in\mathbb{Z}_{\geqq 1}, (\sigma_1-1)(\sigma_2-1)\cdots (\sigma_n-1)x=0
, \forall \sigma_1,\cdots,\sigma_n\in G_K\}$$ which is $E$-vector space with $G_K$-action, hence
$F$ is a left exact functor form the category of $E$-$B$-pairs to that of $E[G_K]$-modules. By this definition, we
obtain the following lemma, which is the $B$-pair version of Lemma 2.3.8 of \cite{Bel-Ch09}.
\begin{lemma}\label{14.5}
Let $\delta:K^{\times}\rightarrow E^{\times}$ be a continuous homomorphism, then
$F(W(\delta))\not= 0$ if and only if $\mathrm{H}^0(G_K, W(\delta))$.
\end{lemma}

Using this lemma, we prove relative representability of $D_{W,\mathcal{T}}$.

\begin{prop}\label{15}
Let $W$ be a trianguline representations with a triangulation $\mathcal{T}$
such that the parameter $\{\delta_i\}_{\i=1}^n$ satisfies $\delta_j/\delta_i\not=\prod_{\sigma\in \mathcal{P}}
\sigma^{k_{\sigma}}$ for any $1\leqq i<j\leqq n$ and for any $\{k_{\sigma}\}_{\sigma\in\mathcal{P}}\in \prod_{\sigma\in\mathcal{P}}\mathbb{Z}_{\leqq 0}$,
then the natural map of functors $D_{W,\mathcal{T}}\rightarrow D_{W}$ is relatively representable.
\end{prop}
\begin{proof}
By $\S 23$ of \cite{Ma97}, it suffices to check that the map $D_{W,\mathcal{T}}\rightarrow D_W$  satisfies the fallowing three conditions (1), (2), (3).

 (1): For any map $A\rightarrow A'$ in $\mathcal{C}_E$ and $W_{A}\in D_{W,\mathcal{T}}(A)$, then
 $W_A\otimes_A A'\in D_{W,\mathcal{T}}(A')$.

 (2): For any maps $A'\rightarrow A$ and $A"\rightarrow A$ in $\mathcal{C}_E$
  and $W^{'"}\in D_W(A'\times_A A")$, if $W^{'"}\otimes_{A'\times_A A"} A'\in D_{W,\mathcal{T}}(A')$ and
  $W^{'"}\otimes_{A'\times_A A"}A"\in D_{W,\mathcal{T}}(A")$, then $W^{'"}\in D_{W,\mathcal{T}}(A'\times_A A")$.

  (3): For any inclusion $A\hookrightarrow A'$ in $\mathcal{C}_E$ and $W_{A}\in D_W(A)$, if $W_A\otimes_A A'\in
   D_{W,\mathcal{T}}(A')$, then $W_A\in D_{W,\mathcal{T}}(A)$.

   The condition (1) is trivial. For (2), let $W^{'"}\in D_{W}(A'\times_A A")$ be such that $W_{A'}:=W^{'"}\otimes_{A'\times_A A"}A'
   \in D_{W,\mathcal{T}}(A')$ and $W_{A"}:=W^{'"}\otimes_{A'\times_A A"}A"\in D_{W,\mathcal{T}}(A")$. We put $W_A:=W^{'"}\otimes_{A'\times_A A"}A$. In the same way
    as in the proof of Proposition $\ref{9}$, we have an isomorphism
   $W^{'"}\isom W_{A'}\times_{W_A} W_{A"}$.
   By Lemma $\ref{14}$, the triangulations of $W_A$ induced from $W_{A'}$ and $W_{A"}$ are same, hence
   these triangulations induce a triangulation of $W^{'"}\isom W_{A'}\times_{W_A} W_{A"}$, i.e. $W^{'"}\in D_{W,\mathcal{T}}(A'\times_A A")$.

   Finally, we prove the condition (3). The proof is essentially same as that of Proposition 2.3.9
   of \cite{Bel-Ch09}, but here we give the proof for convenience of readers.
   Let $W\in D_W(A)$ and $A\hookrightarrow A'$ be an inclusion such that
   $W_A\otimes_A A'\in D_{W,\mathcal{T}}(A')$. Let $0\subseteq W_{1,A'}\subseteq\cdots\subseteq W_{n-1,A'}\subseteq W_{A}
   \otimes_A A'$ be a triangulation lifting of $\mathcal{T}$. By the induction on the rank of $W$, it suffices to show that there exists
   a rank one sub $A$-$B$-pair $W_{1,A}\subseteq W_A$ such that $W_{1,A}\otimes_{A}A'=W_{1,A'}$ and that $W_A/W_{1,A}$ is an $A$-$B$-pair. By Proposition \ref{5}, there exists a continuous homomorphism $\delta_{1,A'}:K^{\times}\rightarrow
   A^{' \times}$ such that $W_{1,A'}\isom W(\delta_{1,A'})$. Twisting $W$ by $\delta_1^{-1}$, we may assume that
   $\delta_{1,A'}\equiv 1$( mod $\mathfrak{m}_{A'}$). Under this assumption, we apply the functor $F(-)$.
   Because $\delta_{1,A'}$ is unitary,  there exists a continuous character $\tilde{\delta}_{1,A'}:G_K^{\mathrm{ab}}\rightarrow
   A^{' \times}$ such that $W(\delta_{1,A'})\isom W(A'(\tilde{\delta}_{1,A'}))=(B_e\otimes_{\mathbb{Q}_p}A'(\tilde{\delta}_{1,A'}),
   B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}A'(\tilde{\delta}_{1,A'}))$, hence $W_{1,A',e}\cap W^+_{1,A',\mathrm{dR}}
   =A'(\tilde{\delta}_{1,A'})$. Moreover, because the image of $\delta_{1,A'}$ is in $1+\mathfrak{m}_{A'}$, we also have
   $F(W_{1,A'})=A'(\tilde{\delta}_{1,A'})$. Next, because $(W_A\otimes_A A')/W_{1,A'}$ is a successive extension of $W(\delta_{i}\delta^{-1}_{1})$ ($i\geqq 2$) as an $E$-$B$-pair,
   the left exactness of $F$ implies that $F((W_A\otimes_A A')/W_{1,A'})=0$ by Lemma \ref{14.5}.
   Applying $F$ to the short exact sequence $0\rightarrow
   W_{1,A'}\rightarrow W_A\otimes_A A'\rightarrow (W_A\otimes_A A')/W_{1,A'}\rightarrow 0$, we obtain
   $A'(\tilde{\delta}_{1,A'})\isom F(W_{1,A'})=F(W_A\otimes_A A')$. In the same way, we obtain
   $E=F(W_1)=F(W)$. Then, by left exactness and by considering the length,
   we can show that $F(W_A)$ is a free $A$-module of rank one and that the natural map $F(W_A)\rightarrow F(W)$ ( induced by the
   natural quotient map $W_A\rightarrow W$) is surjection and that the natural map $F(W_A)\otimes_A A'\rightarrow
   F(W_A\otimes_A A')$ is isomorphism. If we define $\tilde{\delta}_{1,A}:G_K^{\mathrm{ab}}\rightarrow A^{\times}$ such that
   $F(W_A)\isom A(\tilde{\delta}_{1,A})$ and define $W_{1,A}$ as the image of the natural map $(B_e\otimes_{\mathbb{Q}_p}F(W_A),
   B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}F(W_A))\rightarrow W_A$ induced form $F(W)\hookrightarrow W_{A,e}, F(W)\hookrightarrow
   W^+_{A,\mathrm{dR}}$, then we can check that $W_{1,A}$ is a rank one $A$-$B$-pair and that the quotient $W_A/W_{1,A}$ is
   also an $A$-$B$-pair and that $W_{1,A}\otimes_A A'\isom W_{1,A'}$, which proves the condition (3), hence we finish to prove this proposition.

     \end{proof}
     \begin{corollary}\label{16}
     Let $W$ be a trianguline $E$-$B$-pair with a triangulation $\mathcal{T}$ such that $\mathrm{End}_{G_K}(W)=E$
      and the parameter $\{\delta_i\}_{i=1}^n$ of $\mathcal{T}$ satisfies
      $\delta_j/\delta_i\not= \prod_{\sigma\in \mathcal{P}}\sigma^{k_{\sigma}}$  for any $1\leqq i<j\leqq n$ and
      for any $\{k_{\sigma}\}_{\sigma\in \mathcal{P}}
      \in \prod_{\sigma\in\mathcal{P}}\mathbb{Z}_{\leqq 0}$,
      then the functor $D_{W,\mathcal{T}}$ is pro-representable by a quotient $R_{W,\mathcal{T}}$ of $R_W$.
      \end{corollary}
      \begin{proof} This follows from Proposition $\ref{9}$ and Proposition $\ref{15}$.
      \end{proof}

 Next, we prove  the formally smoothness of the functor $D_{W,\mathcal{T}}$.
 \begin{prop}\label{17}
 Let $W$ be a trianguline $E$-$B$-pair of rank $n$ with a triangulation $\mathcal{T}$ such that
  whose parameter $\{\delta_i\}_{i=1}^{n}$ satisfies $\delta_i/\delta_j\not=
  |\mathrm{N}_{K/\mathbb{Q}_p}|\prod_{\sigma\in\mathcal{P}}\sigma^{k_{\sigma}} $  for any $1\leqq i< j\leqq n$ and
  for any
  $\{k_{\sigma}\}_{\sigma\in\mathcal{P}}\in \prod_{\sigma\in\mathcal{P}}\mathbb{Z}_{\geqq 1}$,
  then the functor $D_{W,\mathcal{T}}$ is formally smooth.
  \end{prop}
  \begin{proof}
  We prove this proposition by the induction of the rank of $W$.
  When $W$ is rank one, then $D_{W,\mathcal{T}}=D_W$ and
 $\mathrm{ad}(W)=B_E$ is the trivial $E$-$B$-pair.
 Hence $\mathrm{H}^2(G_K,\mathrm{ad}(W))\allowbreak=0$ by Proposition $\ref{i}$. Hence,
 $D_{W,\mathcal{T}}$ is formally smooth by Proposition $\ref{12}$.
 Let's assume that the proposition is proved for all the rank $n-1$ $E$-$B$-pairs,
  let $W$ be a rank $n$ $E$-$B$-pair with a triangulation
  $\mathcal{T}:0\subseteq W_1\subseteq \cdots \subseteq W_{n-1}\subseteq W_n=W$
  whose parameter $\{\delta_i\}_{i=1}^n$ satisfying the condition as above.
  Let $A'\rightarrow A$ be a small extension in $\mathcal{C}_E$ and let
  $W_A$ be a trianguline deformation of $(W,\mathcal{T})$ with a triangulation
  $\mathcal{T}_A:0\subseteq W_{1,A}\subseteq \cdots \subseteq W_{n-1,A}\subseteq W_{n,A}=W_A$
  which is a lift of $\mathcal{T}$. It suffices to prove that there exists a split trianguline $A'$-$B$-pair
  $W_{A'}$ with a triangulation $0\subseteq W_{1,A'}\subseteq \cdots \subseteq W_{n-1.A'}\subseteq
  W_{n,A'}=W_{A'}$ which is a lift of $W_A$ and $\mathcal{T}_A$. We take a lifting as follows.
  First, because $W_{n-1}$ is a trianguline $E$-$B$-pair of rank $n-1$ satisfying the condition as above,
  hence there exists a rank $n-1$ trianguline $A'$-$B$-pair $W_{n-1,A'}$
  with a triangulation $0\subseteq W_{1,A'}\subseteq \cdots \subseteq W_{n-2,A'}\subseteq W_{n-1,A'}$
  which is a lift of $W_{n-1,A}$ and $0\subseteq W_{1,A}\subseteq\cdots\subseteq W_{n-1,A}$ by the induction hypothesis.
  If we put $\mathrm{gr}_nW_A:=W_{A}/W_{n-1,A}$, then by the rank one case and by Proposition $\ref{5}$, there exists
  a continuous homomorphism $\delta_{n,A'}:K^{\times}\rightarrow A^{' \times}$ such that the rank one $A'$-$B$-pair
  $W(\delta_{n,A'})$ satisfies $W(\delta_{n,A'})\otimes_{A'}A=W(\delta_{n,A})\isom \mathrm{gr}_nW_A$,
  where $\delta_{n,A}:K^{\times}\rightarrow A^{\times}$ is the reduction of $\delta_{n,A'}$.
  We can see the isomorphism class $[W_A]$ as an element in $\mathrm{Ext}^1(W(\delta_{n,A}), W_{n-1,A})\isom \mathrm{H}^1(G_K,
  W_{n-1,A}(\delta_{n,A}^{-1}))$. If we take the long exact sequence associated
  to
   \begin{equation*}
  0\rightarrow I\otimes_E W_{n-1}(\delta_n^{-1})\rightarrow W_{n-1,A'}(\delta^{-1}_{n,A'})\rightarrow
  W_{n-1,A}(\delta^{-1}_{n,A})\rightarrow 0
  \end{equation*}

   where $I\subseteq A'$ is the kernel of $A'\rightarrow A$,  then we obtain a long exact sequence
  \[
  \begin{array}{ll}
  \cdots \rightarrow \mathrm{H}^1(G_K,W_{n-1,A'}(\delta^{-1}_{n,A'}))& \rightarrow \mathrm{H}^1(G_K, W_{n-1,A}(\delta^{-1}_{n,A}))\\
                                        &\rightarrow I\otimes_E \mathrm{H}^2(G_K, W_{n-1}(\delta^{-1}_n))\rightarrow\cdots
  \end{array}
  \]
  By the assumption on $\{\delta_i\}_{i=1}^n$ and by Proposition $\ref{i}$, we have $\mathrm{H}^2(G_K, W_{n-1}(\delta^{-1}_n))=0$.
  Hence we can take a $[W_{A'}]\in\mathrm{Ext}^1(W(\delta_{n,A'}),W_{n-1,A'})\isom\mathrm{H}^1(G_K, W_{n-1,A'}(\delta^{-1}_{n,A'}))$
  which is a trianguline lift of $[W_A]$. This proves the proposition.
  \end{proof}

  Next, we calculate the dimension of $D_{W,\mathcal{T}}$. For this, we interpret the tangent space
  $D_{W,\mathcal{T}}(E[\varepsilon])$ in terms of Galois cohomology of $B$-pair as in Lemma $\ref{10}$.
  Let $W$ be a trianguline $E$-$B$-pair with a triangulation $\mathcal{T}:0\subseteq W_1\subseteq \cdots\subseteq
  W_{n-1}\subseteq W_n=W$ whose parameter is $\{\delta_i\}_{i=1}^n$,
  then we define an $E$-$B$-pair $\mathrm{ad}_{\mathcal{T}}(W)$ by
  $$\mathrm{ad}_{\mathcal{T}}(W):=\{f\in\mathrm{ad}(W)| f(W_i)\subseteq W_i \text{ for any } 1\leqq i\leqq n \}.$$
  \begin{lemma}\label{18}
  Let $W$ be a trianguline $E$-$B$-pair, then there exists
  a canonical bijection of sets $$D_{W,\mathcal{T}}(E[\varepsilon])
  \isom \mathrm{H}^1(G_K, \mathrm{ad}_{\mathcal{T}}(W)).$$
  In particular, if $D_{W,\mathcal{T}}$ has a canonical structure of $E$-vector space
  $($c.f.  the condition $(2)$ in Schlessinger's criterion $\ref{8}$$)$, then this bijection is an $E$-linear isomorphism.
  \end{lemma}
  \begin{proof}
The construction of the map $D_{W,\mathcal{T}}(E[\varepsilon])\rightarrow
\mathrm{H}^1(G_K,\mathrm{ad}_{\mathcal{T}}(W))$ is same as in the proof of
Lemma $\ref{10}$. We put $\mathrm{ad}_{\mathcal{T}}(W):=(\mathrm{ad}_{\mathcal{T}}(W_e), \mathrm{ad}_{\mathcal{T}}(W^+_{\mathrm{dR}}))$. Let $W_{E[\varepsilon]}:=(W_{e,E[\varepsilon]},W^+_{\mathrm{dR},E[\varepsilon]})$ be a trianguline deformation
of $(W,\mathcal{T})$ over $E[\varepsilon]$ whose triangulation $\mathcal{T}_{E[\epsilon]}$ is $0\subseteq W_{1,E[\varepsilon]},\subseteq\cdots\subseteq
W_{n-1,E[\varepsilon]}\subseteq W_{n,E[\varepsilon]}=W_{E[\varepsilon]}$, then we can take a splitting
$W_{e,E[\varepsilon]}=W_e e_1\oplus W_ee_2$ as a filtered $B_e\otimes_{\mathbb{Q}_p}E$-module such that
$W_e e_1=\varepsilon W_{e,E[\varepsilon]}$ and the natural map $W_ee_2\hookrightarrow W_{e,E[\varepsilon]}
\rightarrow W_{e,E[\varepsilon]}/\varepsilon W_{e,E[\varepsilon]}\isom W_e$ is such that $ye_2\mapsto y$ for any $y\in W_e$.
If we define $c_e:G_K\rightarrow \mathrm{Hom}_{B_e\otimes_{\mathbb{Q}_p}E}(W_e, W_e)$ in the same way as in the proof
of Lemma $\ref{10}$, we can check that the image of $c_e$ is contained in $\mathrm{ad}_{\mathcal{T}}(W_e)$. In the same way,
we can define $c_{\mathrm{dR}}:G_K\rightarrow \mathrm{ad}_{\mathcal{T}}(W^+_{\mathrm{dR}})$ from a
filtered splitting $W^+_{\mathrm{dR},E[\varepsilon]}=W^+_{\mathrm{dR}}e_1\oplus W^+_{\mathrm{dR}}e'_2$.
Moreover, we can define $c\in \mathrm{ad}_{\mathcal{T}}(W_{\mathrm{dR}})$ by $ye_2=c(y)e_1+ye'_2$ for any $y\in W_{\mathrm{dR}}$.
Then the map $$D_{W,\mathcal{T}}(E[\varepsilon])\rightarrow \mathrm{H}^1(G_K, \mathrm{ad}_{\mathcal{T}}(W)):[(W_{E[\varepsilon]},
\mathcal{T}_{E[\varepsilon]})]\mapsto [(c_e,c_{\mathrm{dR}},c)]$$ defines a bijection and, when $D_{W,\mathcal{T}}(E[\varepsilon])$
has a canonical $E$-vector space structure, this is an $E$-linear isomorphism.
\end{proof}

We calculate the dimension of $R_{W,\mathcal{T}}$.
\begin{prop}\label{19}
Let $W$ be a split trianguline $E$-$B$-pair of rank $n$ with a triangulation
$\mathcal{T}:0\subseteq W_1\subseteq \cdots\subseteq W_{n-1}\subseteq W_n=W$.
We assume that $(W,\mathcal{T})$ satisfies the following conditions,
\begin{itemize}
\item[(0)]$\mathrm{End}_{G_K}(W)=E$,
\item[(1)]For any $1\leqq i < j\leqq n$,  $\delta_j/\delta_i\not= \prod_{\sigma\in\mathcal{P}}\sigma^{k_{\sigma}}$
for any $\{k_{\sigma}\}_{\sigma\in\mathcal{P}}\in \prod_{\sigma\in\mathcal{P}}\mathbb{Z}_{\leqq 0}$,
\item[(2)]For any $1\leqq i < j\leqq n$,
$\delta_i/\delta_j\not= |\mathrm{N}_{K/\mathbb{Q}_p}|\prod_{\sigma\in\mathcal{P}}\sigma^{k_{\sigma}}$
for any $\{k_{\sigma}\}_{\sigma\in\mathcal{P}}\in \prod_{\sigma\in\mathcal{P}}\mathbb{Z}_{\geqq 1}$,
\end{itemize}
then the universal trianguline deformation ring $R_{W,\mathcal{T}}$ is a quotient ring of $R_W$ such
that $$R_{W,\mathcal{T}}\isom E[[T_1,\cdots, T_{d_n}]]\text{ where }d_n:=\frac{n(n+1)}{2}[K:\mathbb{Q}_p]+1.$$
\end{prop}
\begin{proof}
From Proposition $\ref{15}$ and Proposition $\ref{17}$ and Lemma $\ref{18}$, it suffices to show that $\mathrm{dim}_E\mathrm{H}^1(G_K, \mathrm{ad}_{\mathcal{T}}(W))=d_n$.
We prove this by the induction on the rank $n$ of $W$.
When $n=1$, then $\mathrm{ad}_{\mathcal{T}}(W)=\mathrm{ad}(W)=B_E$, hence the proposition
follows from Proposition $\ref{i}$. Let $W$ be a rank $n$ trianguline $E$-$B$-pair as above, let $\mathcal{T}_{n-1}:0\subseteq
W_{1}\subseteq \cdots \subseteq W_{n-2}\subseteq W_{n-1}$ be the triangulation of $W_{n-1}(\subseteq W)$.
Then for any $f\in \mathrm{ad}_{\mathcal{T}}(W)$, the restriction of $f$
to $W_{n-1}$ is an element of $\mathrm{ad}_{\mathcal{T}_{n-1}}(W_{n-1})$ and this defines a short exact sequence
of $E$-$B$-pair
\begin{equation*}
0\rightarrow \mathrm{Hom}(W(\delta_n), W)\rightarrow \mathrm{ad}_{\mathcal{T}}(W)\rightarrow
\mathrm{ad}_{\mathcal{T}_{n-1}}(W_{n-1})\rightarrow 0.
\end{equation*}
From this, we have
\[
\begin{array}{ll}
\mathrm{rank}(\mathrm{ad}_{\mathcal{T}}(W))&=\mathrm{rank}(\mathrm{ad}_{\mathcal{T}_{n-1}}(W_{n-1}))
+n \\
&=1+2+\cdots +n=\frac{n(n+1)}{2}.
\end{array}
\]
  From this and Theorem $\ref{h}$,
it suffices to show that $\mathrm{H}^0(G_K,\mathrm{ad}_{\mathcal{T}}(W))=E$ and $\mathrm{H}^2(G_K,\mathrm{ad}_{\mathcal{T}}(W))=0$
. For $H^0$, this follows from
$$E\subseteq \mathrm{H}^0(G_K, \mathrm{ad}_{\mathcal{T}}(W))\subseteq \mathrm{H}^0(G_K,
\mathrm{ad}(W))=E.$$ We prove $\mathrm{H}^2(G_K, \mathrm{ad}_{\mathcal{T}}(W))=0$ by the induction of the rank of $W$.
When $n=1$, this follows from Proposition $\ref{i}$. When $W$ is rank $n$, then from the above short exact sequence, we have the following
 long exact sequence
 \begin{equation*}
 \cdots \rightarrow \mathrm{H}^2(G_K, \mathrm{Hom}(W(\delta_n), W))\rightarrow \mathrm{H}^2(G_K, \mathrm{ad}_{\mathcal{T}}(W))
 \rightarrow \mathrm{H}^2(G_K, \mathrm{ad}_{\mathcal{T}_{n-1}}(W_{n-1}))\rightarrow 0,
\end{equation*}
then we have $\mathrm{H}^2(G_K, \mathrm{Hom}\allowbreak (W(\delta_n), W))=0$ by Proposition $\ref{i}$ and by the assumption on $\{\delta_i\}_{i=1}^n$.
Hence, by the induction hypothesis, we have $\mathrm{H}^2(G_K, \mathrm{ad}_{\mathcal{T}}(W))=0$. We finish the proof of this
proposition.

\end{proof}

\subsection{Deformations of benign $B$-pairs}

In this final subsection, we study benign representations (more generally benign $B$-pairs )
which is a class of  potentially crystalline and trianguline representations and have some very good properties for trianguline
deformations and play a crucial role in Zariski density problem of modular Galois (or crystalline)
representations in some deformation spaces of global (or local) $p$-adic Galois representations.
This  class was defined by Kisin in the case of $K=\mathbb{Q}_p$ and the rank $W$ is 2 in \cite{Ki03} and \cite{Ki10}.
He studied some deformation theoretic properties of this class in \cite{Ki03} and used these in a crucial way in his
proof of Zariski density of two dimensional crystalline representations of $G_{\mathbb{Q}_p}$.
For higher dimensional and the $\mathbb{Q}_p$ case, Bella\"iche-Chenevier (\cite{Bel-Ch09}) and Chenevier (\cite{Ch09b}) were
the first ones who noticed the importance of benign representations in the study of $p$-adic families
of trianguline representations. In particular, Chenevier (\cite{Ch09b}, where he calls ``generic" instead of benign)
discovered and proved a crucial theorem concerning to tangent spaces of the universal deformation
rings of benign representations. For the two dimensional case, this was discovered by Kisin (\cite{Ki10})
implicitly when $K=\mathbb{Q}_p$.
In fact, by using this theorem, Chenevier (\cite{Ch09b}) proved  many kinds of theorems concerning
Zariski density of modular Galois representations in some deformation spaces of global $p$-adic representations.

The aim of this subsection is to generalize the definition of benign representations and the
Chenevier's theorem for any $K$-case.

\subsubsection{Benign $B$-pairs}
Let $P(X)\in \mathcal{O}_K[X]$ be a polynomial such that $P(X)\equiv \pi_K X$ (mod deg $2$)
and that $P(X)\equiv X^q$ (mod $\pi_K$), where $q:=p^f$ and $f:=[K_0:\mathbb{Q}_p]$. We take the Lubin-Tate's formal group
law $\mathcal{F}$ of $K$ such that $[\pi_K]=P(X)$, where $[-]:\mathcal{O}_K\isom \mathrm{End}(\mathcal{F})$.  We denote by $K_n$  the abelian extension of $K$
generated by $[\pi_K^n]$-torsion points of $\mathcal{F}$ for any $n$, then we have a canonical isomorphism $\chi_{\mathrm{LT},n}:\mathrm{Gal}(K_n/K)\isom( \mathcal{O}_K^{\times}/
\pi^n\mathcal{O}_K)^{\times}$. We put $K_{\mathrm{LT}}:=\cup_{n=1}^{\infty}K_n$ and $G_n:=\mathrm{Gal}(K_n/K)$.

In \cite{Ki10}, \cite{Bel-Ch09} or \cite{Ch09b} etc, benign representation is defined as a special class of crystalline representations.
But, as we show in the sequel, we can easily generalize the main theorem to some potentially crystalline representations.
Hence, before defining benign representations, we first define the following class of potentially crystalline representations.

\begin{defn}
Let $W$ be an $E$-$B$pair. We say that $W$ is crystabelline if $W|_{G_{L}}$ is a crystalline $E$-$B$-pair of
$G_{L}$ for a finite abel extension $L$ of $K$.
\end{defn}
\begin{rem}
Because a finite abel extension $L$ of $K$ is contained in $K_mL'$ for some $m\geqq 0$, where $L'$ is a finite unramified extension of $K$, by using Hilbert 90, we can easily show that $W$ is crystabelline if and only if
$W|_{G_{K_m}}$ is crystalline for some $m\geqq 0$.
\end{rem}

Let $W$ be a crystabelline $E$-$B$-pair of rank $n$ such that $W|_{G_{K_m}}$ is crystalline for some $m$.
Let $D^{K_m}_{\mathrm{cris}}(W):=(B_{\mathrm{cris}}\otimes_{\mathbb{Q}_p}W)^{G_{K_m}}$
be the rank $n$ $E$-filtered ($\varphi,G_m$)-module over $K$. Because $K_m$ is totally ramified over $K$, this is
a free $K_0\otimes_{\mathbb{Q}_p}E$-module of rank $n$.
We take an embedding $\sigma:K_0\hookrightarrow \overline{E}$. This defines a map
$\sigma:K_0\otimes_{\mathbb{Q}_p}E\rightarrow \overline{E}:x\otimes y\rightarrow \sigma(x)y$.
From this, we can define the $\sigma$-component  $$D^{K_m}_{\mathrm{cris}}(W)_{\sigma}:=D^{K_m}_{\mathrm{cris}}(W)
\otimes_{K_0\otimes_{\mathbb{Q}_p}E, \sigma}\overline{E},$$ this has a $\overline{E}$-linear
$\varphi^f$-action and a $\overline{E}$-linear $G_m$-action. Let $\{\alpha_1,\cdots,\alpha_n\}$ be a solution in $\overline{E}$
(with multiplicities) of $\mathrm{det}_{\overline{E}}(T\cdot \mathrm{id}-\varphi ^f |_{D^{K_m}_{\mathrm{cris}}
(W)_{\sigma}})\in \overline{E}[T]$. Because $\varphi^f$ and the $G_m$-action commute, any generalized $\varphi^f$-eigenvector subspaces
 of $D^{K_m}_{\mathrm{cris}}(W)_{\sigma}$ are preserved by the action of $G_m$.
 Hence we can take a $\overline{E}$-basis $e_{1,\sigma},\cdots,e_{n,\sigma}$ of $D^{K_m}_{\mathrm{cris}}(W)_{\sigma}$ such that $e_{i,\sigma}$ is a generalized eigenvector of
 $\varphi^f$ with an eigenvalue $\alpha_i\in \overline{E}^{\times}$ and $G_m$ acts on $e_{i,\sigma}$ by a character $\tilde{\delta}_i:G_m\rightarrow \overline{E}^{\times}$ for any $i$.  We change numbering of $\{\alpha_1,\cdots,\alpha_n\}$ so that $e_{1,\sigma}, e_{2,\sigma},\cdots,e_{n,\sigma}$ gives a $\varphi^f$-Jordan decomposition of $D^{K_m}_{\mathrm{cris}}(W)_{\sigma}$ by this order.
 Because $\{\sigma, \varphi^{-1}\sigma,\cdots, \varphi^{-(f-1)}\sigma\}=\mathrm{Hom}_{\mathbb{Q}_p}(K_0, \overline{E})$ and
$$\varphi^i:D^{K_m}_{\mathrm{cris}}(W)_{\sigma}\isom D^{K_m}_{\mathrm{cris}}(W)_{\varphi^{-i}\sigma}:x\otimes y\mapsto \varphi^i(x)\otimes y$$ ( for any $x\in D^{K_m}_{\mathrm{cris}}(W)$ and $y\in \overline{E}$ ) is a $\overline{E}[\varphi^f,G_m]$-isomorphism,
the set $\{\alpha_1,\cdots,\alpha_n\}$ doesn't depend on the choice of $\sigma:K_0\hookrightarrow \overline{E}$.
If we put $$e_i:=e_{i,\sigma}+\varphi(e_{i,\sigma})+\cdots +\varphi^{f-1}(e_{i,\sigma})\in D^{K_m}_{\mathrm{cris}}(W)\otimes_{E}\overline{E},$$ we have $$D^{K_m}_{\mathrm{cris}}(W)\otimes_E\overline{E}=K_0\otimes_{\mathbb{Q}_p}\overline{E}e_1\oplus\cdots \oplus K_0\otimes_{\mathbb{Q}_p}\overline{E}e_n$$ such that the subspace $K_0\otimes_{\mathbb{Q}_p}\overline{E}e_1\oplus\cdots\oplus K_0\otimes_{\mathbb{Q}_p}\overline{E}e_i$ is preserved by
$\varphi$ and by the $G_m$-action for any $i$. Moreover, if we take a sufficiently large finite extension $E'$ of $E$, then we can
take $e_i\in D^{K_m}_{\mathrm{cris}}(W)\otimes_E E'$ such that $$D^{K_m}_{\mathrm{cris}}(W)\otimes_E E'=
K_0\otimes_{\mathbb{Q}_p}E'e_1\oplus \cdots\oplus K_0\otimes_{\mathbb{Q}_p}E'e_n$$ and $\alpha_i\in E'$ and $\tilde{\delta}_i:G_m\rightarrow E^{'\times}$ for any $i$.

Using these arguments, we first study a relation between crystabelline $E$-$B$-pairs and trianguline $E$-$B$-pairs.
\begin{lemma}\label{20.5}
Let $W$ be an $E$-$B$-pair of rank $n$. The following conditions are equivalent,
\begin{itemize}
\item[(1)]$W$ is crystabelline,
\item[(2)]$W$ is trianguline $($ i.e, $W\otimes_E E'$ is a
split trianguline $E'$-$B$-pair for a finite extension $E'$ of $E$$)$ and potentially crystalline.
\end{itemize}
\end{lemma}
\begin{proof}
First we assume that $W$ is crystabelline. By the above argument, for a sufficiently large finite extension
$E'$ of $E$, we have $D^{K_m}_{\mathrm{cris}}(W)\otimes_E E'=K_0\otimes_{\mathbb{Q}_p}E'e_1\oplus
\cdots \oplus K_0\otimes_{\mathbb{Q}_p}E'e_n$ as above and $K_0\otimes_{\mathbb{Q}_p}E'e_1
\oplus \cdots\oplus K_0\otimes_{\mathbb{Q}_p}E'e_i$ is a sub $E'$-filtered ($\varphi,G_m$)-module
of $D^{K_m}_{\mathrm{cris}}(W\otimes_EE')$ for any $i$. Hence $W\otimes_E E'$ is
split trianguline and potentially crystalline by Theorem $\ref{e}$.

Next we assume that $W$ is trianguline and potentially crystalline. By extending the coefficient, we may assume that $W$ is split trianguline.
We take a triangulation $0\subseteq W_1\subseteq \cdots \subseteq W_n=W$ of $W$.
Because sub objects and quotients of crystalline $B$-pairs are again crystalline, $W_i$ and $W_i/W_{i-1}$ are
all potentially crystalline. Because  $W_i/W_{i-1}$ are rank one, $W_i/W_{i-1}|_{G_m}$ are crystalline for any $i$ for sufficiently large $m$. We claim that $W|_{G_m}$ is also crystalline.
We prove this claim by the induction on rank $n$ of $W$. When $n=1$, this is trivial.
We assume that the claim is proved for the rank $n-1$ case, hence $W_{n-1}|_{G_m}$ is crystalline.
If we put $W/W_{n-1}\isom W(\delta_n)$, we have $[W]\in \mathrm{H}^1(G_K, W_{n-1}(\delta_n^{-1}))$. By the assumption,
there exists a finite Galois extension $L$ of $K_m$ such that $[W]$ is contained in
$\mathrm{Ker}(\mathrm{H}^1(G_K, W_{n-1}(\delta_n^{-1}))\rightarrow \mathrm{H}^1(G_L, B_{\mathrm{cris}}\otimes_{B_e}
(W_{n-1}(\delta_n^{-1}))_e))$.  Hence, it suffices to prove that the natural map $\mathrm{H}^1(G_{K_m}, B_{\mathrm{cris}}\otimes_{B_e}(W_{n-1}(\delta_n^{-1}))_e)\rightarrow \mathrm{H}^1(G_L, B_{\mathrm{cris}}\otimes_{B_e}(W_{n-1}(\delta_n^{-1}))_e)$ is injective. By inflation
 restriction sequence, the kernel of this map is $\mathrm{H}^1(\mathrm{Gal}(L/K_m), D^L_{\mathrm{cris}}(W_{n-1}(\delta_n^{-1})))=0$.
 Hence $W|_{G_m}$ is crystalline, i.e. $W$ is crystabelline.

\end{proof}

From here, we consider a crystabelline $E$-$B$-pair $W$ such that
$D^{K_m}_{\mathrm{cris}}(W)\isom K_0\otimes_{\mathbb{Q}_p}Ee_1\oplus \cdots \oplus
K_0\otimes_{\mathbb{Q}_p}Ee_n$ such that
$K_0\otimes_{\mathbb{Q}_p}Ee_i$ is preserved by ($\varphi,G_m$) and $\varphi^f(e_i)=\alpha_ie_i$ for
some $\alpha_i\in E^{\times}$ such that $\alpha_i\not= \alpha_j$ for any $i\not= j$.
Let $\mathfrak{S}_n$ be the $n$-th permutation group.
For any $\tau\in \mathfrak{S}_n$, we define a filtration by $E$-filtered ($\varphi,G_m$)-modules on $D^{K_m}_{\mathrm{cris}}(W)$ as follows,

$$\mathcal{F}_{\tau}:0\subseteq F_{\tau,1}\subseteq \cdots\subseteq F_{\tau, n-1}
\subseteq F_{\tau,n}=D^{K_m}_{\mathrm{cris}}(W)$$  such that
$$F_{\tau,i}:=K_0\otimes_{\mathbb{Q}_p}Ee_{\tau(1)}\oplus \cdots\oplus K_0\otimes_{\mathbb{Q}_p}Ee_{\tau(i)}$$ for any $1\leqq i \leqq n$
, where the filtration on $F_{\tau,i}$ is induced from that on $D^{K_m}_{\mathrm{cris}}(W)$.
We put $\mathrm{gr}_{\tau,i}D^{K_m}_{\mathrm{cris}}(W):=F_{\tau,i}/F_{\tau,i-1}$ for any
$1\leqq i\leqq n$. By Theorem $\ref{e}$, there exists a
filtration $$\mathcal{T}_{\tau}:0\subseteq W_{\tau,1}\subseteq\cdots \subseteq W_{\tau,n-1}\subseteq W_{\tau,n}=W$$
such that $W_{\tau,i}|_{G_m}$ is crystalline and $$D^{K_m}_{\mathrm{cris}}(W_{\tau,i})=F_{\tau,i}.$$
$W_{\tau,i}/W_{\tau,i-1}$ is a rank one crystabelline $E$-$B$-pair such that
$D^{K_m}_{\mathrm{cris}}(W_{\tau,i}/W_{\tau,i-1})\allowbreak\isom \mathrm{gr}_{\tau,i}D^{K_m}_{\mathrm{cris}}(W)$. Hence, by Lemma 4.1 of \cite{Na09} and by its proof,
there exists $\{k_{(\tau,i),\sigma}\}_{\sigma\in\mathcal{P}}\in\prod_{\sigma\in \mathcal{P}}\mathbb{Z}$ and
homomorphisms $\tilde{\delta}_{i}:K^{\times}\rightarrow E^{\times}$  satisfying $\tilde{\delta}_{i}|_{1+\pi_K^m\mathcal{O}_K}=1$ and $\tilde{\delta}_{i}(\pi_K)=1$,
such that
$W_{\tau,i}/W_{\tau,i-1}\isom W(\delta_{\alpha_{\tau(i)}}\tilde{\delta}_{\tau(i)}\prod_{\sigma\in\mathcal{P}}\sigma^{k_{(\tau,i),\sigma}})$
for any $1\leqq i\leqq n$, where $\delta_{\alpha_i}:K^{\times}\rightarrow E^{\times}$ is the homomorphism such that
$\delta_{\alpha_i}|_{\mathcal{O}_K^{\times}}=1$ and $\delta_{\alpha_i}(\pi_K)=\alpha_i$. For any $\sigma\in\mathcal{P}$, the set $\{k_{(\tau,1),\sigma},k_{(\tau,2),\sigma},\cdots,k_{(\tau,n),\sigma}\}$ is independent of $\tau\in \mathfrak{S}_n$  because these numbers are
the $\sigma$-part of the Hodge-Tate weights of $W$. We denote this set (with multiplicities) by $\{k_{1,\sigma},k_{2,\sigma},\cdots, k_{n,\sigma}\}$
such that $k_{1,\sigma}\geqq k_{2,\sigma}\geqq\cdots\geqq k_{n,\sigma}$ for any $\sigma\in\mathcal{P}$. Under these notations, we define the notion of benign $E$-$B$-pair as follows.
\begin{defn}\label{26}
Let $W$ be a rank $n$ crystabelline $E$-$B$-pair as above.
We say that $W$ is a benign

$E$-$B$-pair if the following conditions hold:
\begin{itemize}
\item[(1)] For any $i\not= j$, we have $\alpha_i/\alpha_j\not=1, p^f, p^{-f}$.
\item[(2)] For any $\sigma\in\mathcal{P}$, we have $k_{1,\sigma}>k_{2,\sigma}>\cdots >k_{n-1,\sigma}>k_{n,\sigma}$.
\item[(3)] For any

$\tau\in \mathfrak{S}_n$ and for any $\sigma\in \mathcal{P}$, we have
$k_{(\tau,i),\sigma}=k_{i,\sigma}$ for any $1\leqq i\leqq n$.
\end{itemize}
We say that the triangulation $\mathcal{T}_{\tau}$ is non-critical if $\mathcal{T}_{\tau}$ satisfies the condition (3).
\end{defn}

\begin{rem}
By definition, if $W$ is a benign, then we have $W_{\tau,i}/W_{\tau,i-1}
\isom W(\delta_{\alpha_{\tau(i)}}\tilde{\delta}_{\tau(i)}\prod_{\sigma\in\mathcal{P}}\sigma^{k_{i,\sigma}})$
for any $\tau\in\mathfrak{S}_n$ and $1\leqq i\leqq n$.
\end{rem}

\begin{lemma}\label{21}
Let $W$ be a benign $E$-$B$-pair.
If $W_1$ is a saturated sub $E$-$B$-pair of $W$, then
$W_1$ and $W/W_1$ are also benign $E$-$B$-pairs.
\end{lemma}
\begin{proof}
This follows from the definition and the fact that
sub objects and quotients of crystabelline $E$-$B$-pairs
are crystabelline $E$-$B$-pairs.
\end{proof}

\subsubsection{Deformations of benign $B$-pairs}
\begin{lemma}\label{21.5}
Let $W$ be a potentially crystalline $E$-$B$-pair satisfying the condition
$(1)$ of Definition $\mathrm{\ref{26}}$, then
we have $\mathrm{H}^2(G_K,\mathrm{ad}(W))=0$ and
$(W,\mathcal{T}_{\tau})$ satisfies the conditions in Proposition $\ref{19}$ except the condition
$\mathrm{End}_{G_K}(W)=E$ for any $\tau\in \mathfrak{S}_n$.
\end{lemma}
\begin{proof}
That $\mathrm{H}^2(G_K,\mathrm{ad}(W))=0$ follows from the condition (1) of Definition \ref{26} and from (2) of Proposition \ref{i}
because $\mathrm{ad}(W)$ is split trianguline whose graded components are of the forms $(W_{\tau,i}/W_{\tau,i-1})\otimes (W_{\tau,j}/W_{\tau,j-1})^{\vee}$ for any fixed
$\tau\in \mathfrak{S}_n$.
 Other statements follow from the condition (1) of Definition \ref{26}.
\end{proof}

\begin{lemma}\label{22}
Let $W$ be a benign $E$-$B$-pair of rank $n$,
then $\mathrm{End}_{G_K}(W)=E$.
\end{lemma}
\begin{proof}
We prove this by the induction on $n$, the rank of $W$.
If  $n=1$, $\mathrm{End}_{G_K}(W)=\mathrm{H}^0(G_K, B_E)=E$.
We assume that the lemma is proved for the rank $n-1$ case, let
$W$ be a rank $n$ benign $E$-$B$-pair.
We take an element  $\tau\in\mathfrak{S}_n$ and consider the filtration
$\mathcal{T}_{\tau}:0\subseteq W_{\tau,1}\subseteq\cdots \subseteq W_{\tau,n-1}\subseteq W_{\tau,n}=W$.
By Lemma \ref{21}, $W_{\tau,n-1}$ is benign of rank $n-1$, hence by the induction hypothesis,
we have $\mathrm{End}_{G_K}(W_{\tau,n-1})=E$.
Let $f:W\rightarrow W$ be a non-zero morphism of $E$-$B$-pairs. By (1) of Definition \ref{26}
and by Proposition $\ref{i}$, we have $\mathrm{Hom}_{G_K}(W_{\tau,n-1}, W/W_{\tau,n-1})=0$. Hence we have
 $f(W_{\tau,n-1})
\subseteq W_{\tau,n-1}$. Because $\mathrm{End}_{G_K}(W_{\tau,n-1})=E$, we have $f|_{W_{\tau,n-1}}=a\cdot\mathrm{id}_{W_{\tau,n-1}}$
for some $a\in E$. If $a=0$, then $f:W\rightarrow W$ factors through a non-zero morphism $f':W/W_{\tau,n-1}\rightarrow W$.
Because $\mathrm{Hom}_{G_K}(W/W_{\tau,n-1}, W_{\tau,n-1})=0$ by (1) of Definition \ref{26} and by Proposition $\ref{i}$,
the natural map $\mathrm{Hom}_{G_K}(W/W_{\tau,n-1},W)\hookrightarrow \mathrm{Hom}_{G_K}(W/W_{\tau,n-1}, W/W_{\tau,n-1})=E$
is injective, hence the composition of $f'$ with the natural projection $W\rightarrow W/W_{\tau,n-1}$ induces  an isomorphism
$W/W_{\tau,n-1}\isom W/W_{\tau,n-1}$. This implies that the short exact sequence
$0\rightarrow W_{\tau,n-1}\rightarrow W\rightarrow W/W_{\tau,n-1}\rightarrow 0$ splits.
If we take a section $s:W/W_{\tau,n-1}\hookrightarrow W$, then we can choose $\tau'\in\mathfrak{S}_n$
such that $W_{\tau',1}=s(W/W_{\tau,n-1})$, then
this $\tau'$ doesn't satisfy the condition (3) in the definition of benign $B$-pairs. It's contradiction.
Hence the above $a$ must not be zero. If $a\not=0$, consider the map $f-a\cdot\mathrm{id}_W\in \mathrm{End}_{G_K}(W)$,
then the same argument as above implies that $f=a\cdot\mathrm{id}_W$. Hence we obtain $\mathrm{End}_{G_K}(W)=E$.
\end{proof}
\begin{corollary}\label{23}
Let $W$ be a benign $E$-$B$-pair of rank $n$.
The functor $D_{W}$ is pro-representable by $R_W$ which is
formally smooth of dimension $n^2[K:\mathbb{Q}_p]+1$.
For any $\tau\in\mathfrak{S}_n$, the functor $D_{W,\mathcal{T}_{\tau}}$ is pro-representable
by a quotient $R_{W,\mathcal{T}_{\tau}}$ of $R_W$ which is formally smooth
of dimension $\frac{n(n+1)}{2}[K:\mathbb{Q}_p]+1$.
\end{corollary}
\begin{proof}
This follows from Proposition $\ref{19}$.
\end{proof}

Next, we want to consider the relation between $R_{W}$ and $R_{W,\mathcal{T}_{\tau}}$ for
all $\tau\in \mathfrak{S}_n$.
In particular, we want to compare the tangent space of $R_{W}$ and the sum of tangent spaces
of $R_{W,\mathcal{T}_{\tau}}$ for all $\tau\in \mathfrak{S}_n$.
For this, first we need to recall  the potentially crystalline  deformation functor.
\begin{defn}
Let $W$ be a potentially crystalline $E$-$B$-pair.
We define the potentially crystalline deformation functor $D_{W}^{\mathrm{cris}}$ which is a sub functor of $D_W$
defined by $$D_{W}^{\mathrm{cris}}(A):=\{[W_A]\in D_{W}(A)| W_A \text{ is potentially crystalline }\}$$
for any $A\in \mathcal{C}_E$.
\end{defn}
\begin{lemma}\label{24}
Let $W$ be a potentially crystalline $E$-$B$-pair.
If $\mathrm{End}_{G_K}(W)=E$, then $D_{W}^{\mathrm{cris}}$ is pro-representable by
a quotient $R_{W}^{\mathrm{cris}}$ of $R_W$ which is formally smooth of dimension
equal to $$\mathrm{dim}_E(D_{\mathrm{dR}}(\mathrm{ad}(W))/\mathrm{Fil}^0D_{\mathrm{dR}}(\mathrm{ad}(W))) +
\mathrm{dim}_E(\mathrm{H}^0(G_K, \mathrm{ad}(W))). $$
\end{lemma}
\begin{proof}
For pro-representability, by Proposition $\ref{12}$, it suffices to relatively representability of
$D_{W}^{\mathrm{cris}}\hookrightarrow D_W$ as in the proof of Proposition $\ref{15}$.
In this case, the conditions (1) and (2) are trivial and (3) follows from the fact
that the sub object of potentially crystalline $E$-$B$-pair is again potentially crystalline.
Formally smoothness follows from Proposition 3.1.2 and Lemma 3.2.1 of \cite{Ki08} by using the deformations
of filtered ($\varphi, G_K$)-modules. For the dimension, we take a finite Galois extension $L$ of $K$ such that
$W|_{G_L}$ is crystalline. By the same argument as in the proof of Lemma $\ref{20.5}$,
 any $W_A\in D^{\mathrm{cris}}_W(A)$ is crystalline when restricted to $G_L$. It's easy to check
that the map $D_{W}(E[\varepsilon])\isom \mathrm{H}^1(G_K, \mathrm{ad}(W))$
induces isomorphism $D_{W}^{\mathrm{cris}}(E[\varepsilon])\isom \mathrm{Ker}(\mathrm{H}^1(G_K,
\mathrm{ad}(W))\rightarrow \mathrm{H}^1(G_L, B_{\mathrm{cris}}\otimes_{B_e} \mathrm{ad}(W)_e))$. In the same way as in
the proof of Lemma $\ref{20.5}$, the natural map $\mathrm{H}^1(G_K, B_{\mathrm{cris}}\otimes_{B_e}\mathrm{ad}(W)_e)\rightarrow \mathrm{H}^1(G_L, B_{\mathrm{cris}}\otimes_{B_e}\mathrm{ad}(W)_e)$ is
injective. Hence we have an isomorphism $$D^{\mathrm{cris}}_W(E[\varepsilon])\isom \mathrm{Ker}(\mathrm{H}^1(G_K, \mathrm{ad}(W))
\rightarrow \mathrm{H}^1(G_K, B_{\mathrm{cris}}\otimes_{B_e}\mathrm{ad}(W)_e)).$$ We can calculate the dimension of this
group in the same way as in the proof of  Proposition 2.7 \cite{Na09}.
\end{proof}
\begin{corollary}\label{25}
Let $W$ be a benign $E$-$B$-pair of rank $n$,
then $R_{W}^{\mathrm{cris}}$ is formally smooth of dimension
$\frac{(n-1)n}{2}[K:\mathbb{Q}_p]+1$.
\end{corollary}
\begin{proof}
This follows from Lemma $\ref{22}$ and $$\mathrm{dim}_E(D_{\mathrm{dR}}(\mathrm{ad}(W))/\mathrm{Fil}^0D_{\mathrm{dR}}(\mathrm{ad}(W)))
=\frac{(n-1)n}{2}[K:\mathbb{Q}_p],$$ which follows from the condition (2) in Definition $\ref{26}$.
\end{proof}

\begin{defn}\label{34}
Let $W$ be a benign $E$-$B$-pair of rank $n$ such that $W|_{G_{K_m}}$ is crystalline.
For any $\tau\in\mathfrak{S}_n$, we define a rank one saturated
crystabelline $E$-$B$-pair $W'_{\tau}\subseteq W$ such that $D^{K_m}_{\mathrm{cris}}(W'_{\tau})=K_0\otimes_{\mathbb{Q}_p}E
e_{\tau(n)} \subseteq D^{K_m}_{\mathrm{cris}}(W)$ and define a sub functor
$D^{\mathrm{cris}}_{W,\tau}$ of $D_W$ by
 \[
\begin{array}{ll}
D^{\mathrm{cris}}_{W,\tau}(A):=\{[W_A]\in & D_W(A)|\text{ there exists a rank one crystabelline
saturated}\\
&\text{ sub }A\text{-}B\text{-pair } W'_A\subseteq W_A \text{ such that } W'_A\otimes_A E=W'_{\tau} \}.
\end{array}
\]
\end{defn}

\begin{lemma}\label{27}
Under the above condition. The functor $D^{\mathrm{cris}}_{W,\tau}$ is pro-representable by
a quotient $R^{\mathrm{cris}}_{W,\tau}$ of $R_W$ which is formally smooth and it's dimension
is equal to $n(n-1)[K:\mathbb{Q}_p] +1$.
\end{lemma}
\begin{proof}
The proof of relatively representability of $D^{\mathrm{cris}}_{W,\tau}\hookrightarrow D_W$
 and the proof of formally smoothness are easily followed from the combination of proofs
 of Proposition $\ref{15}$ and Proposition $\ref{17}$ and Lemma $\ref{24}$.
 Here, we only prove dimension formula.
 Let $\mathrm{ad}_{\tau}(W):=\{f\in\mathrm{ad}(W)| f(W'_{\tau})\subseteq
 W'_{\tau}\}$, then we have the following short exact sequence,
 \begin{equation*}
 0\rightarrow \mathrm{Hom}(W/W'_{\tau}, W)\rightarrow \mathrm{ad}_{\tau}(W)
 \rightarrow \mathrm{ad}(W'_{\tau})\rightarrow 0.
 \end{equation*}
 Taking long exact sequence and by Proposition $\ref{i}$, we have
 the following short exact sequence
 \begin{equation*}
 0\rightarrow \mathrm{H}^1(G_K, \mathrm{Hom}(W/W'_{\tau},W))\rightarrow
 \mathrm{H}^1(G_K, \mathrm{ad}_{\tau}(W))\rightarrow \mathrm{H}^1(G_K,
 \mathrm{ad}(W'_{\tau}))\rightarrow 0.
 \end{equation*}
 We define $\mathrm{H}^1_{f,\tau}(G_K, \mathrm{ad}_{\tau}(W))$
 by the inverse image of $\mathrm{H}^1_f(G_K,\mathrm{ad}(W'_{\tau}))$
 in $\mathrm{H}^1(G_K,\allowbreak \mathrm{ad}_{\tau}(W))$. Hence we obtain
 a short exact sequence
 \begin{equation*}
 0\rightarrow \mathrm{H}^1(G_K, \mathrm{Hom}(W/W'_{\tau},W))
 \rightarrow \mathrm{H}^1_{f,\tau}(G_K, \mathrm{ad}_{\tau}(W))
 \rightarrow \mathrm{H}^1_f(G_K, \mathrm{ad}(W'_{\tau}))\rightarrow 0.
 \end{equation*}
 In the same way as in Lemma $\ref{24}$, we can show that the natural map $D_{W}(E[\varepsilon])\rightarrow \mathrm{H}^1(G_K, \mathrm{ad}(W))$
 induces an isomorphism $$D^{\mathrm{cris}}_{W,\tau}(E[\varepsilon])\isom \mathrm{H}^1_{f,\tau}(G_K,
 \allowbreak \mathrm{ad}_{\tau}(W)).$$
 By Theorem $\ref{h}$ and Proposition $\ref{i}$, we can calculate that
 $$\mathrm{dim}_E\mathrm{H}^1(G_K, \mathrm{Hom}\allowbreak (W/W'_{\tau},W))=n(n-1)[K:\mathbb{Q}_p].$$
 Because $\mathrm{ad}(W'_{\tau})=B_E$ is trivial, hence we have $\mathrm{H}^1_{f}(G_K, \mathrm{ad}(W'_{\tau}))
 =1$ by Proposition 2.7 of \cite{Na09}. Hence we have
 $$\mathrm{dim}_E\mathrm{H}^1_{f,\tau}(G_K, \mathrm{ad}_{\tau}(W))\allowbreak =n(n-1)[K:\mathbb{Q}_p] +1.$$
 This proves that $R_{W,\tau}^{\mathrm{cris}}$ is of dimension $n(n-1)[K:\mathbb{Q}_p]+1$.

\end{proof}
\begin{lemma}\label{28}
Let $W$ be a benign $E$-$B$-pair of rank $n$.
Let $W_A$ be a deformation of $W$ over $A$ which is potentially crystalline,
then $[W_A]\in D_{W,\mathcal{T}_{\tau}}(A)$ and $[W_A]\in D^{\mathrm{cris}}_{W,\tau}(A)$ for
any $\tau\in \mathfrak{S}_n$.
\end{lemma}
\begin{proof}
Let $W_A$ be as above. If $W|_{G_{K_m}}$ is crystalline,
then $W_A|_{G_{K_m}}$ is crystalline by the proof of Lemma $\ref{24}$.
Hence it suffices to show that we can write by $D^{K_m}_{\mathrm{cris}}(W_A)\isom
K_0\otimes_{\mathbb{Q}_p}Ae_1\oplus K_0\otimes_{\mathbb{Q}_p}Ae_2\oplus\cdots
\oplus K_0\otimes_{\mathbb{Q}_p}Ae_n$ such that $K_0\otimes_{\mathbb{Q}_p}Ae_i$ is
preserved by ($\varphi,G_{m}$) and $\varphi^f(e_i)=\tilde{\alpha}_ie_i$ for a lift $\tilde{\alpha}_i\in A^{\times}$
of $\alpha_i\in E^{\times}$ for any $1\leqq i\leqq n$.  For proving this claim, first we note that
$D^{K_m}_{\mathrm{cris}}(W_A)$ is a free $K_0\otimes_{\mathbb{Q}_p}A$-module of rank $n$ and
$D^{K_m}_{\mathrm{cris}}(W_A)\otimes_A E\isom D^{K_m}_{\mathrm{cris}}(W)$ by
Proposition 1.3.4 and Proposition 1.3.5 \cite{Ki09} and then, for any $\sigma:K_0\hookrightarrow E$, we can write $D^{K_m}_{\mathrm{cris}}(W_A)_{\sigma}=
Ae_{1,\sigma}\oplus \cdots \oplus Ae_{n,\sigma}$ such that
$\varphi^f(e_{i,\sigma})\equiv \alpha_ie_{i,\sigma}$ (mod $\mathfrak{m}_A$) for any $1\leqq i\leqq n$.
By an easy linear algebra, we can take an $A$-basis $e'_{1,\sigma}, e'_{2,\sigma}, \cdots, e'_{n,\sigma}$ of
$D^{K_m}_{\mathrm{cris}}(W_A)_{\sigma}$ such that $\varphi^f(e'_{i,\sigma})=\tilde{\alpha}_ie'_{i,\sigma}$
for a lift $\tilde{\alpha}_i\in A^{\times}$ of $\alpha_i$ for any $i$. Because $\varphi$ and $G_{m}$ commute and $\alpha_i\not=\alpha_j$,
hence $Ae'_{i,\sigma}$ is preserved by $G_{m}$. If we take $e_i:=e'_{i,\sigma}+\varphi(e'_{i,\sigma})+\cdots
\varphi^{f-1}(e'_{i,\sigma})\in D^{K_m}_{\mathrm{cris}}(W_A)$, then we have $D^{K_m}_{\mathrm{cris}}(W_A)=
K_0\otimes_{\mathbb{Q}_p}Ae_1\oplus \cdots \oplus K_0\otimes_{\mathbb{Q}_p}Ae_n$ satisfying
the property of the claim.

\end{proof}

\begin{lemma}\label{29}
Let $W$ be a benign $E$-$B$-pair of rank $n$ and $\tau\in \mathfrak{S}_n$ such that
$W|_{G_{K_m}}$ is crystalline.
Let $[W_A]\in D_{W,\mathcal{T}_{\tau}}(A)$ be a trianguline deformation over $A$
with a lifting of  triangulation $0\subseteq W_{1,A}\subseteq W_{2,A}\subseteq \cdots W_{n,A}=W_A$.
If $W_{i,A}/W_{i-1,A}$ is Hodge-Tate for any $1\leqq i\leqq n$, then $W_A|_{G_{K_m}}$ is crystalline.
\end{lemma}
\begin{proof}
First, we prove that $(W_{i,A}/W_{i-1,A})|_{G_{K_m}}$ is crystalline with Hodge-Tate weight $\{k_{i,\sigma}\}_{\sigma\in \mathcal{P}}$.
Because $W_{i,A}/W_{i-1,A}$ is written as successive extensions of $W_{\tau,i}/W_{\tau,i-1}$, $W_{i,A}/W_{i-1,A}$
has Hodge-Tate weight $\{k_{i,\sigma}\}_{\sigma\in\mathcal{P}}$. Twisting $W_A$ by a crystalline character $\delta_{\alpha_{\tau(i)}}^{-1}
\prod_{\sigma\in\mathcal{P}}\sigma^{-k_{i,\sigma}}:K^{\times}\rightarrow A^{\times}$, we may assume that
$W_{i,A}/W_{i-1, A}$ is an \'etale Hodge-Tate $A$-$B$-pair of rank one with Hodge-Tate weight zero and is
a deformation of an \'etale potentially unramified $E$-$B$-pair $W(\tilde{\delta}_{\tau(i)})$. By Sen's theorem
(\cite{Se73} or Proposition 5.24 of \cite{Be02}),
$W_{i,A}/W_{i-1,A}$ is potentially unramified, hence there exists a unitary homomorphism $\delta:K^{\times}\rightarrow A^{\times}$
such that $\delta|_{\mathcal{O}_K^{\times}}$ has a finite image and $W_{i,A}/W_{i-1,A}\isom W(\delta)$ and
$\delta$ is a lift of $\tilde{\delta}_{\tau(i)}$. Because $(1+\mathfrak{m}_A)\cap A^{\times}_{\mathrm{torsion}}=\{1\}$,
we have $\delta|_{\mathcal{O}_K^{\times}}=\tilde{\delta}_{\tau(i)}|_{\mathcal{O}_K^{\times}}:\mathcal{O}_K^{\times}\rightarrow A^{\times}$, hence $W_{i,A}/W_{i-1,A}|_{G_{K_m}}$ is crystalline.
Next, we prove that $W_A|_{G_{K_m}}$ is crystalline by the induction on the rank of $W$.
When $n=1$, we just have proved this.
Assume that the lemma is proved for the rank $n-1$ case, then
$W_{n-1, A}|_{G_{K_m}}$ is crystalline. If we put $W_A/W_{n-1,A}\isom W(\delta_{A,n})$, then we have
$[W_A]\in \mathrm{H}^1(G_K, W_{n-1,A}(\delta_{A,n}^{-1}))$. By considering Hodge-Tate weight of
$W_{A}$ and the condition (3) of Definition $\ref{26}$, we obtain $\mathrm{Fil}^0D_{\mathrm{dR}}(W_{n-1,A}(\delta_{A,n}^{-1}))=0$.
Comparing dimensions and by Proposition 2.7 of \cite{Na09}, we have $\mathrm{H}^1_f(G_K,W_{A,n-1}(\delta_{A,n}^{-1}))=\mathrm{H}^1(G_K,
W_{A,n-1}(\delta_{A,n}^{-1}))$. Hence  $W_A|_{G_{K_m}}$ is crystalline.

\end{proof}

\begin{lemma}\label{30}
Let $W$ be a benign $E$-$B$-pair of rank $n$ and $W_1$ be a rank one crystabelline sub $E$-$B$-pair of $W$,
then the saturation $W_{1}^{sat}:=(W^{sat}_{1,e}, W^{+, sat}_{1,\mathrm{dR}})$ of $W_1$ in $W$ is
crystabelline and the natural map $\mathrm{Hom}_{G_K}( W_{1}^{sat}, W)
\rightarrow \mathrm{Hom}_{G_K}(W_1,W)$ induced by the natural
inclusion $W_1\hookrightarrow W_{1}^{sat}$ is isomorphism
between one dimensional $E$-vector spaces.
\end{lemma}
\begin{proof}
Because $W_{1,e}=W^{sat}_{1,e}$ by Lemma 1.14 of \cite{Na09}, so $W_{1}^{sat}$ is crystabelline.
By the definition of benign $E$-$B$-pairs, the Hodge-Tate weight of $W_{1}^{sat}$
is $\{k_{1,\sigma}\}_{\sigma\in\mathcal{P}}$. Consider the following short exact sequence
of complexes of $G_K$-modules defined in p.890 of \cite{Na09}
\begin{equation*}
0\rightarrow \mathrm{C}^{\bullet}(W\otimes (W_{1}^{sat})^{\vee})\rightarrow \mathrm{C}^{\bullet}(W
\otimes W_{1}^{\vee})\rightarrow
((W\otimes W_{1}^{\vee})^+_{\mathrm{dR}}/
(W\otimes(W_{1}^{sat})^{\vee})^+_{\mathrm{dR}}) [0]\rightarrow 0,
\end{equation*}
where, for an $E$-$B$-pair $W$, we denote by $C^{\bullet}(W)$ the mapping cone of the natural map
$W_e\oplus W^+_{\mathrm{dR}}\rightarrow W_{\mathrm{dR}}:(x,y)\mapsto x-y$.
From this, we obtain an exact sequence
\[
\begin{array}{ll}
0\rightarrow \mathrm{H}^0(G_K, W\otimes (W_1^{sat})^{\vee})&\rightarrow
\mathrm{H}^0(G_K, W\otimes W_1^{\vee}) \\
 &\rightarrow \mathrm{H}^0(G_K, (W\otimes W_1^{\vee})^+_{\mathrm{dR}}/
(W\otimes(W_1^{sat})^{\vee})^+_{\mathrm{dR}})\rightarrow \cdots
\end{array}
\]
By the condition (3) in Definition $\ref{26}$, we have $\mathrm{dim}_E\mathrm{H}^0(G_K, (W\otimes (W_1^{sat})^{\vee})^+_{\mathrm{dR}})
=\mathrm{dim}_E\mathrm{H}^0(G_K, (W\otimes W_1^{\vee})^+_{\mathrm{dR}})=n[K:\mathbb{Q}_p]$, Hence, by a usual argument
of $B^+_{\mathrm{dR}}$-representations, then we have $\mathrm{H}^0(G_K, (W\otimes W_1^{\vee})^+_{\mathrm{dR}}/(W\otimes (W_1^{sat})^{\vee})^+_{\mathrm{dR}})=0$. Hence the map $\mathrm{H}^0(G_K, W\otimes (W_1^{sat})^{\vee}))\isom \mathrm{H}^0(G_K, W\otimes (W_1)^{\vee})
$ is isomorphism. Finally, for the dimension,
we have $\mathrm{dim}_E\mathrm{H}^0(G_K, W\otimes (W_1^{sat})^{\vee})=1$ by the condition (1) in Definition $\ref{26}$ and by Proposition $\ref{i}$ of \cite{Na09}.

\end{proof}
\begin{lemma}\label{31}
Let $W$ be a benign $E$-$B$-pair and let $W_A$ be a deformation
of $W$ over $A$. If there exists a rank one crystabelline sub $A$-$B$-pair
$W_{1,A}\subseteq W_A$  such
that the reduction map $W_1:=W_{1,A}\otimes_A E\hookrightarrow W_A\otimes_{A}E$ remains injective,
then the saturation $W_{1,A}^{sat}$ of $W_{1,A}$ in $W_A$ as an $E$-$B$-pair is
a crystabelline $A$-$B$-pair and $W_A/W^{sat}_{1,A}$ is an $A$-$B$-pair and
$W_{1,A}^{sat}\otimes_A E\isom W_1^{sat} (\subseteq W)$.
\end{lemma}
\begin{proof}
First, by Proposition 2.14 of \cite{Na09}, there exists $\{l_{\sigma}\}_{\sigma\in\mathcal{P}}
\in \prod_{\sigma\in\mathcal{P}}\mathbb{Z}_{\leqq 0}$ such that  $W_1^{sat}\isom W_1\otimes W(\prod_{\sigma\in\mathcal{P}}
\sigma^{l_{\sigma}})$. We claim that the inclusion
$$\mathrm{Hom}_{G_K}(W_{1,A}\otimes W_A(\prod_{\sigma\in\mathcal{P}}\sigma^{l_{\sigma}}), W_A)
\rightarrow \mathrm{Hom}_{G_K}(W_{1,A}, W_A)$$ induced by the natural inclusion $W_{1,A}\hookrightarrow W_{1,A}
\otimes W_A(\prod_{\sigma\in\mathcal{P}}\sigma^{l_{\sigma}})$ is isomorphism and
that these groups are rank one free $A$-modules. By the same argument as in Lemma \ref{30},
the cokernel of $\mathrm{Hom}_{G_K}(W_{1,A}\otimes W_A(\prod_{\sigma\in\mathcal{P}}\sigma^{l_{\sigma}}), W_A)
\rightarrow \mathrm{Hom}_{G_K}(W_{1,A}, W_A)$ is contained in $\mathrm{H}^0(G_K, (W_A\otimes W_{1,A}^{\vee})^+_{\mathrm{dR}}/
(W_A\otimes W_{1,A}^{\vee}\otimes W_A(\prod_{\sigma\in \mathcal{P}}\sigma^{-l_{\sigma}}))^+_{\mathrm{dR}})$. This is zero
by the proof of Lemma $\ref{30}$. Hence the natural inclusion
$$\mathrm{Hom}_{G_K}(W_{1,A}\otimes W_A(\prod_{\sigma\in\mathcal{P}}\sigma^{l_{\sigma}}), W_A)
\isom \mathrm{Hom}_{G_K}(W_{1,A}, W_A)$$ is isomorphism. Next, we prove
that $\mathrm{Hom}_{G_K}(W_{1,A},\allowbreak W_A)$ is a free $A$-module of rank one by the induction on the length of $A$.
When $A=E$, this claim is proved in Lemma $\ref{30}$.
Assume that $A$ is of length $n$ and assume that the claim is
proved for $W_{A'}:=W\otimes_A A'$ for a small extension $A\rightarrow A'$. We denote by $I$ the kernel of $A\rightarrow A'$, $W_{1,A'}:=W_{1,A}\otimes_A A'$. From the exact sequence
\begin{equation*}
0\rightarrow I\otimes_E \mathrm{Hom}_{G_K}(W_1, W)
\rightarrow \mathrm{Hom}_{G_K}(W_{1,A}, W_A)\rightarrow \mathrm{Hom}_{G_K}(W_{1,A'}, W_{A'})
\end{equation*}
 and the induction hypothesis, we have
$\mathrm{length}\mathrm{Hom}_{G_K}(W_{1,A}, W_A)\leqq
\mathrm{length}A.$ On the other hand, the fact that the given inclusion $\iota:W_{1,A}\hookrightarrow W_A$ remains
injective after tensoring $E$ and the fact that $\mathrm{dim}_E\mathrm{Hom}_{G_K}(W_1, W)=1$ and the induction hypothesis
imply that the map $A\rightarrow \mathrm{Hom}_{G_K}(W_{1,A}, W_A): a\mapsto a\cdot\iota $ is injection. Hence we have
$\mathrm{length}(A)= \mathrm{length} \mathrm{Hom}_{G_K}(W_{1,A}, W_A)$. These facts prove the claim for $A$.
From this claim, the given inclusion $\iota:W_{1,A}\hookrightarrow W_A$  factors through a map
$$\tilde{\iota}:W_{1,A}':=W_{1, A}\otimes W_A(\prod_{\sigma}\sigma^{l_{\sigma}})
\rightarrow W_A$$ and this map is injection because the injection of morphisms of $B$-pair determined only by the injection
of $W_e$-part of $B$-pairs and $W_{1,A,e}=(W_{1,A}\otimes W_A(\prod_{\sigma\in\mathcal{P}}\sigma^{l_{\sigma}}))_e$.
Under this situation, we claim that this $\tilde{\iota}$ gives an isomorphism $W_{1,A}^{sat}\isom W_{1,A}'$ and claim that this satisfies all the properties in this lemma. By the induction on the length of $A$,
we may assume that this claim is proved for $A'$.  First, we prove that $W_{A}/W_{1,A}'$ is an $E$-$B$-pair. For proving this, by Lemma 2.1.4 of \cite{Be08},
it suffices to show that $W_{A,\mathrm{dR}}^{+}/W_{1,A,\mathrm{dR}}^{'+}$ is a free $B_{\mathrm{dR}}^+$-module. By the snake lemma, we have the following short exact sequence
\begin{equation*}
0\rightarrow I\otimes_E W^+_{\mathrm{dR}}/W^{sat +}_{1,\mathrm{dR}}\rightarrow
W^+_{A,\mathrm{dR}}/W^{' +}_{1, A,\mathrm{dR}}\rightarrow W^+_{A',\mathrm{dR}}/(W_{1,A,\mathrm{dR}}^{' +}\otimes_A A')\rightarrow 0.
\end{equation*}
From this and the induction hypothesis, $W^+_{A,\mathrm{dR}}/W^{' +}_{1,A,\mathrm{dR}}$ is a free $B^+_{\mathrm{dR}}$-module.
Finally, we prove the $A$-flatness of $W_{A}/W_{1,A}^{'}$. This follows from the fact that  the map
$\tilde{\iota}\otimes \mathrm{id}_E:W_{1,A}^{'}\otimes_A E\hookrightarrow W_{A}\otimes_A E$ is saturated (we can see this
from the proof of the above first claim). Hence $W_A/W_{1,A}^{'}$ is an $A$-$B$-pair. We finish the proof of this lemma.

\end{proof}

\begin{lemma}\label{32}
Let $W$ be a benign $E$-$B$-pair of rank $n$.
For any $\tau\in\mathfrak{S}_n$, we have $$D_{W,\mathcal{T}_{\tau}}(E[\epsilon])
+D^{\mathrm{cris}}_{W,\tau}(E[\varepsilon])=D_{W}(E[\varepsilon]).$$
\end{lemma}
\begin{proof}
By Corollary $\ref{23}$ and  Lemma $\ref{24}$ and Lemma $\ref{27}$, we have
$$\mathrm{dim}_ED_{W}(E[\varepsilon])\allowbreak +\mathrm{dim}_ED_W^{\mathrm{cris}}(E[\varepsilon])
=\mathrm{dim}_E D_{W,\mathcal{T}_{\tau}}(E[\varepsilon])+\mathrm{dim}_ED^{\mathrm{cris}}_{W,\tau}(E[\varepsilon]).$$
Hence it suffices to show that
$$D_{W,\mathcal{T}_{\tau}}(A)\cap D^{\mathrm{cris}}_{W,\tau}(A)=D_{W}^{\mathrm{cris}}(A)$$ for
any $A\in \mathcal{C}_E$. First, the fact that $D^{\mathrm{cris}}_W(A)\subseteq D_{W,\mathcal{T}_{\tau}}(A)\cap D_{W,\tau}(A)$
follows from Lemma $\ref{28}$.

We prove that $D_{W,\mathcal{T}_{\tau}}(A)\cap D^{\mathrm{cris}}_{W,\tau}(A)\subseteq D_{W}^{\mathrm{cris}}(A)$ by the induction on
the rank $n$ of $W$. When  $n=1$, this is trivial.
Let $W$ be rank $n$ and assume that the lemma is proved for the rank  $n-1$ case.
Let $[W_A]\in D_{W,\mathcal{T}_{\tau}}(A)\cap D^{\mathrm{cris}}_{W,\tau}(A)$, then, by the definition of $D_{W,\mathcal{T}_{\tau}}$ and $D^{\mathrm{cris}}_{W,\tau}$, there exists an $A$-triangulation
$0\subseteq W_{1,A}\subseteq W_{2,A}\subseteq \cdots \subseteq W_{n-1,A}\subseteq W_{n,A}=W_A$
such that $W_{i,A}\otimes_A E\isom W_{\tau,i}$ for any $i$ and there exists a saturated crystabelline rank one $A$-$B$-pair
$W'_{1,A}\subseteq W_{A}$ such that $W'_{1,A}\otimes_A E\isom W'_{\tau}$.
First, we claim that the composition of $W'_{1,A}\hookrightarrow W_A$ and $W_A\rightarrow W_A/W_{1,A}$
is  injection. Because $\mathrm{Ker}(W'_{1,A}\rightarrow W_A/W_{1,A})$ is a sub $E$-$B$-pair of
$W_{1, A}$ and because we have
$\mathrm{Hom}_{G_K}(\mathrm{Ker}(W'_{1, A}\rightarrow W_A/W_{1,A}), W_{1,A})=0$ by the condition (1) of  Definition $\ref{26}$ and by Proposition 2.14 of \cite{Na09}, hence $W'_{1,A}\rightarrow
W_A/W_{1,A}$ is injection. Hence, by Lemma $\ref{31}$, the saturation $(W'_{1,A})^{sat}$ of $W'_{1,A}$ in $W_A/W_{1,A}$
is a rank one crystabelline $A$-$B$-pair satisfying the similar conditions as those of $W'_{1,A}\hookrightarrow W_A$.
Hence, by the induction hypothesis, $W_A/W_{1,A}$ is crystabelline and, by the condition (3) of Definition $\ref{26}$, Hodge-Tate weight of
$W_A/W_{1,A}$ is $\{k_{2,\sigma},k_{3,\sigma},\cdots,k_{n,\sigma}\}_{\sigma\in\mathcal{P}}$ (with multiplicity $[A:E]$).
Moreover, by (3) of Definition $\ref{26}$, $W'_{1,A}$ has Hodge-Tate weight $\{k_{1,\sigma}\}_{\sigma\in\mathcal{P}}$
(with multiplicity). Because $k_{1,\sigma}\not= k_{i,\sigma}$ for any $i\not=1$ and there is no extension between different
Hodge-Tate weight objects by a theorem of Tate, these imply that $W_A$ is a Hodge-Tate $E$-$B$-pair. Hence,
by Lemma $\ref{29}$, $W_A$ is crystabelline. Hence we have that $[W_A]\in D_{W}^{\mathrm{cris}}(A)$.

\end{proof}
The following is the main theorem of $\S$ 2, which is a crucial theorem for the applications to Zariski density.
This theorem was first discovered by Chenevier (Theorem 3.19 of \cite{Ch09b}) for the $\mathbb{Q}_p$ case.
\begin{defn}
For $R=R_W$ or $R_{W,\mathcal{T}_{\tau}}$, we denote by
$$t(R):=\mathrm{Hom}_E(\mathfrak{m}_R/\mathfrak{m}^2_R,E)$$ the tangent
space of $R$. We have a natural inclusion $t(R_{W,\mathcal{T}_{\tau}})\hookrightarrow
t(R_W)$ for any $\tau\in \mathfrak{S}_n$.
\end{defn}

\begin{thm}\label{33}
Let $W$ be a benign $E$-$B$-pair of rank $n$.
We have an equality $$\sum_{\tau\in\mathfrak{S}_n}t(R_{W,\mathcal{T}_{\tau}})=t(R_W).$$
\end{thm}
\begin{proof}
We prove this by the induction on $n$.
When $n=1$, then the theorem is trivial.
Assume that the theorem is true for the rank $n-1$ case.
Let $W$ be a rank $n$ benign $E$-$B$-pair.
We take an element $\tau\in\mathfrak{S}_n$. We define a sub functor $D_{W,\tau}$ of $D_W$ by
 \[
\begin{array}{ll}
&D_{W,\tau}(A):=\{[W_A]\in D_{W}(A)| \text{ there exists a rank one sub } A\text{-}B\text{-pair }W_{1,A}\subseteq W_A\\
&\text{ such
that the quotient }W_A/W_{1,A}\text{ is an }A\text{-}B\text{-pair and }W_{1,A}\otimes_A E\isom W'_{\tau}\},
\end{array}
\]
 where $W'_{\tau}$ is defined in Definition $\ref{34}$.
 $D^{\mathrm{cris}}_{W,\tau}$ is a sub functor of $D_{W,\tau}$
and we can show in the same way that $$D_{W,\tau}(E[\varepsilon])\isom \mathrm{H}^1(G_K, \mathrm{ad}_{\tau}(W)),$$
 where $$\mathrm{ad}_{\tau}(W)
:=\{f \in \mathrm{ad}(W)| f(W'_{\tau})\subseteq W'_{\tau} \}.$$ Hence, by Lemma $\ref{32}$, we have
$$\mathrm{H}^1(G_K, \mathrm{ad}_{\tau}(W))+
\mathrm{H}^1(G_K, \mathrm{ad}_{\mathcal{T}_{\tau}}(W))=\mathrm{H}^1(G_K, \mathrm{ad}(W)).$$
Because, for any $\tau'\in\mathfrak{S}_{\tau,n}:=\{\tau'\in\mathfrak{S}_n| \tau'(1)=\tau(n)\}$,  we have
a natural inclusion $D_{W,\mathcal{T}_{\tau'}}\subseteq D_{W,\tau}$, hence we have natural maps $\mathrm{H}^1(G_K, \mathrm{ad}_{\mathcal{T}_{\tau'}}(W))
\rightarrow \mathrm{H}^1(G_K, \mathrm{ad}_{\tau}(W))$. Hence,  it suffices to prove that
the map $$\oplus_{\tau'\in\mathfrak{S}_{\tau,n}}\mathrm{H}^1(G_K, \mathrm{ad}_{\mathcal{T}_{\tau'}}(W))\rightarrow
\mathrm{H}^1(G_K, \mathrm{ad}_{\tau}(W))$$ is surjection. We prove this surjection as follows.
First, by the definition, we have the following short exact sequences of $E$-$B$-pairs for any $\tau'\in\mathfrak{S}_{\tau,n}$,
\begin{equation}
0\rightarrow \mathrm{Hom}(W/W'_{\tau}, W)\rightarrow \mathrm{ad}_{\tau}(W)\rightarrow \mathrm{ad}(W'_{\tau})
\rightarrow 0,
\end{equation}
\begin{equation}
0\rightarrow \{f\in\mathrm{ad}_{\mathcal{T}_{\tau'}}(W)| f(W'_{\tau})=0\}\rightarrow \mathrm{ad}_{\mathcal{T}_{\tau'}}(W)\rightarrow
\mathrm{ad}(W'_{\tau})\rightarrow 0.
\end{equation}
Moreover, we have
\begin{equation}
0\rightarrow \mathrm{Hom}(W/W'_{\tau}, W'_{\tau})\rightarrow \mathrm{Hom}(W/W'_{\tau}, W)\rightarrow \mathrm{ad}(W/W'_{\tau})\rightarrow 0,
\end{equation}
\begin{equation}
0\rightarrow \mathrm{Hom}(W/W'_{\tau}, W'_{\tau})\rightarrow \{f\in\mathrm{ad}_{\mathcal{T}_{\tau'}}(W)|f(W'_{\tau})=0\}
\rightarrow \mathrm{ad}_{\mathcal{T}_{\bar{\tau}'}}(W/W'_{\tau})\rightarrow 0.
\end{equation}
Here, for any $\tau'\in\mathfrak{S}_{\tau,n}$, we denote by $\mathcal{T}_{\bar{\tau}'}:0\subseteq W_{\tau',2}/W'_{\tau}\subseteq W_{\tau',3}/W'_{\tau}
\subseteq \cdots \subseteq W_{\tau',n-1}/W'_{\tau}\subseteq W/W'_{\tau}$ the induced triangulation from $\mathcal{T}_{\tau'}$
on $W/W'_{\tau}$. We have $\mathrm{H}^2(G_K, \mathrm{Hom}(W/W'_{\tau}, W'_{\tau}))=0$ by the condition $(1)$ of Definition $\ref{26}$ and
Proposition $\ref{i}$. We have  $\mathrm{H}^2(G_K, \mathrm{ad}_{\mathcal{T}_{\tau}}(W))=0$ from the proof of Proposition
$\ref{19}$. Hence, from the short exact sequence (4) above, we have $\mathrm{H}^2(G_K, \{f\in\mathrm{ad}_{\tau'}(W)|f(W'_{\tau})=0\})=0$.
From this and from (1) and (2) above, for proving the surjection of $\oplus_{\tau'\in \mathfrak{S}_{\tau,n}}\mathrm{H}^1(G_K,\mathrm{ad}_{\mathcal{T}_{\tau'}}(W))\allowbreak
\rightarrow \mathrm{H}^1(G_K, \mathrm{ad}_{\tau}(W))$, it suffices to prove
the surjection of $\oplus_{\tau'\in \mathfrak{S}_{\tau,n}}\mathrm{H}^1(G_K, \{f\in\mathrm{ad}_{\tau'}(W)|f(W'_{\tau})=0\})
\rightarrow \mathrm{H}^1(G_K, \mathrm{Hom}(W/W'_{\tau}, W))$. By (3) and (4) above and by $\mathrm{H}^2(G_K, \mathrm{Hom}(W/W'_{\tau}, W'_{\tau}))=0$,
this surjection follows from the surjection of $\oplus_{\tau'\in\mathfrak{S}_{\tau,n}}\mathrm{H}^1(G_K, \mathrm{ad}_{\mathcal{T}_{\bar{\tau}'}}(W/W'_{\tau}))\rightarrow \mathrm{H}^1(G_K, \mathrm{ad}(W/W'_{\tau}))$, which  is the induction hypothesis. Hence we have finished the proof of this theorem

\end{proof}

\section{Construction of $p$-adic families of two dimensional trianguline representations.}
In this section, we generalize Kisin's construction of $p$-adic families of two dimensional
trianguline representations for any $p$-adic field. In the next section, generalizing Kisin's method in \cite{Ki10},
we use these families to prove Zariski density of two dimensional crystalline representations for any
$p$-adic field.

\subsection{Almost $\mathbb{C}_p$-representations}
In this subsection, we recall some rings of Lubin-Tate's $p$-adic periods defined by Colmez ( \cite{Co02}) and
the definition of almost $\mathbb{C}_p$-representations defined by Fontaine (\cite{Fo03}).
Using these, we prove some propositions concerning
to Banach $G_K$-modules which we need for  the construction of
$p$-adic families of trianguline representations.

Let $\pi_K$ be a fixed uniformizer of $K$. $P(X)\in \mathcal{O}_K[X]$ be a monic polynomial of degree $q:=p^f$ such
that $P(X)\equiv \pi_K X$ (mod deg $2$) and $P(X)\equiv X^q$ (mod $\pi_K$). From $P(X)$, we obtain the Lubin-Tate formal
group law $\mathcal{F}_{\pi_K}$ of $\mathcal{O}_K$ on which the multiplication by $\pi_K$ is given by $[\pi_K]=P(X)$.
Let $\chi_{\mathrm{LT}}:G_K\rightarrow \mathcal{O}_K^{\times}$ be the Lubin-Tate character associated to $\pi_K$.
Let $A_{\mathrm{inf},K}:=\mathbb{A}^+\otimes_{\mathcal{O}_{K_0}}\mathcal{O}_K$. This ring is equipped with the weak topology on which
$G_K$ acts continuously. $A_{\mathrm{inf},K}$ also has a $\mathcal{O}_K$-linear continuous $\varphi_K:=\varphi^f$-action.
Any element of $A_{\mathrm{inf},K}$ can be written uniquely of the form $\sum_{k=0}^{\infty}[x_k]\pi_K^k$ ($x_k\in \tilde{\mathbb{E}}^+$).
We put $B_{\mathrm{inf},K}:=A_{\mathrm{inf},K}[p^{-1}]$. By Lemma 8.3 of  \cite{Co02}, for any element $x\in \tilde{\mathbb{E}}^+$,
there exists unique element $\{x\}\in A_{\mathrm{inf},K}$ such that $\{x\}$ is a lift of $x$ and $\varphi_K(\{x\})=[\pi_K](\{x\})(=P(\{x\}))$.
 We fix $\{\omega_n\}_{n\geqq 0}$ such that $\omega_1\in \mathfrak{m}_{\overline{K}}$ is a primitive $[\pi_K]$-torsion point of $\mathcal{F}_{\pi_K}$ and
 $[\pi_K](\omega_{n+1})=\omega_n$ for any $n\geqq 0$, then $(\bar{\omega}_n)_{n\geqq 0}$ defines an element in $\tilde{\mathbb{E}}^+\isom \varprojlim_{n}\mathcal{O}_{\overline{K}}/\pi_K\mathcal{O}_{\overline{K}}$ where the projective limit is given by $q$-th power Frobenius map. We define $\omega_K:=\{(\bar{\omega}_n)_{n \geqq 0}\}\in A_{\mathrm{inf},K}$. By the definition of $\{-\}$ and by the uniqueness of $\{-\}$,
 the actions of $G_K$ and $\varphi_K$ on $\omega_K$ are given by $g(\omega_K)=[\chi_{\mathrm{LT}}(g)](\omega_K)$ for any $g\in G_K$ (which converges for the weak topology)
 and by $\varphi_K(\omega_K)=[\pi_K](\omega_K)$.
 We put  $\tilde{\pi}_K:=(\bar{\pi}_n)_{n\geqq 0}\in \tilde{\mathbb{E}}^+$ where $\pi_n\in \mathcal{O}_{\overline{K}}$ satisfies that
 $\pi_0=\pi_K$ and
 $\pi_{n+1}^{q}=\pi_n$ for any $n$, then we define $A_{\mathrm{max},K}:=\widehat{A_{\mathrm{inf},K}[\frac{[\tilde{\pi}_K]}{\pi_K}]}$
 the $p$-adic
 completion of $A_{\mathrm{inf},K}[\frac{[\tilde{\pi}_K]}{\pi_K}]$.
 We define $B^+_{\mathrm{max},K}=A_{\mathrm{max},K}[p^{-1}]$, this is a $K$-Banach space with continuous actions of $G_K$ and $\varphi_K$.  By the definition, we have a canonical isomorphism
 $K\otimes_{K_0}B^+_{\mathrm{max},\mathbb{Q}_p}\isom B^+_{\mathrm{max},K}$ (Remark 7.13 of \cite{Co02}).
 By Lemma 8.8 and Proposition 8.9 of \cite{Co02}, there exists a power series $F_K(X)\in K[[X]]$ (Lubin-Tate's logarithm)
 such that $F_K(X)\circ [a]=aF_K(X)$ for any $a\in \mathcal{O}_K$ and $t_K:=F_K(\omega)$ converges in $A_{\mathrm{max},K}$ such
 that $\varphi_K(t_K)=\pi_Kt_K, g(t_K)=\chi_{\mathrm{LT}}(g)t_K$ for any $g\in G_K$. We define $B_{\mathrm{max},K}:=B^+_{\mathrm{max},K}[t_K^{-1}]$.
  We define $B^+_{\mathrm{dR}}:=\varprojlim_{n}B^+_{\mathrm{inf},K}/
( \mathrm{Ker}(\theta))^n$ equipped with the projective limit topology of $K$-Banach spaces $B^+_{\mathrm{inf},K}/(\mathrm{Ker}(\theta))^n$ whose
$\mathcal{O}_K$-lattice is defined as the image of $A_{\mathrm{inf},K}\rightarrow B^+_{\mathrm{inf},K}/(\mathrm{Ker}(\theta))^n$.
By Proposition 7.12 of \cite{Co02}, this $B^+_{\mathrm{dR}}$ is canonically topologically isomorphic to the usual $B^+_{\mathrm{dR}}$.  We define $B_{\mathrm{dR}}:=B^+_{\mathrm{dR}}
[t^{-1}]=B^+_{\mathrm{dR}}[t_K^{-1}]$.

Under this situation, we define a functor from the category of $\varphi_K$-modules to
the category of almost $\mathbb{C}_p$-representations defined by Fontaine. We can see this
construction as a very special case of a generalization of Berger's results (\cite{Be09}) to the case of Lubin-Tate period rings.

\begin{defn}
We say that $D$ is a $\varphi_K$-module over $K$ if $D$ is a finite dimensional
$K$-vector space with a $K$-linear isomorphism $\varphi_K:D\isom D$.
\end{defn}
Let $D$ be a $\varphi_K$-module over $K$, we extend the action of $\varphi_K$ to $\hat{K}^{\mathrm{ur}}\otimes_KD$ by
$\varphi_K(a\otimes x):=\varphi_K(a)\otimes\varphi_K(x)$, where $\hat{K}^{\mathrm{ur}}$ is the $p$-adic completion of
the maximal unramified extension $K^{\mathrm{ur}}$ of $K$ and $\varphi_K\in \mathrm{Gal}(K^{\mathrm{ur}}/K)$ is the lift of
$q$-th Frobenius in $\mathrm{Gal}(\overline{\mathbb{F}}/\mathbb{F}_q)$. The Deudonn\'e-Manin theorem
gives a decomposition $\hat{K}^{\mathrm{ur}}\otimes_{K}D=\oplus_{s\in\mathbb{Q}}D_{s}$, where for any
$s=\frac{a}{h}\in\mathbb{Q}$ such that
$(a,h)\in \mathbb{Z}\times\mathbb{Z}_{\geqq 1}$ are co-prime, $D_{s}$ is zero or a finite direct sum of $D_{a,h}:=\hat{K}^{\mathrm{ur}}e_1\oplus \hat{K}^{\mathrm{ur}}e_2\oplus
\cdots \oplus\hat{K}^{\mathrm{ur}}e_h$ such that $\varphi_K(e_1)=e_2,\varphi_K(e_2)=e_3,\cdots, \varphi_K(e_{h-1})=e_h, \varphi_K(e_h)=\pi_K^ae_1$. We define the set of slopes of $D$ as the set of $s\in\mathbb{Q}$ such that $D_s\not=0$.
We define a $\hat{K}^{\mathrm{ur}}$-semi-linear $G_K$-action on $\hat{K}^{\mathrm{ur}}\otimes_KD$ by
$g(a\otimes x):=g(a)\otimes x$ for any $g\in G_K$, $a\in \hat{K}^{\mathrm{ur}}$, $x\in D$, then $D_s$ is preserved by this $G_K$-action
for any $s\in \mathbb{Q}$ because the actions of $G_K$ and $\varphi_K$ commute each other. For any $s=\frac{a}{h}$, if we define
$D'_{s}:=\{x\in D_s| \varphi^h_K(x)=\pi_K^a x \}$, then we have $D_s=\hat{K}^{\mathrm{ur}}\otimes_{K^{\mathrm{ur}}_h}D'_{s}$ and
$D'_{s}$ is preserved by $G_K$ and $\varphi_K$, where $K^{\mathrm{ur}}_h$ is the unramified extension of $K$ of degree $h$.

The notion of almost $\mathbb{C}_p$-representations was defined by Fontaine (\cite{Fo03}).

\begin{defn}
Let $W$ be a $\mathbb{Q}_p$-Banach space equipped with a continuous
$\mathbb{Q}_p$-linear $G_K$-action. We say that $W$ is an almost $\mathbb{C}_p$-representation
if there exists $\mathbb{Q}_p$-representations $V_1$, $V_2$ and an integer $d\in \mathbb{Z}_{\geqq 0}$ such that
$V_1$ is a sub $G_K$-module of $W$ and $V_2$ is a sub $G_K$-module of $\mathbb{C}_p^d$ and there exists an
isomorphism $W/V_1\isom \mathbb{C}_p^d/V_2$ as $\mathbb{Q}_p$-Banach $G_K$-modules.
\end{defn}
\begin{rem}\label{3.3}
By Theorem C of \cite{Fo03}, $B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}$ is an almost $\mathbb{C}_p$-representation for any $k\geqq 0$. By Theorem B of \cite{Fo03}, for any continuous
$\mathbb{Q}_p$-linear $G_K$-morphism  $f:W_1\rightarrow W_2$ between
almost $\mathbb{C}_p$-representations $W_1$, $W_2$, it is known that $\mathrm{Ker}(f)$ and $\mathrm{Coker}(f)$ (as $\mathbb{Q}_p[G_K]$-modules)
are almost $\mathbb{C}_p$-representations and $\mathrm{Im}(f)$ is also an almost $\mathbb{C}_p$-representation
which is a closed subspace of $W_2$.
\end{rem}

Let $D$ be a $\varphi_K$-module over $K$. We will prove that $X_0(D):=(B^+_{\mathrm{max},K}\otimes_KD)^{\varphi_K=1}$
is an almost $\mathbb{C}_p$-representation.

\begin{lemma}\label{5.4}
Let $D$ be a $\varphi_K$-module over $K$.
\begin{itemize}
\item[(1)] $X_0(D)$ is an almost $\mathbb{C}_p$-representation.
\item[(2)]If any slope $s$ of $D$ satisfies $s>0$, then $X_0(D)=0$.
\end{itemize}
\end{lemma}
\begin{proof}
The proof is similar to that of Proposition 2.2 of \cite{Be09}.
If we denote by $\hat{K}^{\mathrm{ur}}\otimes_{K}D=\oplus_{s\in\mathbb{Q}}D_s$ as above, then we have
$B^+_{\mathrm{max},K}\otimes_KD=\oplus_{s\in \mathbb{Q}}B^+_{\mathrm{max},K}\otimes_{\hat{K}^{\mathrm{ur}}}D_{s}$
as a  $\varphi_K$-module and, for any $s=\frac{a}{h}$,  $B^+_{\mathrm{max},K}\otimes_{\hat{K}^{\mathrm{ur}}}D_{s}=B^+_{\mathrm{max},K}\otimes_{K^{\mathrm{ur}}_h}D'_{s}$ is preserved by the actions of $G_K$ and
$\varphi_K$. Hence, it suffices to show that, for any $s=\frac{a}{h}$, $(B^+_{\mathrm{max},K}\otimes_{K_h^{\mathrm{ur}}}D'_{s})^{\varphi_K=1}$ is an almost
$\mathbb{C}_p$-representation and is zero if $a>0$. By the definition of $D'_s$, we have a canonical inclusion
$(B^+_{\mathrm{max},K}\otimes_{K_h^{\mathrm{ur}}}D'_{s})^{\varphi_K=1}\subseteq B_{\mathrm{max},K}^{+, \varphi^h_K=\pi_K^{-a}}
\otimes_{K^{\mathrm{ur}}_h}D'_s$.

By 8.5 of \cite{Co02}, for $a>0$ we have $B_{\mathrm{max},K}^{+, \varphi_K^h=\pi_K^{-a}}=0$ and, for
$a\leqq 0$ we have a short exact sequence
$$0\rightarrow K_{h}^{\mathrm{ur}}t_{K_{h}^{\mathrm{ur}}}^{-a}\rightarrow B_{\mathrm{max},K}^{+, \varphi_K^h=\pi_K^{-a}}\rightarrow B^+_{\mathrm{dR}}/t^{-a}B^+_{\mathrm{dR}}\rightarrow 0,$$
 where

$t_{K_{h}^{\mathrm{ur}}}\in B^+_{\mathrm{max},K}=B^+_{\mathrm{max},K_{h}^{\mathrm{ur}}}$ is defined from ($K_{h}^{\mathrm{ur}},\pi_K, \varphi_K^h$) in the same way
as in the definition of $t_K$ defined from $(K,\pi_K,\varphi_K)$. Moreover, because $B^+_{\mathrm{dR}}/t^{-a}B^+_{\mathrm{dR}}\otimes_{K_h^{\mathrm{ur}}}D'_s$ is
a $B^+_{\mathrm{dR}}$-representation, so this is also an almost $\mathbb{C}_p$-representation by Theorem 5.13 of \cite{Fo03}.
Hence,  $B_{\mathrm{max},K}^{+, \varphi^h_K=\pi_K^{-a}}
\otimes_{K^{\mathrm{ur}}_h}D'_s$ is also an almost $\mathbb{C}_p$-representation. Because $(B_{\mathrm{max},K}^{+}
\otimes_{K^{\mathrm{ur}}_h}D'_s)^{\varphi_K=1}=\mathrm{Ker}(\varphi_K-1:B_{\mathrm{max},K}^{+, \varphi^h_K=\pi_K^{-a}}
\otimes_{K^{\mathrm{ur}}_h}D'_s\rightarrow B_{\mathrm{max},K}^{+, \varphi^h_K=\pi_K^{-a}}
\otimes_{K^{\mathrm{ur}}_h}D'_s)$, then $(B_{\mathrm{max},K}^{+}
\otimes_{K^{\mathrm{ur}}_h}D'_s)^{\varphi_K=1}$ is also an almost $\mathbb{C}_p$-representation by Remark \ref{3.3}.
\end{proof}

As an application of this lemma, we obtain the following corollary.
We fix an embedding $\sigma:K\hookrightarrow E$. For a
$K$-vector space $M$ and an $E$-vector space $N$, we denote by $M\otimes_{K,\sigma}N$ the tensor product
of $M$ and $N$ over $K$, where we view $N$ as a $K$-vector space by using $\sigma:K\hookrightarrow E$.
\begin{corollary}\label{5.5}
Let $\alpha\in E^{\times}$ be a non-zero element,
then $(B^+_{\mathrm{max},K}\otimes_{K,\sigma}E)^{\varphi_K=\alpha}$ is
an almost $\mathbb{C}_p$-representation and, for any positive integer $k$ such that $k> e_Kv_p(\alpha)$, the natural map
$(B^+_{\mathrm{max},K}\otimes_{K,\sigma}E)^{\varphi_K=\alpha}\rightarrow (B^+_{\mathrm{dR}}/
t^kB^+_{\mathrm{dR}})\otimes_{K,\sigma} E$ is injection, where $e_K$ is the absolute ramified index of $K$.
Moreover, if we denote the cokernel of this inclusion by $U_k$,
we have the following short exact sequence of $E$-Banach almost $\mathbb{C}_p$-representations,
\begin{equation*}
0\rightarrow (B^+_{\mathrm{max},K}\otimes_{K,\sigma}E)^{\varphi_K=\alpha}\rightarrow (B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}})
\otimes_{K,\sigma}E\rightarrow U_k\rightarrow 0.
\end{equation*}
\end{corollary}
\begin{proof}
For $\alpha\in E^{\times}$, we define a $\varphi_K$-module $D_{\alpha}$ over $K$ by
$D_{\alpha}:=Ee$ such that
$\varphi_K(ae)=\alpha^{-1} ae$ for any $a\in E$. $D_{\alpha}$ has unique slope $-e_Kv_p(\alpha)$ and we have a natural
isomorphism $$X_0(D_{\alpha})\isom (B^+_{\mathrm{max},K}\otimes_{K,\sigma}E)^{\varphi_K=\alpha}.$$ Hence, by Lemma \ref{5.4},
$(B^+_{\mathrm{max},K}\otimes_KE)^{\varphi=\alpha}$ is a non-zero almost $\mathbb{C}_p$-representation.
Moreover, by using Proposition 8.10 of \cite{Co02}, we can show that, for any $k\geqq0$, we have
an equality $$(B^+_{\mathrm{max},K}\otimes_{K,\sigma}E)^{\varphi_K=\alpha}
\cap(t^kB^+_{\mathrm{dR}}\otimes_{K,\sigma}E)=(t_K^kB^+_{\mathrm{max},K}\otimes_{K,\sigma}E)^{\varphi_K=\alpha}.$$ This is isomorphic
to $X_0(D_{\alpha\sigma(\pi_K)^{-k}})$ where $D_{\alpha\sigma(\pi_K)^{-k}}=Ee$ is defined by
 $\varphi_K(e)=\alpha^{-1}\sigma(\pi_K)^ke$. Because,
$D_{\alpha\sigma(\pi_K)^{-k}}$ has unique slope $(k-e_Kv_p(\alpha))$, so
we have $X_0(D_{\alpha\sigma(\pi_K)^{-k}})=0$ when $k>e_Kv_p(\alpha)$
by Lemma \ref{5.4}. This implies that the natural map $(B^+_{\mathrm{max},K}\otimes_{K,\sigma}E)^{\varphi_K=\alpha}
\rightarrow (B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}})\otimes_{K,\sigma}E$ is injection. Because both of these
are almost $\mathbb{C}_p$-representations, hence the cokernel $U_k$ is also an almost $\mathbb{C}_p$-representation
 by Remark \ref{3.3} in particular $U_k$ is an $E$-Banach space.
\end{proof}

For two $K$-Banach spaces $M_1$ and $M_2$, we denote by $M_1\hat{\otimes}_KM_2$ the complete tensor product of
$M_1$ and $M_2$ over $K$. Let $R$ be a complete topological $E$-algebra. We say that $R$ is an $E$-Banach algebra if there exists
a map $|-|_R:R\rightarrow \mathbb{R}_{\geqq}$ which satisfies, for any $x,y\in R$, $a\in E$,
\begin{itemize}
\item[(1)]$|1|_R=1$, $|x|_R=0$ if and only if $x=0$,
\item[(2)]$|x+y|_R\leqq \mathrm{max}\{|x|_R, |y|_R\}$,
\item[(3)]$|xy|_R\leqq |x|_R|y|_R$ and $|ax|_R=|a|_p|x|_R$
\end{itemize}
and if  the topology of $R$ is defined by the metric induced from $|-|_R$.

\begin{lemma}\label{5.6}
Let $R$ be an $E$-Banach algebra and $\alpha\in R$ be an element of $R$ such that $\alpha-1$ is
topologically nilpotent, then there exists $u\in (\hat{K}^{\mathrm{ur}}\hat{\otimes}_{K,\sigma}R)^{\times}$ such that $\varphi_K(u)=\alpha u$.
\end{lemma}
\begin{proof}
The proof is same as that of Lemma 3.6 of \cite{Ki03}.

\end{proof}

Here, we recall some terminologies concerning to  Banach modules from $\S2$ of \cite{Bu07}.
 Let $R$ be an $E$-Banach algebra and let $M$ be a topological $R$-module.
 We say that $M$ is a Banach $R$-module if  $M$ is
 a complete topological $R$-module with a map $|-|:M\rightarrow \mathbb{R}_{\geqq 0}$
 satisfying that, for any $m,n\in M$, $a\in R$,
 \begin{itemize}
 \item[(1)]$|m|=0$ if and only if $m=0$,
 \item[(2)]$|m+n|\leqq \mathrm{max}\{|m|, |n|\}$,
 \item[(3)]$|am|\leqq |a|_R|m|$( where $|-|_R$ is an $E$-Banach norm on $R$)
 \end{itemize}
  and the topology on $M$ is defined by the metric induced from $|-|$.
 Let $M$ be a Banach $R$-module. We say that $M$ is potentially orthonormalizable if there exist a map
 $|-|:M\rightarrow \mathbb{R}_{\geqq 0}$ as above and a subset $\{e_i\}_{i\in I}$ of $M$ such that
 \begin{itemize}

 \item[(1)]for any $m\in M$, there exists unique $\{a_i\}_{i\in I}$ ($a_i\in R$) such that
 $a_i\rightarrow 0 (i\rightarrow \infty)$ and that $m=\sum_{i\in I}a_ie_i$,
 \item[(2)] in the situation (1), we have $|m|=
 \mathrm{max}_{i\in I}\{|a_i|_R\}$.
 \end{itemize}
 We say that a Banach $R$-module $M$ has property $(\mathrm{Pr})$ if there exists a Banach $R$-module $N$ such that
 $M\oplus N$ is potentially orthonormalizable.

The following proposition is also a generalization of Corollary 3.7 of \cite{Ki03} which will play a crucial role in the
next subsection.

\begin{prop}\label{5.7}
Let $R$ be an $E$-Banach algebra and let $Y\in R^{\times}$ be a unit of $R$. We assume that
there exists a finite Galois extension $E'$ of $E$ and there exists $\lambda\in (R\otimes_E E')^{\times}$
such that $E'[\lambda]\subseteq R\otimes_{E}E'$ is an \'etale $E'$-algebra
and that $Y\lambda^{-1}-1$ is topologically nilpotent in $R\otimes_E E'$,
then, for sufficient large $k\in \mathbb{Z}_{>0}$, there exists
a Banach $R$-module $U_k$ which has property $(Pr)$ with a continuous $R$-linear $G_K$-action
such that there exists a $G_K$-equivariant short exact sequence of  Banach $R$-modules with property (Pr) ,
\begin{equation*}
0\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R)^{\varphi_K=Y}
\rightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R
\rightarrow U_k\rightarrow 0
\end{equation*}
\end{prop}
\begin{proof}
If we decompose $E'[\lambda]=\prod_{i\in I}E_i\subseteq R\otimes_E E'$ such
that each $E_i$ is a finite extension of $E'$ and if we denote by  $\lambda_i\in E_i$ the image
of $\lambda$ in $E_i$, then we can decompose $R\otimes_E E'=\prod_{i\in I}R_i$ such
that $E_i\subseteq R_i$. By Corollary \ref{5.5}, for any $k\in\mathbb{Z}_{>0}$ such that $k>e_Kv_p(\lambda_i)$ for any $i\in I$, we have a short exact
sequence of $E_i$-Banach spaces
$$ 0\rightarrow (B^+_{\mathrm{max},K}\otimes_{K,\sigma}E_i)^{\varphi_K=\lambda_i}
\rightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\otimes_{K,\sigma}E_i\rightarrow U_{k,i}\rightarrow 0.$$
Hence, if we take the complete tensor product over $E_i$ of this sequence with $R_i$,
for any $i\in I$, we obtain a short exact sequence of potentially orthonormalizable $R_i$-Banach spaces
$$0\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R_i)^{\varphi_K=\lambda_i}
\rightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R_i\rightarrow U_{k,i}\hat{\otimes}_{E_i}R_i\rightarrow 0.$$
By the assumption, the element $Y\lambda_i^{-1}-1$ is topologically nilpotent in $R_i$, hence by Lemma \ref{5.6}, we have
an element $u_i\in( \hat{K}^{\mathrm{ur}}\hat{\otimes}_{K,\sigma}R_i)^{\times}$ such that $\varphi_K(u_i)=Y\lambda_i^{-1} u_i$. Multiplying
$u_i$ to the above short exact sequence, we obtain a short exact sequence
$$0\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R_i)^{\varphi_K=Y}\rightarrow
B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R_i\rightarrow (U_{k,i}\hat{\otimes}_{E_i}R_i)\rightarrow 0.$$
Summing up for all $i\in I$, we obtain
$$0\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}(R\otimes_{E}E'))^{\varphi_K=Y}
\rightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma} (R\otimes_E E')\rightarrow \oplus_{i\in L}(U_{k,i}
\hat{\otimes}_{E_i}R_i)\rightarrow 0.$$
Finally, by taking $\mathrm{Gal}(E'/E)$-fixed part, if we put $U_k:=(\oplus_{i\in I}
(U_{k,i}\hat{\otimes}_{E_i}R_i))^{\mathrm{Gal}(E'/E)}$ (this has property (Pr)), we obtain a short exact sequence
$$0\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R)^{\varphi_K=Y}\rightarrow
B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R\rightarrow U_k\rightarrow 0$$
satisfying all the conditions in the proposition.

\end{proof}

Let $V$ be an $E$-representation, then we define
$$D^+_{\mathrm{cris}}(V):=(B^+_{\mathrm{max}}\otimes_{\mathbb{Q}_p}V)^{G_K},\,\mathrm{Fil}^0D_{\mathrm{cris}}(V):=D_{\mathrm{cris}}(V)\cap \mathrm{Fil}^0D_{\mathrm{dR}}(V),$$ where we write $B^+_{\mathrm{max}}:=B^+_{\mathrm{max},\mathbb{Q}_p}$.
Then, we have a natural inclusion
$D^+_{\mathrm{cris}}(V)\subseteq \mathrm{Fil}^0D_{\mathrm{cris}}(V)$, which is not equal in general.

\begin{lemma}\label{5.7.5}
Let $\alpha\in E^{\times}$ be a non zero element. If a $\varphi$-sub module $D$ of $D_{\mathrm{cris}}(V)^{\varphi^f=\alpha}$
is contained in $\mathrm{Fil}^0D_{\mathrm{dR}}(V)$, then $D$ is also contained in $D^+_{\mathrm{cris}}(V)^{\varphi^f=\alpha}$.

\end{lemma}

\begin{proof}
It suffices to show that
if an element $x\in (B_{\mathrm{max}}\otimes_{\mathbb{Q}_p}E)^{\varphi^f=\alpha}$ satisfies
that $\varphi^i(x)\in B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E$ for any $i\in \mathbb{Z}_{\geqq 0}$, then
$x\in (B^+_{\mathrm{max}}\otimes_{\mathbb{Q}_p}E)^{\varphi^f=\alpha}$.
If we write $x=\frac{a}{t^n}$ for some $a\in (B^+_{\mathrm{max}}\otimes_{\mathbb{Q}_p}E)^{\varphi^f=\alpha p^{fn}}$ and $n\geqq 0$, then we have $\frac{\varphi^i(a)}{p^{ni}t^n}=\varphi^i(x)\in B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E$ for any $0\leqq i\leqq f-1$.
Hence, we have $$\varphi^i(a)\in (B^+_{\mathrm{max}}\otimes_{\mathbb{Q}_p}E)^{\varphi^f=\alpha p^{nf}}\cap t^{n}B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E
=(t_{K_0}^nB^+_{\mathrm{max}}\otimes_{\mathbb{Q}_p}E)^{\varphi^f=\alpha p^{nf}},$$ where the last equality follows from
Proposition 8.10 of \cite{Co02}. Hence, we can write $a=\varphi^{-i}(t_{K_0}^n)a_i$ for some
$a_i\in B^+_{\mathrm{max}}\otimes_{\mathbb{Q}_p}E$ for any $1\leqq i\leqq f-1$ and then we can write by
$a=(\prod_{i=0}^{f-1}\varphi^{-i}(t_{K_0}^n))a'$ for some $a'\in B^+_{\mathrm{max}}\otimes_{\mathbb{Q}_p}E$ by Lemma 8.18 of \cite{Co02}.
Because $\prod_{i=0}^{f-1}\varphi^{-i}(t_{K_0})\in K_0^{\times}t$ by Lemma 8.17 of \cite{Co02}, we have $x=\frac{a}{t^n}\in B^+_{\mathrm{max}}\otimes_{\mathbb{Q}_p}E$.

\end{proof}

\subsection{Construction of the finite slope subspace: for general $p$-adic field case}
In this subsection, we generalize Kisin's construction of $X_{fs}$ for any $p$-adic field.
If we admit Proposition \ref{5.7}, the construction and the proof is almost same as in the $\mathbb{Q}_p$-case, the
only difference is that we need to consider all the embeddings $\sigma:K\hookrightarrow E$.
But, for the convenience of readers and the author, here we reprove this construction in full detail.

Let $X$ be a separated rigid analytic space (in the sense of Tate) over $E$.
Let $M$ be a free $\mathcal{O}_X$-module of rank $d$ for some $d\geqq 1$ equipped with
a continuous $\mathcal{O}_X$-linear $G_K$-action, where ``continuous" means that, for any admissible open affinoid $U=\mathrm{Spm}(R)$ of $X$, the action of $G_K$ on $\Gamma(U, M)$ is continuous on which the topology is defined as the
direct sum topology of $R$. We denote by $M^{\vee}:=\mathrm{Hom}_{\mathcal{O}_X}(M,\mathcal{O}_X)$ the $\mathcal{O}_X$-dual of $M$. For any point
$x\in X$, we denote the local ring at $x$ by $\mathcal{O}_{X,x}$, the maximal ideal at $x$ by $\mathfrak{m}_x$, the residue field at $x$
by $E(x)$ which is a finite extension of $E$ and
we denote by $M(x)$ the fiber of $M$ at $x$ which is a $d$-dimensional
$E(x)$-representation of $G_K$. By Lemma 3.18 of \cite{Ch09a}, we can take
an affinoid covering $ \{U_i\}_{i\in I}$ of $X$ such that $M_i:=\Gamma(U_i,M)$ is a free $R_i:=\Gamma(U_i,\mathcal{O}_X)$-module
which has a $G_K$-stable finite free $R_i^0$-module $M_i^0$ such that $M_i=M^0_i[p^{-1}]$ for a model
$R^0_{i}\subseteq R_i$, where, for an affinoid $A$, a model is defined as a topologically finite generated complete
$\mathcal{O}_E$-sub algebra of $A$ which generates $A$ after inverting $p$.
Then, we can apply the theory of \cite{Be-Co08} of families of $p$-adic representations to $M_i$ (and $M^0_i$) for any
$i\in I$. By \cite{Be-Co08}, there exists a unique monic polynomial $P_{M_i}(T)\in K\otimes_{\mathbb{Q}_p}R_i[T]$ of dimension $d$
, which is the characteristic polynomial
of Sen's operator on $D_{\mathrm{Sen}}^{L}(M_i)$ of Proposition 4.1.2 of \cite{Be-Co08} (where $L$ is a sufficiently large finite extension of $K$), such that for any point $x\in \mathrm{Spm}(R_i)$,
the specialization of $P_{M_i}(T)$ at $x$ gives the Sen's polynomial $P_{M(x)}(T)\in K\otimes_{\mathbb{Q}_p}E(x)[T]$ of $M(x)$.
By the uniqueness of $D_{\mathrm{Sen}}^{L}(M_i)$, $\{P_{M_i}(T)\}_{i\in I}$ glue together to
a monic polynomial $P_M(T)\in K\otimes_{\mathbb{Q}_p}\mathcal{O}_X[T]$. By the canonical decomposition $K\otimes_{\mathbb{Q}_p}E=\oplus_{\sigma\in \mathcal{P}}E:a\otimes b\mapsto (\sigma(a)b)_{\sigma}$, $P_M(T)$ decomposes into the $\sigma$-components
$P_M(T)=(P_M(T)_{\sigma})_{\sigma\in\mathcal{P}}\in \oplus_{\sigma\in \mathcal{P}} \mathcal{O}_X[T]$.

Now we assume that the constant term of $P_M(T)_{\sigma}$ is zero for any $\sigma\in\mathcal{P}$.
We denote by $P_M(T)_{\sigma}=TQ_{\sigma}(T)$ for some $Q_{\sigma}(T)\in \mathcal{O}_X[T]$.

Before stating the theorem, we recall some terminologies
 of rigid geometry  (from $\S$ 5 of \cite{Ki03}) which we need for the construction of $X_{fs}$.

 Let $X=\mathrm{Spm}(R)$ be an affinoid variety over $E$ and
 $U$ be an admissible open in $X$. We say that
 $U$ is scheme theoretically dense in $X$ if there exists a
 Zariski open $V\subseteq \mathrm{Spec}(R)$ which is
 dense in $\mathrm{Spec}(R)$ for the Zariski topology
 and $U=V^{\mathrm{an}}$ where $V^{\mathrm{an}}$ is
the admissible open set of $X$ associated to $V$.
For any rigid analytic variety $X$ over $E$ and an admissible
open $U$ of $X$, we say that $U$ is scheme theoretically dense
in $X$ if there exists an admissible affinoid covering $\{U_i\}_{i\in I}$ of $X$
such that $U\cap U_i$ is scheme theoretically dense in $U_i$ for any $i\in I$.
The typical example is the following. Let $f\in \mathcal{O}_X$ be any element,
then we denote $X_f:=\{x\in X| f(x)\not=0\}$ which is an admissible open in $X$.
If $f$ is a non-zero divisor, then $X_f$ is scheme theoretically dense in $X$.

Next, let $Y\in \mathcal{O}_X^{\times}$ be an invertible function on $X$.
We say that a morphism $f:\mathrm{Spm}(R)\rightarrow X$ is $Y$-small
if there exists a finite extension $E'$ of $E$ and $\lambda\in (R\otimes_{E}E')^{\times}$ such
that $E'[\lambda]\subseteq R\otimes_EE'$ is a finite \'etale $E'$-algebra and that
$Y\lambda^{-1}-1\in R\otimes_EE'$ is topologically nilpotent. A typical example
of $Y$-small morphism is following. For any $x\in X$ and $n\in \mathbb{Z}_{\geqq 1}$, the natural
map $f:\mathrm{Spm}(\mathcal{O}_{X,x}/\mathfrak{m}_{x}^n)\rightarrow X$ is $Y$-small.

The following theorem is the key theorem for the construction of $p$-adic families of
trianguline representations, which is exactly the  generalization of Proposition 5.4 of \cite{Ki03} for any $K$-case.
For an $E$-affinoid $R$, we denote by $B^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}R:=\varprojlim_kB^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}R$ equipped with the projective limit topology.

\begin{thm}\label{5.8}
Let $X$ be a separated rigid analytic variety over $E$, $M$ be a free $\mathcal{O}_X$-module
of rank $d$ with a continuous $\mathcal{O}_X$-linear $G_K$-action. Let $Y\in \mathcal{O}_X^{\times}$
be an invertible function.
We assume that the constant term of $P_M(T)_{\sigma}$ is zero for any $\sigma\in \mathcal{P}$.
Then, there exists unique Zariski closed sub space $X_{fs}$ of $X$ satisfying the following conditions.
\begin{itemize}
\item[(1)]For any $\sigma\in \mathcal{P}$ and $i\in \mathbb{Z}_{\leqq 0}$, $X_{fs, Q_{\sigma}(i)}$ is scheme theoretically dense in $X_{fs}$.
\item[(2)]For any $Y$-small map $f:\mathrm{Spm}(R)\rightarrow X$
which factors through $X_{Q_{\sigma}(i)}$ for any
$\sigma\in\mathcal{P}$ and $i\in\mathbb{Z}_{\leqq0}$, the following two conditions are equivalent.
\begin{itemize}
\item[(i)]$f:\mathrm{Spm}(R)\rightarrow X$ factors through $X_{fs}$.
\item[(ii)] Any $R$-linear $G_K$-equivariant  map $h:M^{\vee}\otimes_{\mathcal{O}_X}R\rightarrow B^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}R$ factors through $h':M^{\vee}\otimes_{\mathcal{O}_X}R\rightarrow K\otimes_{K_0}(B^+_{\mathrm{max}}\hat{\otimes}_{\mathbb{Q}_p}R)^{\varphi^f=Y}$.
\end{itemize}
\end{itemize}
\end{thm}

The proof of this theorem is almost same as that of Kisin's in the  $\mathbb{Q}_p$-case.
But for convenience of readers and of the author, we reprove this in detail.
As in \cite{Ki03}, we prove this theorem by several steps.
First we prove the following.
\begin{lemma}\label{5.9}
Let $X$, $M$ be as above. Let $X'$ be a separated rigid analytic variety over $E$ and
$f:X'\rightarrow X$ be a flat $E$-morphism. If there exists a Zariski closed sub space $X_{fs}\subseteq X$
which satisfies $(1)$ and $(2)$ of the above theorem, then
$X'_{fs}:=X_{fs}\times_XX'\subseteq X'$ also satisfies $(1)$ and $(2)$ for $X'$ and $M':=f^*{M}$ and $Y':=f^{*}(Y)\in\mathcal{O}_{X'}^{\times}$.
\end{lemma}
\begin{proof}
First, the condition (1) is satisfied by $X'_{fs}$ because the notion of scheme theoretically dense is preserved by
flat base changes and because we have $f^*(P_M(T))=P_{f^*M}(T)$. That $X'_{fs}$ satisfies (2) is trivial.
\end{proof}
Next, we prove uniqueness of $X_{fs}$.
\begin{lemma}
If two Zariski closed sub varieties $X_1$ and $X_2$ of $X$ satisfy the conditions $(1)$ and $(2)$, then
$X_1=X_2$.
\end{lemma}
\begin{proof}
By the above Lemma \ref{5.9}, for any affinoid open $U\subseteq X$ and for any $i=1,2$, $X_i\cap U\subseteq U$ satisfies (1) and (2) for
$U$ because the inclusion $U\hookrightarrow X$ is flat. Hence, for an admissible covering $\{U_i\}_{i\in I}$ of
$X$, it suffices to prove that  $X_1\cap U_i=X_2\cap U_i$ for any $i\in I$. Hence we may assume that
$X=\mathrm{Spm}(R)$ is an affinoid. We denote by $X_1=\mathrm{Spm}
(R/I_1)$, $X_2=\mathrm{Spm}(R/I_2)$ for some ideals $I_1, I_2\subseteq R$. If we denote
$X_3:=\mathrm{Spm}(R/I_1\cap I_2)$, then we claim that $X_3$ also satisfies the conditions
(1) and (2) . For (1), by the assumption, we have inclusions $R/I_j\hookrightarrow R/I_j[\frac{1}{Q_{\sigma}(i)}]$ for
any $j=1,2$ and for any $\sigma\in \mathcal{P}, i\in \mathbb{Z}_{\leqq 0}$, hence we have an inclusion
$R/I_1\cap I_2\hookrightarrow R/I_1\cap I_2[\frac{1}{Q_{\sigma}(i)}]$ for any $\sigma\in \mathcal{P}, i\in
\mathbb{Z}_{\leqq0}$, which proves that $X_3$ satisfies (1). For proving that $X_3$ satisfies (2), we take a $Y$-small morphism
$f:\mathrm{Spm}(R')\rightarrow X$ which factors $f:\mathrm{Spm}(R')\rightarrow
X_{Q_{\sigma}(i)}$ for any $\sigma\in \mathcal{P}, i\in \mathbb{Z}_{\leqq 0}$. We denote by $Y':=f^*(Y)\in R^{' \times}$.
If $f$ satisfies (ii) of (2), then by the definition
of $X_1$ and $ X_2$, $f$ factors through $X_1$ and $X_2$, hence $f$ also factors through $X_3$ because $X_1,X_2\subseteq X_3$.
Next we assume that
$f$ satisfies (i) of (2) .
We have a canonical decomposition
\[
\begin{array}{ll}
K\otimes_{K_0} (B^+_{\mathrm{max}}\hat{\otimes}_{\mathbb{Q}_p}R')
^{\varphi^f=Y'}&=((K\otimes_{K_0}B^+_{\mathrm{max}})\hat{\otimes}_{\mathbb{Q}_p}R')^{\varphi^f=Y'}\\
&=(B^+_{\mathrm{max},K}\hat{\otimes}_{\mathbb{Q}_p}R')^{\varphi^f=Y'}=\oplus_{\sigma\in \mathcal{P}}(B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R')^{\varphi_K=Y'}
\end{array}
\]
and $B^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}R'=\oplus_{\sigma\in \mathcal{P}}B^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R'$.
Hence, it suffices to show that, for any $\sigma\in \mathcal{P}$, any $G_K$-equivariant $R'$-linear map
$h:M^{\vee}\otimes_{\mathcal{O}_X}R'\rightarrow B^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R'$ factors through
$M^{\vee}\otimes_{\mathcal{O}_X}R'\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R')^{\varphi_K=Y'}$.
Because $Q_{\sigma}(i)$ is invertible in
$R'$ for
any $\sigma\in \mathcal{P} , i\in \mathbb{Z}_{\leqq 0}$ by the assumption, the natural map

$$(B^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}M\otimes_{\mathcal{O}_X}R')^{G_K}\isom (B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma} M\otimes_{\mathcal{O}_X}R')^{G_K}$$ is isomorphism for any $\sigma\in\mathcal{P}$ and any $k\in\mathbb{Z}_{\geqq 1}$ by Proposition 2.5 of \cite{Ki03}.
Hence, it suffices to show that
for some $k\in\mathbb{Z}_{\geqq 1}$, any $G_K$-equivariant map $h:M^{\vee}\otimes_{\mathcal{O}_X}R'\rightarrow (B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R')$ factors through $M^{\vee}\otimes_{\mathcal{O}_X}R'\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R')^{\varphi_K=Y'}$.
We choose a $k\in\mathbb{Z}_{\geqq 1}$ such that there exists a short exact sequence of
Banach $R'$-modules with property (Pr) as in Proposition \ref{5.7}
whose exactness is preserved by any complete tensor base change,
$$0\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R')^{\varphi_K=Y'}\rightarrow
B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R'\rightarrow U_{k,\sigma}\rightarrow 0.$$

Moreover, if we denote by $\mathrm{Spm}(R'_i):=f^{-1}(X_{i})\subseteq \mathrm{Spm}(R')$ for $i=1,2$, then we have an inclusion
$R'\hookrightarrow R'_1\oplus R'_2$ because $f$ factors through $X_3$. From these, the above short exact sequence can be embedded in the following short exact
sequence
$$0\rightarrow \oplus^2_{i=1}(B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R'_i)^{\varphi_K=Y'}\rightarrow
\oplus^2_{i=1}B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R'_i\rightarrow \oplus^2_{i=1}
U_{k,\sigma}\hat{\otimes}_{R'}R'_i\rightarrow 0.$$
Then, the composition of $h$ with
$B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}
R'\hookrightarrow \oplus^2_{i=1}B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R'_i$ factors
through $M^{\vee}\otimes_{\mathcal{O}_X}R'\hookrightarrow \oplus^2_{i=1}(B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R'_i)^{\varphi_K=Y'}$ by  the definition of $X_i$.
 Hence, by
the above diagram of the two short exact sequences, $h$ also factors through $M^{\vee}\otimes_{\mathcal{O}_X}R'\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}
R_i')^{\varphi_K=Y'}$. Hence, $X_3$ also satisfies (1) and (2).

Hence, for proving the lemma, we may assume that $X_1\subseteq X_2$. We put $W\subseteq X_2$ the support of
$I_1/I_2$ with the reduced structure. If $x\in X_2$ satisfies $Q_{\sigma}(i)(x)\not=0$ for any $\sigma\in \mathcal{P}$,
$i\in\mathbb{Z}_{\leqq 0}$, then, for any $n\geqq 1$, the natural $Y$-small map $\mathrm{Spm}(\mathcal{O}_{X_2,x}/\mathfrak{m}_x^n)\rightarrow
X_2$ factors through $\mathrm{Spm}(\mathcal{O}_{X_2,x}/\mathfrak{m}_x^n)\rightarrow X_1$ by the definition on $X_1$ and $X_2$.
This implies that there exists a map $\mathcal{O}_{X_1,x}\rightarrow \hat{\mathcal{O}}_{X_2,x}$ such that the composition
of this with the natural map $\mathcal{O}_{X_2,x}\rightarrow \mathcal{O}_{X_1,x}$ is the natural map $\mathcal{O}_{X_2,x}
\rightarrow \hat{\mathcal{O}}_{X_2,x}$. This implies that the natural quotient map $\mathcal{O}_{X_2,x}
\rightarrow \mathcal{O}_{X_1,x}$ is isomorphism, hence we have $x\not\in W$. Hence, we have
an inclusion $W\subseteq \cup_{\sigma\in\mathcal{P},i\in\mathbb{Z}_{\leqq 0}}\{x\in X_2| Q_{\sigma}(i)(x)=0\}$.
By Lemma 5.7 \cite{Ki03}, then there exists a $Q\in \mathcal{O}_{X_2}$ a finite product of $Q_{\sigma}(i)$ such
that $X_{2,Q}\subseteq X_2\setminus W=X_1\setminus W\subseteq X_1 \subseteq X_2$. Then, the condition
(1) for $X_2$ implies that $X_1=X_2$. We finish to prove the lemma.

\end{proof}

Let $\{U_i\}_{i\in I}$ be an admissible affinoid covering of $X$ such that $U_{i,fs}\subseteq U_i$ exists for all $i\in I$.
By the uniqueness of $X_{fs}$, $U_{i,fs}$ glue to a Zariski closed sub space $X'_{fs}\subseteq X$ satisfying that
$X'_{fs}\cap U_i=U_{i,fs}$ for all $i\in I$.
\begin{lemma}\label{5.11}
In the above situation, $X'_{fs}\subseteq X$ satisfies the conditions $(1)$ and $(2)$ in the theorem.
\end{lemma}
\begin{proof}
That $X'_{fs}$ satisfies (1) is trivial. We show that $X'_{fs}$ satisfies (2).
Let $f:\mathrm{Spm}(R)\rightarrow X$ be a $Y$-small map which factors through $X_{Q_{\sigma}(i)}$
for any $\sigma\in \mathcal{P}$ and $i\in \mathbb{Z}_{\leqq 0}$.  Because $X$ is separated,
$f^{-1}(U_{i}):=\mathrm{Spm}(R_i)$ is also an affinoid for any $i\in I$.
First we show that (i) implies (ii).
We assume that $f$ factors through $X'_{fs}$. Let $h:M^{\vee}\otimes_{\mathcal{O}_X}R\rightarrow B^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R$
be a $R$-linear $G_K$-equivariant map. By Proposition 2.5 of \cite{Ki03}, it suffices to show that $h:M^{\vee}\otimes_{\mathcal{O}_X}R
\rightarrow B^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R\rightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R$
factors through $M^{\vee}\otimes_{\mathcal{O}_X}R\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R)^{\varphi_K=Y}$  for some
 $k\in\mathbb{Z}_{\geqq 1}$.
We choose a $k\in\mathbb{Z}_{\geqq 1}$ such that there exists a short exact sequence of
 Banach $R$-modules with property (Pr) as in Proposition \ref{5.7}
$$0\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R)^{\varphi_K=Y}\rightarrow
B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R\rightarrow U_{k,\sigma}\rightarrow 0.$$
By the property (Pr), this short exact sequence can be embedded into the following
$$0\rightarrow \prod_{i\in I}(B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R_i)^{\varphi_K=Y}\rightarrow
\prod_{i\in I}B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R_i \rightarrow \prod_{i\in I}
U_{k,\sigma}\hat{\otimes}_R R_i\rightarrow 0.$$

By the assumption, $h_i:M^{\vee}\otimes_{\mathcal{O}_X}R\rightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R
\rightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R_i$ factors through
$M^{\vee}\otimes_{\mathcal{O}_X}R\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R_i)^{\varphi_K=Y}$ for any $i\in I$. Hence,
$ h:M^{\vee}\otimes_{\mathcal{O}_X}R\rightarrow
B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R$ also factors through $M^{\vee}\otimes_{\mathcal{O}_X}R\rightarrow
(B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}\allowbreak R)^{\varphi_K=Y}$, which can be seen from the above diagram
of the two exact sequences.

Next, we assume that, for any $\sigma\in \mathcal{P}$, any
$R$-linear $G_K$-equivariant map $h:M^{\vee}\otimes_{\mathcal{O}_X}R\rightarrow B^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R$
factors through $M^{\vee}\otimes_{\mathcal{O}_X}R\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R)^{\varphi_K=Y}$.
Because $Q_{\sigma}(j)\in R^{\times}$ for any $\sigma\in \mathcal{P}$ and any $j\in \mathbb{Z}_{\leqq 0}$,
we have isomorphisms $(B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}M\otimes_{\mathcal{O}_X}R)^{G_K}
\otimes_R R_i\isom (B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}M\otimes_{\mathcal{O}_X}R_i)^{G_K}$
for any $k\geqq 1$ and for any $i\in I$ by Corollary 2.6 of \cite{Ki03}. Hence, any $R_i$-linear $G_K$-equivariant maps $h_i:M^{\vee}\otimes_{\mathcal{O}_X}R\rightarrow
B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R_i$  factor through $M^{\vee}\otimes_{\mathcal{O}_X}R\rightarrow
(B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R_i)^{\varphi_K=Y}$ for any $i\in I$. This implies that
$f|_{\mathrm{Spm}(R_i)}:\mathrm{Spm}(R_i)\rightarrow U_i$ factors through $U_{i,fs}$ for any
$i\in I$. Hence, $f$ also factors through $X'_{fs}$.
\end{proof}

By this lemma, it suffices to construct $X_{fs}$ for a sufficiently small affinoid open
$X=\mathrm{Spm}(R)$. We may assume that $|Y|$ satisfies $|Y||Y^{-1}|< \frac{1}{|\pi_K|_p}$, where
$|-|:R\rightarrow \mathbb{R}_{\geqq 0}$ is a map which define the topology of $R$ as in $\S3.2$.
Then, we construct $X_{fs}\subseteq \mathrm{Spm}(R)$ as follows. First, we
construct an ideal of $R$ which determines $X_{fs}$.
Let $\lambda\in \overline{E}$ be any element such
that $|Y^{-1}|^{-1}\leqq |\lambda|_p\leqq |Y|$ and $E'$ be a finite Galois extension of $E$ which contains $\lambda$.
By Corollary \ref{5.5},
we can take a sufficiently large $k\in\mathbb{Z}_{\geqq 1}$ such that, for any $\lambda$ as above and for any $\sigma\in \mathcal{P}$, there exists a
short exact sequence of $E'$-Banach spaces
$$0\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}E')^{\varphi_K=\sigma(\pi_K)\lambda}
\rightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}E'\rightarrow U_{k,\sigma,\lambda}\rightarrow 0.$$
For any $x\in \tilde{\mathbb{E}}^+$ such that $v(x)>0$, we define  an element
$$P(x,\frac{Y}{\sigma(\pi_K)\lambda}):=
\sum_{n\in \mathbb{Z}}\varphi_K^n([x])(\frac{Y}{\sigma(\pi_K)\lambda})^n\allowbreak\in (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R\otimes_E E')^{\varphi_K=
\frac{\sigma(\pi_K)\lambda}{Y}}.$$
This element converges because
$$|\frac{\sigma(\pi_K)\lambda}{Y}|\leqq |\sigma(\pi_K)|_p|\lambda|_p|Y^{-1}|<
|\sigma(\pi_K)|_p|\lambda|_p|Y|^{-1}|\pi_K|_p^{-1}\leqq 1$$ and because $\varphi_K^n([x])(\frac{Y}{\sigma(\pi_K)\lambda})^n\rightarrow 0$ ($n\rightarrow +\infty$) (see Corollary 4.4 of \cite{Ki03}). For any $\sigma\in\mathcal{P}$ and for any $R$-linear $G_K$-equivariant map
$h:M^{\vee}\rightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R$, we consider
the composition of this map with
$$B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R\rightarrow
B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R\otimes_E E':v\mapsto P(x,\frac{Y}{\sigma(\pi_K)\lambda})v$$
and with $B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R\otimes_E E'\rightarrow
U_{k,\sigma,\lambda}\hat{\otimes}_{E'}(R\otimes_E E')$ which is the base change of the surjection
$B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}E'\rightarrow U_{k,\sigma,\lambda}$ in the above short exact sequence.
We denote this composition by
$$h_{x,\lambda}:M^{\vee}\rightarrow
U_{k,\sigma,\lambda}\hat{\otimes}_{E'} (R\otimes_E E').$$ If we fix an orthonormalizable $E'$-base $\{e_i\}_{i\in I}$ of $U_{k,\sigma,\lambda}$, then, for any
$m\in M^{\vee}$, we can write uniquely by
$$h_{x,\lambda}(m)=\sum_{i\in I}a_{x,\lambda,i}(m)e_i\text{ for some }
\{a_{x,\lambda,i}(m)\}_{i\in I}\subseteq R\otimes_EE'.$$ Then, we define an ideal
$$I(h,x,\lambda,m)\subseteq R\otimes_E E'$$ which is
generated by $a_{x,\lambda,i}(m)$ for all $i\in I$. Because, for any $\tau\in \mathrm{Gal}(E'/E)$, we have $\tau(I(h,x,\lambda,m))
=I(h,x,\tau(\lambda),m)\subseteq R\otimes_EE'$,  the ideal
$$\sum_{\tau\in \mathrm{Gal}(E'/E)}
I(h,x,\tau(\lambda),m) \subseteq R\otimes_E E'$$ descends to an ideal $I'(h,x,\lambda,m)\subseteq R$ and this ideal is
independent of the choice of $E'$. We define an ideal $I$ of $R$  by $$I:=\sum_{h,x,\lambda,m}I'(h,x,\lambda,m)\subseteq R,$$ where we take
all $h,x,\lambda, m$ and $\sigma\in \mathcal{P}$ as above. Next, we consider the kernels of the natural maps
$R/I\rightarrow R/I[\frac{1}{\prod_{l=1}^{n}
Q_{\sigma_l}(i_l)}]$ for any $n\geqq 1$, $\sigma_l\in \mathcal{P}, i_l\in\mathbb{Z}_{\leqq 0}$. Because $R$ is noetherien, there exists
the largest ideal $(I\subseteq )J\subseteq R$ such that $R/I\rightarrow R/I[\frac{1}{\prod_{l=1}^nQ_{\sigma_l}(i_l)}]$ factor through
$R/I\rightarrow R/J$ and the induced maps $R/J\rightarrow R/I[\frac{1}{\prod_{l=1}^nQ_{\sigma_l}(i_l)}]$ are injection for any
$n$ and $\sigma_l$ and $i_l$.
We claim that $X_{fs}=\mathrm{Spm}(R/J),$ which proves the theorem.
\begin{lemma}\label{5.12}
$\mathrm{Spm}(R/J)\subseteq \mathrm{Spm}(R)$ satisfies the conditions $(1)$ and $(2)$ in the
theorem.
\end{lemma}

\begin{proof}
Because $R/J\rightarrow R/J[\frac{1}{Q_{\sigma}(i)}]$ are injection for any $\sigma\in\mathcal{P}$ and $i\in\mathbb{Z}_{\leqq 0}$
by the definition of $J$,
$\mathrm{Spm}(R/J)$ satisfies the condition (1).
We show that $\mathrm{Spm}(R/J)$ satisfies (2).
Let $f:\mathrm{Spm}(R')\rightarrow \mathrm{Spm}(R)$ be a $Y$-small map
which factors through $\mathrm{Spm}(R')\rightarrow \mathrm{Spm}(R)_{Q_{\sigma}(i)}$ for
any $\sigma\in \mathcal{P}$ and $i\in\mathbb{Z}_{\leqq 0}$.
Under this situation, we first prove that (ii) implies (i). We assume that
, for any $\sigma\in\mathcal{P}$, any $h:M^{\vee}\rightarrow B^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R'$ factor through
$M^{\vee}\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R')^{\varphi_K=Y}$.
Then, for any $h:M^{\vee}\rightarrow B^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R$ and $\lambda\in E'$ and
$x\in \tilde{\mathbb{E}}^+$ as in the construction of the ideal $I\subseteq R$, the maps
$$P(x,\frac{Y}{\sigma(\pi_K)\lambda})h\otimes_R\mathrm{id}_{R'}:
M^{\vee}\rightarrow U_{k,\sigma,\lambda}\hat{\otimes}_{E'}(R'\otimes_EE')$$ are zero  because the multiplication by $P(x,\frac{Y}{\sigma(\pi_K)\lambda})$ sends $(B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R)^{\varphi=Y}$ to $(B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R\otimes_EE')^{\varphi_K=\sigma(\pi_K)\lambda}$. Hence, the map $R\rightarrow R'$ factors through $R/I\rightarrow R'$ and then, because $Q_{\sigma}(i)\in R^{' \times}$ for any $\sigma\in\mathcal{P}$ and $i\in\mathbb{Z}_{\leqq 0}$, $R/I\rightarrow R'$ factors through $R/J\rightarrow R'$ by the definition of $J$.

Next, we assume that $f:\mathrm{Spm}(R')\rightarrow \mathrm{Spm}(R)$ factors through $\mathrm{Spm}(R')
\rightarrow \mathrm{Spm}(R/J)$. Let $h:M^{\vee}\rightarrow B^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R'$ be a $R'$-linear
$G_K$-equivariant map. We want to show that this map  factors through $M^{\vee}\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R')^{\varphi_K=Y}$.
By Galois descent, it suffices to show that this factors through $M^{\vee}\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R'\otimes_EE')^{\varphi_K=Y}$ for a sufficiently large finite Galois extension $E'$ of $E$.
Hence, by the definition of $Y$-smallness, we may assume that there exists $\lambda\in E$ such that $Y\lambda^{-1}-1$ is topologically
nilpotent in $R'$. Then, we have $|Y^{-1}|^{-1}\leqq |f^*(Y)^{-1}|_{R'}^{-1}=|\lambda|_p=|f^*(Y)|_{R'}\leqq |Y|$, hence $\lambda$ satisfies the condition in the construction of $I\subseteq R$.
By the definition of $I$, for any $m\in M^{\vee}$, $P(x,\frac{Y}{\sigma(\pi_K)\lambda})h(m)$ is an element
in $(B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R')^{\varphi_K=\sigma(\pi_K)\lambda}$ for any
$x\in \tilde{\mathbb{E}}^+$ such that $v(x)>0$. If we take an element $u\in (\hat{K}^{\mathrm{ur}}\hat{\otimes}_{K,\sigma}R')^{\times, \varphi_K=\frac{\lambda}{Y}}$ as in Lemma \ref{5.6}, then we have
$$t_Kuh(m)\in (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R')^{\varphi_K=
\sigma(\pi_K)\lambda}$$ because $t_Ku\in (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R')^{\varphi_K=\frac{\sigma(\pi_K)\lambda}{Y}}$ and because the $R'$-module generated by the sets $\{P(x,\frac{Y}{\sigma(\pi_K)\lambda})\}_{x\in \tilde{\mathbb{E}}^+, v(x)>0}$ is dense in $(B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R')^{\varphi_K=\frac{\sigma(\pi_K)\lambda}{Y}}$, which can be proved in the same way as Corollary
4.6 of \cite{Ki03} by using Lemma 4.3.1 of \cite{Ke05}, and because
$(B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R')^{\varphi_K=\sigma(\pi_K)\lambda}$ is closed in $B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R'$ by Proposition \ref{5.7}. Hence, we have
\[
\begin{array}{ll}
uh(m)&\in \frac{1}{t_K}(B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}\allowbreak R')^{\varphi_K=\sigma(\pi_K)\lambda}\cap B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R' \\
 &=(B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R')^{\varphi_K=\sigma(\pi_K)\lambda},
 \end{array}
 \] where the last equality follows from Proposition
8.10 of \cite{Co02}. Hence, we obtain $h(m)\in (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}\allowbreak R')^{\varphi_K=Y}.$
We finish to prove the lemma, hence we finish the proof or the theorem.

\end{proof}

We will apply this construction to the rigid analytic space associated to the universal deformation rings of
mod $p$ representations of $G_K$. Before doing this, we reprove some important general
properties of $X_{fs}$ proved in Corollary 5.16 of \cite{Ki03} for general $K$ case.

Let $\mathrm{Spm}(R)\subseteq X_{fs}$ be an affinoid open of $X_{fs}$, we assume that
this inclusion is $Y$-small. By Proposition \ref{5.7}, there exists $k>0$ such that, for any $\sigma\in\mathcal{P}$, there exists a short
exact sequence of Banach $R$-modules with property (Pr)
$$0\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R)^{\varphi_K=Y}
\rightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R\rightarrow U_{k,\sigma}\rightarrow 0.$$
We denote by $M_R$ the restriction of $M$ to $\mathrm{Spm}(R)$.
\begin{prop}\label{5.13}
For any $\sigma\in \mathcal{P}$, let $H_{\sigma}\subseteq R$ be the smallest  ideal of $R$ such that
any $R$-linear $G_K$-equivariant morphisms
$h:M^{\vee}_R\rightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R$ factor through
$M^{\vee}_R\rightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}H_{\sigma}$ and put
$H:=\prod_{\sigma\in \mathcal{P}}H_{\sigma}\subseteq R$,
then the following hold:
\begin{itemize}
\item[(1)]The natural maps $$((B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}M_R)^{\varphi_K=Y})^{G_K}\rightarrow
(B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}M_R)^{G_K}$$ are isomorphism
for any $\sigma\in \mathcal{P}$ $($i.e. the natural map $$K\otimes_{K_0}D^+_{\mathrm{cris}}(M_R)^{\varphi_K=Y}\rightarrow
(B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}\allowbreak M_R)^{G_K}$$ is
isomorphism$)$.
\item[(2)] $\mathrm{Spm}(R)\setminus V(H)$ and $\mathrm{Spm}(R)\setminus V(H_{\sigma})$ ( for any $\sigma\in \mathcal{P}$) are  scheme theoretically dense in $\mathrm{Spm}(R)$, where $V(H_{*}):=\mathrm{Spm}(R/H_{*})$.
\item[(3)]For any $x\in \mathrm{Spm}(R)$, $M(x)$ is a split trianguline $E(x)$-representation, more precisely,
there exists a $\{k_{\sigma}\}_{\sigma\in \mathcal{P}}\in \prod_{\sigma\in \mathcal{P}}\mathbb{Z}_{\geqq 0}$ such that
there exists a short exact sequence of $E(x)$-$B$-pairs
$$0\rightarrow W(\delta_{Y(x)}\prod_{\sigma\in \mathcal{P}}\sigma^{-k_{\sigma}})\rightarrow W(M(x))\rightarrow
W(\mathrm{det}(M(x))\delta^{-1}_{Y(x)}\prod_{\sigma\in \mathcal{P}}\sigma^{k_{\sigma}})\rightarrow 0,$$

where, for any $\lambda\in E(x)^{\times}$, we define a homomorphism $\delta_{\lambda}:K^{\times}\rightarrow E(x)^{\times}$ such
that $\delta_{\lambda}(\pi_K)=\lambda$ and $\delta_{\lambda}|_{\mathcal{O}_K^{\times}}$ is trivial.
\end{itemize}

\end{prop}
\begin{proof}
First, we prove (1).
Let's consider a point $x\in \mathrm{Spm}(R)$ such that $x\in \mathrm{Spm}(R)_{Q_{\sigma}(i)}$ for any
$\sigma\in \mathcal{P}$ and any $i\in\mathbb{Z}_{\leqq 0}$. Then, by the definition of $X_{fs}$, any $G_K$-map
$h:M^{\vee}_R\rightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\otimes_{K,\sigma}\mathcal{O}_{X,x}/\mathfrak{m}_x^n$
factor through $M^{\vee}_R\rightarrow (B^+_{\mathrm{max},K}\otimes_{K,\sigma}\mathcal{O}_{X,x}/\mathfrak{m}_x^n)^{\varphi_K=Y}$
for any $n\geqq 1$.
We denote by $V\subseteq \mathrm{Spm}(R)$ the set of points satisfying the above condition.
By the same argument as in Lemma \ref{5.11},  it suffices to show that the natural
map $R\rightarrow \prod_{x\in V,n\geqq 1}\mathcal{O}_{X,x}/\mathfrak{m}_{x}^n$ is injection.
Let $f\in R$ be an element in the kernel of this map. Let $W\subseteq \mathrm{Spm}(R)$ be
the support of $f$ with the reduced structure,  then we have $W \subseteq \cup_{\sigma\in \mathcal{P},i\leqq 0}
V(Q_{\sigma}(i))$, hence there exists $Q\in R$ a finite product of $Q_{\sigma}(i)$ such that
$W\subseteq V(Q)$ by Lemma 5.7 of \cite{Ki03}. Hence we have  $X_Q\subseteq X\setminus W\subseteq X$. This implies that $f=0\in R[\frac{1}{Q}]$ and then
the condition (1) of Theorem \ref{5.8} implies that $f=0$ in $R$. Hence, the map $R\rightarrow \prod_{x\in V,n\geqq 1}\mathcal{O}_{X,x}/\mathfrak{m}_{x}^n$ is injection.

Next we prove (2).
Let $x\in \mathrm{Spm}(R)$ such that $x\in \mathrm{Spm}(R)_{Q_{\sigma}(i)}$ for any
$\sigma\in \mathcal{P}$ and $i\in\mathbb{Z}_{\leqq 0}$. Then, for any quotient $R'$ of
$\mathcal{O}_{X,x}/\mathfrak{m}_x^n$ and for any $\sigma\in \mathcal{P}$, we have an isomorphism $$(B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}
M_R)^{G_K}\otimes_RR'\isom
(B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\otimes_{K,\sigma}(M_R\otimes_R R'))^{G_K}$$ and this is a free $R'$-module of rank one
by Corollary 2.6 of \cite{Ki03}.
If $H_{\sigma}\mathcal{O}_{X,x}\not=\mathcal{O}_{X,x}$, then
we have $(B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}M_R)^{G_K}\otimes_R\mathcal{O}_{X,x}/H_{\sigma}\mathcal{O}_{X,x}=0$  by the definition of $H_{\sigma}$, this is a contradiction. Hence, we have $H_{\sigma}\mathcal{O}_{X,x}=
\mathcal{O}_{X,x}$ for any $\sigma$ and for such a point $x$, hence we also have $H\mathcal{O}_{X,x}=\mathcal{O}_{X,x}$. This implies that $V(H)\subseteq \cup_{\sigma\in \mathcal{P},i\leqq 0}V(Q_{\sigma}(i))$. Hence,
there exists $Q\in R$ a finite product of $Q_{\sigma}(i)$ such that $\mathrm{Spm}(R)_Q\subseteq \mathrm{Spm}(R)\setminus V(H)
\subseteq \mathrm{Spm}(R)$ by Lemma 5.7
of \cite{Ki03}. Because $\mathrm{Spm}(R)_{Q}$ is scheme theoretically dense, so
$\mathrm{Spm}(R)\setminus V(H)$ is also scheme theoretically dense. Because $V(H)\subseteq V(H_{\sigma})$,
$\mathrm{Spm}(R)\setminus V(H_{\sigma})$ is also scheme theoretically dense.

Finally we prove (3). Let $x$ be any point of $\mathrm{Spm}(R)$. By (2), for any $\sigma\in \mathcal{P}$, there exists $n_{\sigma}\geqq 0$ such that $H_{\sigma}\subseteq \mathfrak{m}_x^{n_{\sigma}}$ and
$H_{\sigma}\not\subseteq \mathfrak{m}_x^{n_{\sigma}+1}$. By the definition of $H_{\sigma}$, there exists a $G_K$-map $h:M^{\vee}_R\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}H_{\sigma})^{\varphi_K=Y}
$ which, by composing with
$B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}H_{\sigma}\rightarrow B^+_{\mathrm{max},K}\otimes_{K,\sigma}\mathfrak{m}_x^{n_{\sigma}}/\mathfrak{m}_x^{n_{\sigma}+1}$, induces a nonzero map $M^{\vee}_R\rightarrow (B^+_{\mathrm{max},K}\otimes_{K,\sigma}\mathfrak{m}_x^{n_{\sigma}}/\mathfrak{m}_x^{n_{\sigma}+1})^{\varphi_K=Y(x)}$. Hence, by taking a
suitable $E(x)$-linear projection $\mathfrak{m}_x^{n_{\sigma}}/\mathfrak{m}_x^{n_{\sigma}+1}\rightarrow E(x)$, we obtain
a non zero $G_K$-map $M^{\vee}_R\rightarrow (B^+_{\mathrm{max},K}\otimes_{K,\sigma}E(x))^{\varphi_K=Y(x)}$.
This implies that $(B^+_{\mathrm{max},K}\otimes_{K,\sigma}M(x))^{\varphi_K=Y(x)}\not= 0$, hence
this also implies that $D^+_{\mathrm{cris}}(M(x))^{\varphi_K=Y(x)}\not= 0$, then $M(x)$ is a split trianguline
$E(x)$-representation as in the statement of (3).

\end{proof}

\subsection{Construction of $p$-adic families of two dimensional trianguline representations for any $p$-adic fields}

In this subsection, we will apply the results of $\S$ 3.2 to the rigid analytic space associated with
a universal deformation ring of mod $p$ Galois representation.

Let $\mathcal{C}_{\mathcal{O}}$ be the category of local Artin $\mathcal{O}$-algebras with the residue field $\mathbb{F}$.
Let $\bar{\rho}:G_K\rightarrow \mathrm{GL}_2(\mathbb{F})$ be a continuous homomorphism, we denote by $\overline{V}$ a two dimensional $\mathbb{F}$-representation defined by $\bar{\rho}$.
 As in the case of $E$-representations, we define a functor
 $D_{\bar{\rho}}:\mathcal{C}_{\mathcal{O}}\rightarrow Sets$ by
 $$D_{\bar{\rho}}(A):=\{\text{ equivalent classes of deformations of }\overline{V}\text{ over }A \}$$
  for any $A\in\mathcal{C}_{\mathcal{O}}$.
 In this paper, for simplicity, we assume that $\overline{V}$ satisfies that $$\mathrm{H}^0(G_K,\mathrm{ad}(\overline{V}))=\mathbb{F}.$$
 Then, $D_{\bar{\rho}}$ is pro-representable by a complete noetherian local $\mathcal{O}$-algebra $R_{\bar{\rho}}$ with the residue
 field $\mathbb{F}$. $R_{\bar{\rho}}\isom \varprojlim_n R_{\bar{\rho}}/\mathfrak{m}^n$ is equipped with the projective limit topology, where
 $R_{\bar{\rho}}/\mathfrak{m}^n$ is equipped with the discrete topology for any $n\in\mathbb{Z}_{\geqq 1}$.
 When $\overline{V}$ does not satisfy $\mathrm{H}^0(G_K,\mathrm{ad}(\overline{V}))=\mathbb{F}$, we can prove
 the same theorems below in the same way if we consider the framed deformations.
  Let $V^{\mathrm{univ}}$ be the universal deformation over $R_{\bar{\rho}}$, which is a rank two free $R_{\bar{\rho}}$-module with a $R_{\bar{\rho}}$-linear continuous
  $G_K$-action.
 Let $\mathfrak{X}(\bar{\rho})$ be the rigid analytic space over $E$ associated
 to $R_{\bar{\rho}}$. Let $\tilde{V}^{\mathrm{univ}}$ be a free $\mathcal{O}_{\mathfrak{X}(\bar{\rho})}$-module associated to $V^{\mathrm{univ}}$, which is
 naturally equipped with an $\mathcal{O}_{\mathfrak{X}(\bar{\rho})}$-linear continuous $G_K$-action induced from that on $V^{\mathrm{univ}}$, where ``continuous" means that $G_K$ acts continuously on $\Gamma(U, \tilde{V}^{\mathrm{univ}})$ for any
 affinoid opens $U=\mathrm{Spm}(R)\subseteq \mathfrak{X}(\bar{\rho})$.
 \begin{rem}\label{a1}
 For a point $x\in \mathfrak{X}(\bar{\rho})$, the fiber $V_x$ of $\tilde{V}^{\mathrm{univ}}$ at $x$ is a  two dimensional
 $E(x)$-representation such that a mod $p$ reduction is isomorphic to
 $\bar{\rho}$ for a $G_K$-stable $\mathcal{O}_{E(x)}$-lattice of $V_x$. Because we assume that $\mathrm{End}_{\mathbb{F}[G_K]}(\bar{\rho})=\mathbb{F}$,
  we also have
 $\mathrm{End}_{E(x)[G_K]}(V_x)=E(x)$ for any $x\in \mathfrak{X}(\bar{\rho})$.
 \end{rem}
 Let $\mathcal{W}_E$ be the rigid analytic space over $E$ which represents the functor $D_{\mathcal{W}_E}$ (from
 the category of rigid analytic spaces over $E$ to the category of groups) defined by, for any
 rigid analytic space $Y$ over $E$,
 $$D_{\mathcal{W}_E}(Y):=\{\delta: \mathcal{O}_K^{\times}\rightarrow \Gamma(Y,\mathcal{O}_Y^{\times}) \text{ continuous homomorphisms }\},$$ where ``continuous" is the same
 meaning as in the definition of $\tilde{V}^{\mathrm{univ}}$.
 It is known that $\mathcal{W}_E$ is the rigid analytic space associated to the Iwasawa algebra $\mathcal{O}[[\mathcal{O}_{K}^{\times}]]$, which is non-canonically isomorphic to a finite (this number is equal to the number of torsion points in $\mathcal{O}_K^{\times}$) union of $[K:\mathbb{Q}_p]$-dimensional
 open unit disc over $E$. We denote by $$\delta^{\mathrm{univ}}_0:\mathcal{O}_K^{\times}\rightarrow \Gamma(\mathcal{W}_E,\mathcal{O}_{\mathcal{W}_E}^{\times})$$ the universal
 continuous homomorphism, which is the composition of the natural maps $\mathcal{O}_K^{\times}\rightarrow
 \mathcal{O}[[\mathcal{O}_K^{\times}]]^{\times}:a\mapsto [a]$ and $\mathcal{O}[[\mathcal{O}_K^{\times}]]^{\times}\rightarrow \Gamma(\mathcal{W}_E, \mathcal{O}_{\mathcal{W}_E}^{\times})$. Using a fixed $\pi_K$, we extend $\delta^{\mathrm{univ}}_0$ to $K^{\times}$ by
 $$\delta^{\mathrm{univ}}:K^{\times}\rightarrow \Gamma(\mathcal{W}_E, \mathcal{O}_{\mathcal{W}_E}^{\times})\text{ such
 that }\delta^{\mathrm{univ}}(\pi_K)=1, \, \delta^{\mathrm{univ}}|_{\mathcal{O}_K^{\times}}=\delta^{\mathrm{univ}}_0.$$ By local class field theory, we can uniquely extend $\delta^{\mathrm{univ}}$ to a character
 $$\tilde{\delta}^{\mathrm{univ}}:G^{\mathrm{ab}}_K\rightarrow \Gamma(\mathcal{W}_E, \mathcal{O}_{\mathcal{W}_E}^{\times})\text{ such that }\delta^{\mathrm{univ}}=\tilde{\delta}^{\mathrm{univ}}\circ \mathrm{rec}_K.$$
 We denote by
 $$X(\bar{\rho}):=\mathfrak{X}(\bar{\rho})\times_E \mathcal{W}_E\times_E\mathbb{G}^{an}_{m,E}.$$
 Let $Y$ be the canonical parameter of $\mathbb{G}^{an}_{m,E}$.
 We denote the projections by $$p_1:X(\bar{\rho})\rightarrow \mathfrak{X}(\bar{\rho}),\,
 p_2:X(\bar{\rho})\rightarrow \mathcal{W}_E,\, p_3:X(\bar{\rho})\rightarrow \mathbb{G}^{an}_{m,E}.$$
 We denote by  $N:=p_1^*\tilde{V}^{\mathrm{univ}}$ and denote by  $M:=N((p_2^*\tilde{\delta}^{\mathrm{univ}})^{-1})$.
 These are rank two free $\mathcal{O}_{X(\bar{\rho})}$-modules with $\mathcal{O}_{X(\bar{\rho})}$-linear continuous $G_K$-actions.
 A point $x$ of $X(\bar{\rho})$ can be written as a triple $x=([V_x],\delta_x,\lambda_x)$ where,
 $V_x$ is an $E(x)$-representation such that the mod $p$-reduction of a suitable $G_K$-stable
 $\mathcal{O}_{E(x)}$-lattice of $V_x$ is isomorphic to $\bar{V}$ (after scalar extension) and $\delta_x:\mathcal{O}_K^{\times}\rightarrow
 E(x)^{\times}$ is a continuous homomorphism and $\lambda_x\in E(x)^{\times}$ (where $E(x)$ is the residue field of $x$), i.e. in this paper, if we write $x=([V_x],\delta_x,\lambda_x)$ then we assume that these are defined over $E(x)$.
 We denote by
 $$P_M(T)=(P_M(T)_{\sigma})_{\sigma\in \mathcal{P}}=(T^2-a_{1,\sigma}T+a_{0,\sigma})_{\sigma\in \mathcal{P}}
 \in K\otimes_{\mathbb{Q}_p}\mathcal{O}_{X(\bar{\rho})}[T]$$ the Sen's polynomial
 of $M$. We denote by $X_0(\bar{\rho})\subseteq X(\bar{\rho})$ the Zariski closed subspace defined by the ideal generated by
 $a_{0,\sigma}$ for all $\sigma\in\mathcal{P}$. Let $M_0:=M|_{X_0(\bar{\rho})}$ be the restriction of $M$ to $X_0(\bar{\rho})$,
 then we have $$P_{M_0}(T)=(T(T-a_{1,\sigma}))_{\sigma\in \mathcal{P}}\in K\otimes_{\mathbb{Q}_p}\mathcal{O}_{X_0(\bar{\rho})}[T].$$
 We denote by $Q_{\sigma}(T):=T-a_{1,\sigma}\in\mathcal{O}_{X_0(\bar{\rho})}[T]$ for any $\sigma\in\mathcal{P}$.
 Under this situation, we apply Theorem \ref{5.8} to $X_0(\bar{\rho})$ and $M_0$ and $Y:=(p_3^*Y)|_{X_0(\bar{\rho})}$, then we obtain a Zariski closed
 subspace
 $$\mathcal{E}(\bar{\rho}):=X_0(\bar{\rho})_{fs}\subseteq X_0(\bar{\rho}).$$
 For this $\mathcal{E}(\bar{\rho})$, we have a following theorem, which is a modified version of Proposition 10.4 of \cite{Ki03}.
 For any $\lambda\in \overline{E}^{\times}$, we define a unramified continuous homomorphism
 $\delta_{\lambda}:K^{\times}\rightarrow \overline{E}^{\times}$ such that $\delta_{\lambda}(\pi_K):=\lambda$ and
 $\delta_{\lambda}|_{\mathcal{O}_K^{\times}}$ is trivial. For a point $\delta\in \mathcal{W}_E(\overline{E})$, i.e. for a
 continuous homomorphism $\delta:\mathcal{O}_K^{\times}\rightarrow \overline{E}^{\times}$, we denote by the same letter
 $\delta:K^{\times}\rightarrow \overline{E}^{\times}$ such that $\delta(\pi_K)=1$ and $\delta|_{\mathcal{O}_K^{\times}}=\delta$.

 \begin{thm}\label{5.14}
 $\mathcal{E}(\bar{\rho})$ has the following properties.
 \begin{itemize}
 \item[(1)]For any point $x:=([V_x],\delta_x,\lambda_x)\in \mathcal{E}(\bar{\rho})$,
 there exists $\{k_{\sigma}\}_{\sigma\in \mathcal{P}}\in \prod_{\sigma\in \mathcal{P}}\mathbb{Z}_{\geqq 0}$
 and, if we put $\delta_1:=\delta_x\delta_{\lambda_x}\prod_{\sigma\in \mathcal{P}}\sigma^{-k_{\sigma}}$,
  there exists a short exact sequence of $E(x)$-$B$-pairs.
 $$0\rightarrow W(\delta_1)\rightarrow W(V_x)\rightarrow W(\mathrm{det}(V_x)\delta_1^{-1})\rightarrow 0.$$
 \item[(2)]Conversely, if a point $x:=([V_x],\delta_x,\lambda_x)\in X(\bar{\rho})$ satisfies the following conditions (i) and (ii), then
 $x\in \mathcal{E}(\bar{\rho})$.
 \begin{itemize}
 \item[(i)] $V_x$ is a split trianguline $E(x)$-representation
  with a triangulation
  $$\mathcal{T}_x:0\subseteq W(\delta_x\delta_{\lambda_x})\subseteq W(V_x).$$

 \item[(ii)]$(V_x,\mathcal{T}_x)$ satisfies all the assumptions in Proposition $\mathrm{\ref{19}}$.

 \end{itemize}

\end{itemize}
\end{thm}
\begin{proof}
(1) follows from (3) of Proposition \ref{5.13}.

We prove (2). Extending the scalar from $E$ to $E(x)$, we may assume that $E(x)=E$. Let $x:=([V_x],\delta_x,\lambda_x)\in X(\bar{\rho})$ be an $E$-rational  point satisfying all the conditions in (2),

then the trianguline deformation functor $D_{V_x,\mathcal{T}_x}$ is represented by
a formally smooth quotient $R_{V_x,\mathcal{T}_x}$ of the universal deformation ring $R_{V_x}$ of $V_x$ by Proposition \ref{19}. Moreover, we have a canonical isomorphism $R_{V_x}\isom \widehat{\mathcal{O}}_{\mathfrak{X}(\bar{\rho}),p_1(x)}$
and then $V_x^{\mathrm{univ}}:=\tilde{V}^{\mathrm{univ}}\otimes_{\mathcal{O}_{\mathfrak{X}(\bar{\rho})}}\widehat{\mathcal{O}}_{\mathfrak{X}(\bar{\rho}),p_1(x)}$ is
the universal deformation of $V_x$ by Proposition 9.5 of \cite{Ki03}. Taking a quotient, we have a map
$\widehat{\mathcal{O}}_{X(\bar{\rho}),x}
\isom \widehat{\mathcal{O}}_{\mathfrak{X}(\bar{\rho}),p_1(x)}\hat{\otimes}_{E}\widehat{\mathcal{O}}_{\mathcal{W}_E,p_2(x)}\hat{\otimes}_{E}
\widehat{\mathcal{O}}_{\mathbb{G}^{an}_{m,E},p_3(x)}\rightarrow R_{V_x,\mathcal{T}_x}\hat{\otimes}_{E}\widehat{\mathcal{O}}_{\mathcal{W}_E,p_2(x)}\hat{\otimes}_{E}\widehat{\mathcal{O}}_{\mathbb{G}^{an}_{m,E},p_3(x)}.$ By the definition of $R_{V_x,\mathcal{T}_x}$, there exists
a continuous homomorphism $\delta_{\mathcal{T}_x}:K^{\times}\rightarrow R^{\times}_{V_x,\mathcal{T}_x}$ which gives the universal
triangulation, i.e. we have the following triangulations of $V_x^{\mathrm{univ}}\otimes_{R_{V_x}}R_{V_x,\mathcal{T}_x}/\mathfrak{m}^n$,
$$\mathcal{T}_{\mathrm{univ},n}: 0\subseteq W(\delta_{\mathcal{T}_x,n})\subseteq W(V_x^{\mathrm{univ}}\otimes_{R_{V_x}}R_{V_x,\mathcal{T}_x}/\mathfrak{m}^n)$$ where $\mathfrak{m}$ is the maximal ideal of $R_{V_x,\mathcal{T}_x}$ and $\delta_{\mathcal{T}_x,n}$ is the composition of
$\delta_{\mathcal{T}_x}$ with the natural quotient map $R_{V_x,\mathcal{T}_x}\rightarrow R_{V_x,\mathcal{T}_x}/\mathfrak{m}^n$ for any
$n\geqq 1$. We put $\lambda_{\mathcal{T}_x}:=\delta_{\mathcal{T}_x}(\pi_K)\in R_{V_x,\mathcal{T}_x}^{\times}$.
On the other hand, we denote the composition
of the universal
homomorphism $\delta_0^{\mathrm{univ}}:\mathcal{O}_K^{\times}\rightarrow \mathcal{O}_{\mathcal{W}_E}^{\times}$ with
$\mathcal{O}^{\times}_{\mathcal{W}_E}\rightarrow \widehat{\mathcal{O}}_{\mathcal{W}_E,p_2(x)}^{\times}$ by $\delta^{\mathrm{univ}}_{p_2(x)}:\mathcal{O}_K^{\times}\rightarrow \widehat{\mathcal{O}}_{\mathcal{W}_E,p_2(x)}^{\times}$. The $E$-algebra
$\widehat{\mathcal{O}}_{\mathcal{W}_E,p_2(x)}$ is topologically generated by $\{\delta^{\mathrm{univ}}_{p_2(x)}(a)-\delta_x(a) |a\in\mathcal{O}_K^{\times} \}$.
We take a quotient $\bar{R}$ of $R_{V_x,\mathcal{T}_x}\hat{\otimes}_{E}\widehat{\mathcal{O}}_{\mathcal{W}_E,p_2(x)}\hat{\otimes}_{E}
\widehat{\mathcal{O}}_{\mathbb{G}^{an}_{m,E},p_3(x)}$ by the ideal generated by $\delta_{\mathcal{T}_x}(a)\otimes 1\otimes 1-1\otimes\delta^{\mathrm{univ}}_{p_2(x)}(a)\otimes 1$ (any $a\in \mathcal{O}_K^{\times}$)
 and $\lambda_{\mathcal{T}_x}\otimes 1\otimes 1-1\otimes 1\otimes Y$, where $Y$ is the canonical coordinate of $\mathbb{G}^{an}_{m,E}$. Then, we can see that
the composition of  $R_{V_x,\mathcal{T}_x}\rightarrow R_{V_x,\mathcal{T}_x}\hat{\otimes}_{E(x)}\widehat{\mathcal{O}}_{\mathcal{W}_E,p_2(x)}
\hat{\otimes}\widehat{\mathcal{O}}_{\mathbb{G}^{an}_{m,E},E(x)}: z\mapsto z\otimes 1\otimes 1$ with the natural quotient map
$R_{V_x,\mathcal{T}_x}\hat{\otimes}_{E(x)}\widehat{\mathcal{O}}_{\mathcal{W}_E,p_2(x)}
\hat{\otimes}_{E(x)}\widehat{\mathcal{O}}_{\mathbb{G}^{an}_{m,E},p_3(x)}\rightarrow \bar{R}$ is an isomorphism
$R_{V_x,\mathcal{T}_x}\isom \bar{R}$ and, if we denote by $\bar{\delta}^{\mathrm{univ}}_{p_2(x)}:\mathcal{O}_K^{\times}\rightarrow \bar{R}^{\times}$
and $\bar{Y}\in \bar{R}^{\times}$ the reduction of $1\otimes \delta^{\mathrm{univ}}_{p_2(x)}\otimes 1$ and $1\otimes 1\otimes Y$, then the universal triangulation on $R_{V_x,\mathcal{T}_x}$
is transformed to (we drop the notation $n\in \mathbb{Z}_{\geqq 1}$)
$$0\subseteq W(\bar{\delta}^{\mathrm{univ}}_{p_2(x)}\delta_{\bar{Y}})\subseteq W((p_1^*\tilde{V}^{\mathrm{univ}})\otimes_{\mathcal{O}_{X(\bar{\rho})}}\bar{R}).$$

We put $V_{\bar{R}}:=(p_1^*\tilde{V}^{\mathrm{univ}})\otimes_{\mathcal{O}_{X(\bar{\rho})}}\bar{R}$ and put
 $\bar{R}_n:=\bar{R}/\mathfrak{m}^n$ and $V_{\bar{R}_n}:=V_{\bar{R}}\otimes_{\bar{R}}\bar{R}_n$.
Under this situation, first we claim that  the natural map $\mathrm{Spm}(\bar{R}_n)\rightarrow X(\bar{\rho})$ factors through
$X_0(\bar{\rho})$ for any $n\geqq 1$. This follows immediately  from
the facts that $W(V_{\bar{R}_n}(\tilde{\bar{\delta}}^{\mathrm{univ} -1}_{p_2(x)}))$ has a triangulation $0\subseteq W(\bar{\delta}_{\bar{Y}_n})\subseteq
W(V_{\bar{R}_n}(\tilde{\bar{\delta}}_{p_2(x)}^{\mathrm{univ} -1}))$ and that $W(\bar{\delta}_{\bar{Y}_n})$ is crystalline with Hodge-Tate weight zero, where $\bar{Y}_n\in \bar{R}_n$ is the reduction of $\bar{Y}$. From this,  $D_{\mathrm{cris}}(W(\bar{\delta}_{\bar{Y}_n}))=D_{\mathrm{cris}}(W(\bar{\delta}_{\bar{Y}_n}))^{\varphi_K=\bar{Y}_n}\cap \mathrm{Fil}^0D_{dR}(W(\bar{\delta}_{\bar{Y}_n}))$ is a $\varphi$-stable
free rank one $K_0\otimes_{\mathbb{Q}_p}\bar{R}_n$ sub module of $D_{\mathrm{cris}}(V_{\bar{R}_n}(\tilde{\bar{\delta}}_{p_2(x)}^{\mathrm{univ} -1}))^{\varphi_K=\bar{Y}_n}$ which is contained in $\mathrm{Fil}^0D_{\mathrm{dR}}(V_{\bar{R}_n}(\tilde{\bar{\delta}}_{p_2(x)}^{\mathrm{univ} -1}))$. Hence, by Lemma \ref{5.7.5}, we have a natural inclusion
$$D_{\mathrm{cris}}(W(\delta_{\bar{Y}_n})) \subseteq D^+_{\mathrm{cris}}
(V_{\bar{R}_n}(\tilde{\bar{\delta}}_{p_2(x)}^{\mathrm{univ} -1}))^{\varphi_K=\bar{Y}_n}\,\, \cdots (1).$$

Next, we take an affinoid open $\mathrm{Spm}(R)\subseteq X_0(\bar{\rho})$ which
contains $x$ and
satisfies the condition in the construction of $X_{fs}$ ( see the paragraph after the proof of Lemma \ref{5.11}). Let $J$ be
the ideal of $R$ which determines $\mathrm{Spm}(R)_{fs}$. We claim that the natural map $R\rightarrow \bar{R}$ factors through
$R/J\rightarrow \bar{R}$, which proves that $x\in \mathcal{E}(\bar{\rho})(E)$ because $x$ is the point determined by
the map $R\rightarrow \bar{R}\rightarrow \bar{R}/\mathfrak{m}$.
By construction of $J$, it suffices to show the following lemma.
\end{proof}

\begin{lemma}\label{5.15}
In the above situation, the following hold:
\begin{itemize}
\item[(i)]For any $k\geqq 1$ and for any $\sigma\in \mathcal{P}$, the natural map
$$\varprojlim_n(B^+_{\mathrm{max},K}\otimes_{K,\sigma}V_{\bar{R}_n}(\tilde{\bar{\delta}}_{p_2(x)}^{\mathrm{univ} -1}))^{\varphi_K=\bar{Y},G_K}\rightarrow
\varprojlim_n(B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\otimes_{K,\sigma}V_{\bar{R}_n}(\tilde{\bar{\delta}}_{p_2(x)}^{\mathrm{univ} -1}))^{G_K}$$ is surjection.
\item[(ii)]For any $\sigma\in \mathcal{P}$ and $i\in\mathbb{Z}_{\leqq 0}$, $Q_{\sigma}(i)$ is nonzero in $\bar{R}$.
\end{itemize}
\end{lemma}
\begin{proof}
Because $\bar{R}\isom R_{V_x,\mathcal{T}_x}$ is domain, (i) follows from (ii) and from the above inclusion (1) by the same argument as in the proof of Proposition 2.8 of \cite{Ki03}.
We prove (ii). We can write $Q_{\sigma}(T)=T-\bar{a}_{1,\sigma}$ for the reduction $\bar{a}_{1,\sigma}\in\bar{R}$
of $a_{1,\sigma}\in \mathcal{O}_{X(\bar{\rho})}$.
Because, for any $n\in\mathbb{Z}_{\geqq 1}$, $V_{\bar{R}_n}(\tilde{\bar{\delta}}^{\mathrm{univ} -1}_{p_2(x)})$ has a triangulation $0\subseteq W(\delta_{\bar{Y}_n})\subseteq
W(V_{\bar{R}_n}(\tilde{\bar{\delta}}^{\mathrm{univ} -1}_{p_2(x)}))$, $\bar{a}_{1,\sigma}\in \bar{R}_n$ is the $\sigma$-part of the Hodge-Tate weight
of $\mathrm{det}(V_{\bar{R}_n})(\tilde{\bar{\delta}}_{p_2(x)}^{\mathrm{univ}-2})$. By Lemma \ref{5.16} below, if we denote by $\delta_0:=(\mathrm{det}(V_x)|_{\mathcal{O}_K^{\times}})/\delta_x^2:\mathcal{O}_K^{\times}\rightarrow E^{\times}$, $\bar{a}_{1,\sigma}\in \bar{R}$ is the image of
the $\sigma$-part of Hodge-Tate weight $a^{\mathrm{univ}}_{\sigma}\in R_{\delta_0}$
of the universal deformation $\delta_{\delta_0}^{\mathrm{univ}}:\mathcal{O}_K^{\times}
\rightarrow R_{\delta_0}^{\times}$ by the injection $R_{\delta_0}\hookrightarrow R_{V_x,\mathcal{T}_x}\isom \bar{R}$ induced by a morphism
$f:D_{V_x,\mathcal{T}_x}\rightarrow D_{\delta_0}$ defined below, where the
injection follows from Lemma \ref{5.16} below. Hence, for any $i\in\mathbb{Z}_{\leqq 0}$,
$Q_{\sigma}(i)=(i-\bar{a}_{1,\sigma})\not=0\in\bar{R}$ by
Lemma \ref{5.17} below.
\end{proof}
Let $\delta_0:\mathcal{O}_K^{\times}\rightarrow E^{\times}$ be a continuous homomorphism.
We define a functor $D_{\delta_0}:\mathcal{C}_E\rightarrow Sets$ by
$$D_{\delta_0}(A):=\{\delta_A:\mathcal{O}_K^{\times}\rightarrow A^{\times}:\text{ continuous homomorphisms }\delta_A \text{ (mod } \mathfrak{m}_A)=\delta_0\}.$$
 It is easy to show that this functor is
pro-representable by a ring $R_{\delta_0}$ which is isomorphic to $E[[T_1,T_2,\cdots,T_d]]$ where $d:=[K:\mathbb{Q}_p]$.
Let $W$ be a rank two split trianguline $E$-$B$-pair with a triangulation
$\mathcal{T}:0\subseteq W(\delta_1)\subseteq W$ such that $W/W(\delta_1)\isom W(\delta_2)$ for some
continuous homomorphisms $\delta_1,\delta_2:K^{\times}\rightarrow E^{\times}$.
We put $\delta_0:=(\delta_2/\delta_1)|_{\mathcal{O}_K^{\times}}$.
We define a morphism of functors $f:D_{W,\mathcal{T}}\rightarrow D_{\delta_0}$ as follows.
Let $[(W_A,\psi_A,\mathcal{T}_A)]\in D_{W,\mathcal{T}}(A)$ be an equivalent class of trianguline deformation
of $(W,\mathcal{T})$ over $A$ with a triangulation $\mathcal{T}_A:0\subseteq W(\delta_{1,A})\subseteq W_A$ such that
$W_A/W(\delta_{1,A})\isom W(\delta_{2,A})$, then we define $f$ by
$$f([(W_A,\psi_A,\mathcal{T}_A]):=
(\delta_{2,A}/\delta_{1,A})|_{\mathcal{O}_K^{\times}}\in D_{\delta_0}(A).$$

 \begin{lemma}\label{5.16}
Let $W$ be a two dimensional split trianguline $E$-$B$-pair  with a triangulation
$\mathcal{T}:0\subseteq W(\delta_1)\subseteq W$ such that $W/W(\delta_1)\isom W(\delta_2)$.
If  $\mathrm{H}^2(G_K, W(\delta_1/\delta_2))\allowbreak=0$, then the morphism of functors
$f:D_{W,\mathcal{T}}\rightarrow D_{\delta_0}$ defined above is formally smooth.
\end{lemma}
\begin{proof}
Let $A\in \mathcal{C}_E$ and $I$ be an ideal of $A$ such that $I\mathfrak{m}_A=0$. If we are given
$[(W_{A/I},\psi_{A/I},\mathcal{T}_{A/I})]\in D_{W,\mathcal{T}}(A/I)$ such that
$\mathcal{T}_{A/I}:0\subseteq W(\delta_{1,A/I})\subseteq W_{A/I}$ and $W_{A/I}/W(\delta_{1,A/I})\isom W(\delta_{2,A/I})$
and are given $\delta_A\in D_{\delta_0}(A)$ such that $\delta_A\otimes_A A/I=(\delta_{2,A/I}/\delta_{1,A/I})|_{\mathcal{O}_K^{\times}}$.
Because $D_{\delta_0}$ is formally smooth for any $\delta_0$, there exists
a $\delta_{1,A}:K^{\times}\rightarrow A^{\times}$ such that $\delta_{1,A}\otimes_A A/I=\delta_{1,A/I}$.
We take a lift $\lambda\in A^{\times}$ of $\delta_{2,A/I}(\pi_K)\in (A/I)^{\times}$ and define
$\delta_{2,A}:K^{\times}\rightarrow A^{\times}$ by $\delta_{2,A}(\pi_K)=\lambda$ and $\delta_{2,A}|_{\mathcal{O}_K^{\times}}
=\delta_A\delta_{1,A}|_{\mathcal{O}_K^{\times}}$, then we have a short exact sequence of $E$-$B$-pairs
$$0\rightarrow W(\delta_1/\delta_2)\otimes_E I\rightarrow W(\delta_{1,A}/\delta_{2,A})\rightarrow
W(\delta_{1,A/I}/\delta_{2,A/I})\rightarrow 0.$$
Because $\mathrm{H}^2(G_K, W(\delta_1/\delta_2))=0$ by the assumption,
 we have a surjection $\mathrm{H}^1(G_K, W(\allowbreak \delta_{1,A}/\delta_{2,A}))\rightarrow
\mathrm{H}^1(G_K, W(\delta_{1,A/I}/\delta_{2,A/I}))$. This surjection means that there exists
a $[(W_A,\psi_A,\mathcal{T}_A)]\in D_{W,\mathcal{T}}(A)$ which is a lift of $[(W_{A/I},\psi_{A/I},\mathcal{T}_{A/I})]\in
D_{W,\mathcal{T}}(A/I)$
satisfying that $f([W_A,\psi_A,\mathcal{T}_A])=(\delta_{2,A}/\delta_{1,A})|_{\mathcal{O}_K^{\times}}=\delta_A$.
We finish the proof of lemma.
\end{proof}
Let $A\in \mathcal{C}_E$ and $\delta:\mathcal{O}_K^{\times}\rightarrow A^{\times}$ be a continuous
homomorphism, then it is known that this is locally $\mathbb{Q}_p$-analytic by Proposition 8.3 of \cite{Bu07}. Then, for any $\sigma\in \mathcal{P}$,
we define the $\sigma$-component of Hodge-Tate weight of $\delta$ by $\frac{\partial\delta(x)}{\partial \sigma(x)}|_{x=1}\in A$, which
is equal to the $\sigma$-part of Hodge-Tate weight of $A(\tilde{\delta})$ where $\tilde{\delta}:G^{\mathrm{ab}}_K\rightarrow A^{\times}$ is
any characters such that $\tilde{\delta}\circ\mathrm{rec}_K|_{\mathcal{O}_K^{\times}}=\delta$ by Proposition 3.3 of \cite{Na11}.
\begin{lemma}\label{5.17}
Let $\delta_0:\mathcal{O}_K^{\times}\rightarrow E^{\times}$ be a continuous homomorphism.
Let $R_{\delta_0}$  be the universal deformation ring of $D_{\delta_0}$.
Let $\delta_0^{\mathrm{univ}}:\mathcal{O}_K^{\times}\rightarrow R_{\delta_0}^{\times}$ be
the universal deformation of $\delta_0$. Then, for any $\sigma\in \mathcal{P}$,
the $\sigma$-part of Hodge-Tate weight $a^{\mathrm{univ}}_{\sigma}\in R_{\delta_0}$ of $\delta_0^{\mathrm{univ}}$ $($ i.e. the projective limit
of those of $\delta_0^{\mathrm{univ}}\otimes_{R_{\delta_0}}R_{\delta_0}/\mathfrak{m}^n$ for all $n\in\mathbb{Z}_{\geqq 1}$$)$ is not contained in
$E$, i.e. not constant.
\end{lemma}
\begin{proof}
Let $a:=\{a_{\sigma}\}_{\sigma\in \mathcal{P}}\in \prod_{\sigma\in \mathcal{P}}E$ be any element, then
we define a deformation of $\delta_0$ over $E[\varepsilon]$ by
$$\delta_a:\mathcal{O}_K^{\times}
\rightarrow E[\varepsilon]^{\times}: \delta_a(x):=\delta_0(x)(1+(\sum_{\sigma\in \mathcal{P}}a_{\sigma}
\mathrm{log}(\sigma(x)))\varepsilon).$$ The $\sigma$-part of Hodge-Tate weight
of $\delta_a$ is $\frac{\partial\delta_0(x)}{\partial\sigma(x)}|_{x=1}+a_{\sigma}\varepsilon$. The lemma follows from this.
\end{proof}
\begin{corollary}\label{5.16.5}
Let $x=[V_x]\in \mathfrak{X}(\bar{\rho})$ be a point such that $V_x$ is a crystabelline $E(x)$-trianguline representation satisfying the conditions
$(1)$ of Definition $\mathrm{\ref{26}}$.

Then, the point  $x_{\tau}:=([V_x],\delta_{\tau,1}|_{\mathcal{O}_K^{\times}},\delta_{\tau,1}(\pi_K))\in X(\bar{\rho})$ is
contained in $\mathcal{E}(\bar{\rho})$ for any $\tau\in \mathfrak{S}_2$, where
we denote the triangulation $\mathcal{T}_{\tau}$
by $0\subseteq W(\delta_{\tau,1})\subseteq W(V_x)$.

\end{corollary}
\begin{proof}
This follows from (2) of the above theorem and from Lemma \ref{21.5}.
\end{proof}

Next, we describe the local structure of $\mathcal{E}(\bar{\rho})$ at the points satisfying the
conditions (i), (ii) in (2) of Theorem \ref{5.14} by the universal trianguline deformation rings.

 We prove the following theorem, which is a generalization of Proposition 10.6 of \cite{Ki03}.
\begin{thm}\label{5.18}
Let $x:=([V_x],\delta_x,\lambda_x)\in \mathcal{E}(\bar{\rho})$ be a  point such that  the conditions (i), (ii) in $(2)$ of Theorem
$\mathrm{\ref{5.14}}$ hold.

Then, we have a canonical $E(x)$-algebra isomorphism
$$\widehat{\mathcal{O}}_{\mathcal{E}(\bar{\rho}),x}\isom R_{V_x,\mathcal{T}_x}.$$
In particular, $\mathcal{E}(\bar{\rho})$ is smooth of dimension $3[K:\mathbb{Q}_p] +1$ at $x$.
\end{thm}
\begin{proof}

We may assume that $E=E(x)$.
We have already  showed that
the natural map $\widehat{\mathcal{O}}_{X(\bar{\rho}),x}\rightarrow \bar{R}\isom R_{V_x,\mathcal{T}_x}$ factors through $\widehat{\mathcal{O}}_{\mathcal{E}(\bar{\rho}),x}\rightarrow
\bar{R}\isom R_{V_x,\mathcal{T}_x}$ in the proof of Theorem \ref{5.14}.

We prove the existence of the  inverse map $R_{V_x,\mathcal{T}_x}\isom \bar{R}\rightarrow \hat{\mathcal{O}}_{\mathcal{E}(\bar{\rho}),x}$.

Because $x$ is an $E$-rational point, then we can take a $Y$-small
affinoid neighborhood $\mathrm{Spm}(R)$ of $x$ in $\mathcal{E}(\bar{\rho})$. By Proposition \ref{5.7} and Proposition
\ref{5.13}, there exists
a sufficiently large $k>0$ such that, for any $\sigma\in\mathcal{P}$,
there exists a short exact sequence of Banach $R$-modules with property (Pr) $$0\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R)^{\varphi_K=Y} \rightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}R \rightarrow U_{k,\sigma}\rightarrow 0$$ and that we have a natural isomorphism $$K\otimes_{K_0}D^+_{\mathrm{cris}}(V_R(\tilde{\delta}^{\mathrm{univ}  -1}))^{\varphi_K =Y}\isom
(B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\allowbreak \hat{\otimes}_{\mathbb{Q}_p}V_R(\tilde{\delta}^{\mathrm{univ} -1}))^{G_K}$$ and that, for any $\sigma\in \mathcal{P}$, we have an ideal $H_{\sigma}\subseteq R$ defined in Proposition \ref{5.13} satisfying
that $\mathrm{Spm}(R)\setminus V(H)$ ($H:=\prod_{\sigma\in \mathcal{P}}H_{\sigma}$) and $\mathrm{Spm}(R)\setminus V(H_{\sigma})$ are scheme theoretically dense in $\mathrm{Spm}(R)$.

Under this situation,  we prove the existence of the inverse.
First, we claim that $D^+_{\mathrm{cris}}(V_x(\tilde{\delta}_x^{-1}))^{\varphi_K=\lambda_x}$ is a free $K_0\otimes_{\mathbb{Q}_p}E$-module of rank one. By the definition and by Lemma \ref{5.7.5}, this module has a sub module $D_{\mathrm{cris}}(W(\delta_{\lambda_x}))=D_{\mathrm{cris}}(W(\delta_{\lambda_x}))^{\varphi_K=\lambda_x}\cap \mathrm{Fil}^0D_{\mathrm{dR}}(W(\delta_{\lambda_x}))$ which
is rank one. Hence $D^+_{\mathrm{cris}}(V_x(\tilde{\delta}_x^{-1}))^{\varphi_K=\lambda_x}$ is rank one or two. If
this is rank two, then $V_x(\tilde{\delta}_x^{-1})$ is crystalline with Hodge-Tate weight $\{0,k_{\sigma}\}_{\sigma\in \mathcal{P}}$ such that
$k_{\sigma}\in \mathbb{Z}_{\leqq 0}$ for any $\sigma\in \mathcal{P}$ with unique relative Frobenius eigenvalue $\lambda_x$. But then, this implies that
$\delta_2/\delta_1=\prod_{\sigma\in \mathcal{P}}\sigma^{k_{\sigma}}$, this contradicts the assumption on $(V_x,\mathcal{T}_x)$.
Hence
$D^+_{\mathrm{cris}}(V_x(\tilde{\delta}_x^{-1}))^{\varphi_K=\lambda_x}$ is rank one and the inclusion $D_{\mathrm{cris}}(W(\delta_{\lambda_x}))\hookrightarrow D^+_{\mathrm{cris}}(V_x(\tilde{\delta}_x^{-1}))^{\varphi_K=\lambda_x}$ is isomorphism.

In the same way as in the proof of
Proposition 10.6 of \cite{Ki03}, we first take the blow up $\tilde{T}$ of  $\mathrm{Spm}(R)$ along $H$.

By the definition of blow up, for any point $\tilde{x}\in \tilde{T}$ above $x\in\mathrm{Spm}(R)$ and for any $\sigma\in \mathcal{P}$,
there exists an element
$f_{\sigma}\in H_{\sigma}$ such that $f_{\sigma}$ is a non zero divisor of $\widehat{\mathcal{O}}_{\tilde{T},\tilde{x}}$ and
$H_{\sigma}\widehat{\mathcal{O}}_{\tilde{T},\tilde{x}}=f_{\sigma}\widehat{\mathcal{O}}_{\tilde{T},\tilde{x}}$. By the definition of $H_{\sigma}$, for any $\sigma\in \mathcal{P}$ and for any $\tilde{x}\in \tilde{T}$ above $x$, there exists a $G_K$-equivariant map $V_R(\tilde{\delta}^{\mathrm{univ}  -1})^{\vee}\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R)^{\varphi_K=Y}$such that the composite with

\[
\begin{array}{ll}
(B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}R)^{\varphi_K=Y}&\rightarrow (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}
f_{\sigma}\widehat{\mathcal{O}}_{\tilde{T},\tilde{x}})^{\varphi_K=Y} \\
&\isom (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}\widehat{\mathcal{O}}_{\tilde{T},\tilde{x}})^{\varphi_K=Y}\rightarrow (B^+_{\mathrm{max},K}\otimes_{K,\sigma}E(\tilde{x}))^{\varphi_K=Y(\tilde{x})}
\end{array}
\]
is not zero, where the isom $(B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}f_{\sigma}\widehat{\mathcal{O}}_{\tilde{T},\tilde{x}})^{\varphi_K=Y}\isom (B^+_{\mathrm{max},K}\hat{\otimes}_{K,\sigma}\widehat{\mathcal{O}}_{\tilde{T},\tilde{x}} )^{\varphi_K=Y}$
is given by $a\mapsto \frac{a}{f_{\sigma}}$. Using this map and the fact that $D^+_{\mathrm{cris}}(V_x(\tilde{\delta}_x^{-1}))^{\varphi^f=\lambda_x}$ is rank one, by the induction on $n$ that $D^+_{\mathrm{cris}}(V_R(\tilde{\delta}^{\mathrm{univ} -1})\otimes_R
\widehat{\mathcal{O}}_{\tilde{T},\tilde{x}}/\mathfrak{m}^n_{\tilde{x}})^{\varphi^f=Y}$ is a free $K_0\otimes_{\mathbb{Q}_p}\widehat{\mathcal{O}}_{\tilde{T},\tilde{x}}/\mathfrak{m}^n_{\tilde{x}}$-module of rank one and the natural base change
map $$D^+_{\mathrm{cris}}(V_R(\tilde{\delta}^{\mathrm{univ} -1})\otimes_R
\widehat{\mathcal{O}}_{\tilde{T},\tilde{x}}/\mathfrak{m}^n_{\tilde{x}})^{\varphi^f=Y}\otimes_{\mathcal{O}_{\tilde{T},\tilde{x}}/\mathfrak{m}^n_{\tilde{x}}} E(\tilde{x})\isom D^+_{\mathrm{cris}}(V_x(\tilde{\delta}_x^{-1})\otimes_{E}E(\tilde{x}))^{\varphi_K=Y(x)}$$
is isomorphism. Because $\mathrm{Fil}^1(K\otimes_{K_0}D^+_{\mathrm{cris}}(V_x(\tilde{\delta}_x^{-1}))^{\varphi_K=Y(x)})
=\mathrm{Fil}^1D_{\mathrm{dR}}(W(\delta_{\lambda_x}))=0$, then $D^+_{\mathrm{cris}}(V_R(\tilde{\delta}^{\mathrm{univ} -1})\otimes_R
\widehat{\mathcal{O}}_{\tilde{T},\tilde{x}}/\mathfrak{m}^n_{\tilde{x}})^{\varphi^f=Y}$ is a ($\widehat{\mathcal{O}}_{\tilde{T},\tilde{x}}/\mathfrak{m}^n_{\tilde{x}}$)-filtered $\varphi$-module of rank one such that $\mathrm{Fil}^0=K\otimes_{K_0}D^+_{\mathrm{cris}}(V_R(\delta^{\mathrm{univ} -1})\otimes_R
\widehat{\mathcal{O}}_{\tilde{T},\tilde{x}}/\mathfrak{m}^n_{\tilde{x}})^{\varphi^f=Y}$ and $\mathrm{Fil}^1=0$.
By Lemma \ref{s4},

this shows that $V_R\otimes_R \widehat{\mathcal{O}}_{\tilde{T},\tilde{x}}$ is
the projective limit of split trianguline $(\mathcal{O}_{\tilde{T},\tilde{x}}/\mathfrak{m}^n_{\tilde{x}})$-representations with triangulations
$0\subseteq W(\bar{\delta}_n^{\mathrm{univ}}\delta_{\bar{Y}_n})\subseteq W(V_R\otimes_R \widehat{\mathcal{O}}_{\tilde{T},\tilde{x}}/\mathfrak{m}^n_{\tilde{x}})$ ( for any $n\in\mathbb{Z}_{\geqq 1}$) which are
trianguline deformations of $(V_x,\mathcal{T}_x)\otimes_{E} E(\tilde{x})$, hence the natural map
$R_{V_x}\rightarrow \widehat{\mathcal{O}}_{\mathcal{E}(\bar{\rho}),x}\rightarrow \widehat{\mathcal{O}}_{\tilde{T},\tilde{x}}$ factors
through $R_{V_x}\rightarrow R_{V_x,\mathcal{T}_x}$ for any $\tilde{x}\in \tilde{T}$ above $x$. Moreover, because
the natural map $$\widehat{\mathcal{O}}_{\mathcal{E}(\bar{\rho}),x}\hookrightarrow \prod_{\tilde{x}\in \tilde{T}, p(\tilde{x})=x}\widehat{\mathcal{O}}_{\tilde{T},\tilde{x}}$$ is injection by Lemma 10.7 of \cite{Ki03} and by (2) of Proposition \ref{5.13} ( where $p:\tilde{T}\rightarrow \mathrm{Spm}(R)$ is the projection ) ,
the map $R_{V_x}\rightarrow \widehat{\mathcal{O}}_{\mathcal{E}(\bar{\rho}),x}$ also
factors through $R_{V_x,\mathcal{T}_x}\rightarrow \widehat{\mathcal{O}}_{\mathcal{E}(\bar{\rho}),x}$.
By this natural construction, we can easily check that this is the inverse of the map giving the above.
We finish to prove the existence of the isomorphism $\widehat{\mathcal{O}}_{\mathcal{E}(\bar{\rho}),x}
\isom R_{V_x,\mathcal{T}_x}$ for such points. Because this isomorphism is preserved by the base change from $E$ to any
finite extension $E'$ by Lemma \ref{5.9}, the smoothness around these points
follows from this isomorphism and from Lemma 2.8 of \cite{BLR95}.
\end{proof}

\section{Zariski density of two dimensional crystalline representations.}
In this final section, as an application of Theorem \ref{33} (in the two dimensional case) and of Theorem \ref{5.18},
we prove Zariski density of two dimensional crystalline representations for any $p$-adic field.
\smallskip

We define a map $\pi:\mathcal{E}(\bar{\rho})\rightarrow \mathcal{W}_E\times_E\mathcal{W}_E$ by
$([V_x],\delta_x,\lambda_x)\mapsto (\delta_x, \mathrm{det}(V_x)|_{\mathcal{O}_K^{\times}}/\delta_x)$.

\begin{prop}\label{6.1}
For any point $x\in \mathcal{E}(\bar{\rho})$ which satisfies all the conditions of Theorem $\ref{5.18}$,
the map $\pi:\mathcal{E}(\bar{\rho})\rightarrow \mathcal{W}_E\times_E\mathcal{W}_E$ is smooth at $x$.
\end{prop}
\begin{proof}
Let $x:=([V_x],\delta_x,\lambda_x)\in\mathcal{E}(\bar{\rho})$ be such a point. We put $\delta'_x:=\mathrm{det}(V_x)|_{\mathcal{O}_K^{\times}}/\delta_x$.
By the same argument as in Proposition 9.5 of \cite{Ki03}, we have an
isomorphism $\widehat{\mathcal{O}}_{\mathcal{W}_E\times_E\mathcal{W}_E,(\delta_x,\delta'_x)}\isom \widehat{\mathcal{O}}_{\mathcal{W}_E,\delta_x}\hat{\otimes}_{E(x)}
\widehat{\mathcal{O}}_{\mathcal{W}_E,\delta'_x}\isom R_{\delta_x}\hat{\otimes}_{E(x)}R_{\delta'_x}$. Hence, by Theorem \ref{5.18}, the completion of
$\pi$ at $x$ is the morphism
$$\pi:\mathrm{Spf}(\widehat{\mathcal{O}}_{\mathcal{E}(\bar{\rho}),x})\rightarrow \mathrm{Spf}(\widehat{\mathcal{O}}_{\mathcal{W}_E\times_E\mathcal{W}_E,(\delta_x,\delta'_x)})$$ induced by the morphism of functors
$$\pi_x:D_{V_x,\mathcal{T}_x}\rightarrow D_{\delta_x}\times D_{\delta'_x}:[(W_A,\psi_A,\mathcal{T}_A)]\mapsto (\delta_{1,A}|_{\mathcal{O}_K^{\times}},\delta_{2,A}|_{\mathcal{O}_K^{\times}})$$ where $\mathcal{T}_A:0\subseteq W(\delta_{1,A})\subseteq W_A$ and $W_A/W(\delta_{1,A})\isom W(\delta_{2,A})$ for any $A\in\mathcal{C}_{E(x)}$. Then, we can prove formally smoothness of this morphism of functor in the  same way as in Lemma \ref{5.16}.
 Hence, $\pi$ is smooth at $x$  by Proposition  \ref{19} and by Proposition 2.9 of \cite{BLR95}.
\end{proof}

Let $x:=([V_x],\delta_x,\lambda_x)\in \mathcal{E}(\bar{\rho})$ be an $E$-rational  point such
that $V_x$ is a

crystalline split trianguline $E$-representation with a
triangulation $\mathcal{T}_x:0\subseteq W(\delta_x\delta_{\lambda_x})\subseteq W(V_x)$
satisfying the conditions (1) of Definition \ref{26} (Corollary \ref{5.16.5}).

By Proposition \ref{5.13}, for any $Y$-small
affinoid open neighborhood $U=\mathrm{Spm}(R)$ of $x$ in $\mathcal{E}(\bar{\rho})$, there exits $k>0$ and
there exists a short exact sequence of Banach $R$-modules with property (Pr)
$$0\rightarrow K\otimes_{K_0}(B^+_{\mathrm{max}}\hat{\otimes}_{\mathbb{Q}_p}R)^{\varphi_K=Y}
\rightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}R\rightarrow U_{k}\rightarrow 0$$
and we have a natural isomorphism $$K\otimes_{K_0}D^+_{\mathrm{cris}}(V_R(\tilde{\delta}_R^{-1}))^{\varphi_K=Y}\isom
(B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}} \hat{\otimes}_{\mathbb{Q}_p}V_R(\tilde{\delta}_R^{-1}))^{G_K}$$
 and for any $\sigma\in \mathcal{P}$ there exists the smallest
ideal $H_{\sigma}\subseteq R$ satisfying that $$(B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}H_{\sigma}V_R(\tilde{\delta}_R^{-1}))^{G_K}
\isom (B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{K,\sigma}V_R(\tilde{\delta}_R^{-1}))^{G_K}$$
where we put $V_R:=\Gamma(U, p_1^*(\tilde{V}^{\mathrm{univ}}))$ and $\delta_R:\mathcal{O}_K^{\times}\rightarrow R^{\times}$ is the restriction of $p_2^*(\delta^{\mathrm{univ}})$ to $U$. Moreover,
if we put $Q:=\prod_{\sigma\in \mathcal{P},0\leqq i\leqq k}Q_{\sigma}(-i)\in R$, then we have inclusions
$\mathrm{Spm}(R)_Q\subseteq \mathrm{Spm}(R)\setminus V(H_{\sigma})\subseteq \mathrm{Spm}(R)$ by the proof of
Proposition \ref{5.13}. Moreover, shrinking $U$ suitably, we may assume that $v_p(\lambda_y)=v_p(\lambda_x)$ for any
$y=(V_y,\delta_y,\lambda_y)\in U$ and that $\pi|_U$ is smooth by Proposition \ref{6.1}.

Under this situation, we study the map
$\pi|_U:U \rightarrow \mathcal{W}_E\times_E\mathcal{W}_E:
([V_y],\delta_y,\lambda_y)\mapsto (\delta_y, \mathrm{det}(V_y)|_{\mathcal{O}_K^{\times}}/\delta_y)$ around $x$ in detail.
Because $V_x$ is crystalline

, we can write $$\pi(x)=(\prod_{\sigma\in\mathcal{P}}\sigma^{k_{1,\sigma}},\prod_{\sigma\in \mathcal{P}}\sigma^{k_{2,\sigma}})\in \mathcal{W}_E\times_E\mathcal{W}_E$$

for some integers $\{k_{1,\sigma},k_{2,\sigma}\}_{\sigma\in \mathcal{P}}$, then we define a subset
\[
\begin{array}{ll}
(\mathcal{W}_E\times_E&\mathcal{W}_E)_{cl,x}:=\{
(\prod_{\sigma\in \mathcal{P}}\sigma^{n_{\sigma}},\prod_{\sigma\in\mathcal{P}}\sigma^{n_{\sigma}-m_{\sigma}})\in \mathcal{W}_E\times_E
\mathcal{W}_E| n_{\sigma}\in \mathbb{Z}, \\
&m_{\sigma}\in \mathbb{Z}_{\geqq k+1} \,\,\text{for any} \,\,\sigma\in\mathcal{P} \,\,\text{and} \,\,
\sum_{\sigma\in\mathcal{P}}m_{\sigma}\geqq 2e_Kv_p(\lambda_x)+[K:\mathbb{Q}_p]+1\},
\end{array}
\]
where $e_K$ is the absolute ramified index of $K$.
Then, for any  admissible open neighborhood $V$ of $\pi(x)$ in $\mathcal{W}_E\times_E\mathcal{W}_E$, there exists an affinoid open
$V'\subseteq V$ which contains $\pi(x)$ such that $V'_{\mathrm{cl},x}:=(\mathcal{W}_E\times_E\mathcal{W}_E)_{\mathrm{cl},x}\cap V'$ is Zariski dense
in $V'$. Under this situation, we prove the following lemma.

\begin{lemma}\label{6.2}
Let $y:=(\prod_{\sigma\in\mathcal{P}}\sigma^{n_{\sigma}},\prod_{\sigma\in\mathcal{P}}\sigma^{n_{\sigma}-m_{\sigma}})$ be an element in $(\mathcal{W}_E\times_E\mathcal{W}_E)_{\mathrm{cl},x}$ and let $z:=([V_z],\delta_z,\lambda_z)$ be a point in $U\cap \pi^{-1}(y)$, then
$V_z$ is crystalline and split trianguline $E(z)$-representation with a
triangulation
$\mathcal{T}_z:0\subseteq W(\delta_z\delta_{\lambda_z})\subseteq W(V_z)$ which satisfies the conditions $(1)$ and $(2)$
of Definition $\mathrm{\ref{26}}$.

\end{lemma}
\begin{proof}
Let $z$ be such a point. By Corollary 2.6 of \cite{Ki03}, we have a natural isomorphism
 $$(B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}V_R(\tilde{\delta}_R^{ -1}))^{G_K}\otimes_R E(z)
\isom (B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}V_z(\tilde{\delta}_z^{-1}))^{G_K}$$
 and this is
a free $K\otimes_{\mathbb{Q}_p}E(z)$-module of rank one. Because we
have an isomorphism  $$K\otimes_{K_0}D^+_{\mathrm{cris}}(V_R(\tilde{\delta}_R^{ -1}))^{\varphi_K=Y}
\isom (B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}V_R(\tilde{\delta}_R^{-1}))^{G_K}$$
and have an injection
$$K\otimes_{K_0}D^+_{\mathrm{cris}}(V_z(\tilde{\delta}_z^{-1}))^{\varphi_K=\lambda_z}\hookrightarrow (B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}V_z(\tilde{\delta}_z^{-1}))^{G_K}$$ induced from the injection
$K\otimes_{K_0}(B^+_{\mathrm{cris}}\otimes_{\mathbb{Q}_p}E(z))^{\varphi_K=\lambda_z}
\hookrightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}E(z)$. From these facts, we obtain  an isomorphism
$$K\otimes_{K_0}D^+_{\mathrm{cris}}(V_z(\tilde{\delta}_z^{-1}))^{\varphi_K=\lambda_z}\isom (B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}V_z(\tilde{\delta}_z^{-1}))^{G_K} .$$ On the other hand, because the Hodge-Tate weight of $V_z(\tilde{\delta}_z^{-1})$
is $\{0, -m_{\sigma}\}_{\sigma\in\mathcal{P}}$ and $m_{\sigma}\geqq k+1\geqq 1$,
$(t^{k+1}B^+_{\mathrm{dR}}\otimes_{\mathbb{Q}_p}V_z(\tilde{\delta}_z^{-1}))^{G_K}$ is also a free $K\otimes_{\mathbb{Q}_p}E(z)$-module
of rank one. These implies that $D_{\mathrm{dR}}(V_z(\tilde{\delta}_z^{-1}))$ is a free of rank two $K\otimes_{\mathbb{Q}_p}E(z)$-module, i.e.
$V_z(\tilde{\delta}_z^{-1})$ is potentially semi-stable and split trianguline with a triangulation
$\mathcal{T}'_z:0\subseteq W(\delta_{\lambda_z})\hookrightarrow W(V_z(\tilde{\delta}_z^{-1}))$. Moreover, if we write
$\delta_2:=\mathrm{det}(V_z)/\delta_z^2\delta_{\lambda_z}:K^{\times}\rightarrow E(z)^{\times}$, then we have $W(V_z(\tilde{\delta}_z^{-1}))/W(\delta_{\lambda_z})\isom W(\delta_2)$ and $\delta_2|_{\mathcal{O}_K^{\times}}=\prod_{\sigma\in\mathcal{P}}\sigma^{-m_{\sigma}}$ because $z\in \pi^{-1}(y)$, hence these facts imply that $V_z(\tilde{\delta}_z^{-1})$ is semi-stable. Finally, we claim that $V_z(\tilde{\delta}_z^{-1})$ is crystalline. If we assume that $V_z(\tilde{\delta}_z^{-1})$ is semi-stable but not crystalline, then $\varphi^f$-eigenvalue of $W(\delta_2)$ is $\lambda_zp^{f}$ or $\lambda_zp^{-f}$. Because of the weakly admissibility
of $D_{\mathrm{st}}(V_z(\tilde{\delta}_z^{-1}))$, we have an equality $t_N(V_z(\tilde{\delta}_z^{-1}))= t_H(V_z(\tilde{\delta}_z^{-1}))$. On the other hand, because $W(\mathrm{det}(V_z(\tilde{\delta}_z^{-1})))\isom W(\delta_{\lambda_z}\delta_2)$, we have
$t_N(V_z(\tilde{\delta}_z^{-1}))=\frac{2}{f}v_p(\lambda_z)\pm 1=\frac{2}{f}v_p(\lambda_x)\pm 1$ and $t_H(V_z(\tilde{\delta}_z^{-1}))=\frac{1}{[K:\mathbb{Q}_p]}(\sum_{\sigma\in \mathcal{P}}m_{\sigma})$, hence $t_N(V_z(\tilde{\delta}_z^{-1}))<t_H(V_z(\tilde{\delta}_z^{-1}))$ because
$y\in (\mathcal{W}\times_E\mathcal{W}_E)_{\mathrm{cl},x}$, this is a contradiction.

Hence, $V_z(\tilde{\delta}_z^{-1})$ is crystalline and, because $\tilde{\delta}_z$ is crystalline, $V_z$ is
also crystalline. Finally, twisting $\mathcal{T}'_z$ by $\delta_z$, we obtain a triangulation $\mathcal{T}_z:0\subseteq W(\delta_z\delta_{\lambda_z})
\subseteq W(V_z)$ which satisfies (1) and (2) of Definition \ref{26}.
\end{proof}

\begin{lemma}\label{6.3}
Let $z=([V_z],\delta_z,\lambda_z)$ be a point  in $U\cap\pi^{-1}(y)$ as in the above lemma.

 Then we have a natural isomorphism
$\widehat{\mathcal{O}}_{\pi^{-1}(y),z}\isom R^{\mathrm{cris}}_{V_z}$.
\end{lemma}
\begin{proof}
First, by Lemma \ref{21.5} and by Theorem \ref{5.18} and by Lemma \ref{6.2}, we have a triangulation
$\mathcal{T}_z:0\subseteq W(\delta_z\delta_{\lambda_z})\subseteq W(V_z)$ and
the functor $D_{V_z,\mathcal{T}_z}$ is represented by $R_{V_z,\mathcal{T}_z}$ and we have an isomorphism
$\widehat{\mathcal{O}}_{\mathcal{E}(\bar{\rho}),z}\isom R_{V_z,\mathcal{T}_z}$.
Then, the completion at $z$ and $\pi(z)$ of the morphism $\pi_z:\mathcal{E}(\bar{\rho})\rightarrow \mathcal{W}_E\times_E\mathcal{W}_E$
is the morphism
$$\pi:\mathrm{Spf}(R_{V_z,\mathcal{T}_z})\rightarrow \mathrm{Spf}(R_{\delta_z}\hat{\otimes}_{E(z)}R_{\delta'_z})$$
 induced by
$$D_{V_z,\mathcal{T}_z}\rightarrow D_{\delta_z}\times D_{\delta'_z}:[(V_A,\psi_A,\mathcal{T}_A)]\mapsto
(\delta_{1,A}|_{\mathcal{O}_K^{\times}},\delta_{2,A}|_{\mathcal{O}_K^{\times}}),$$
 where $\delta'_z:=\mathrm{det}(V_z)|_{\mathcal{O}_K^{\times}}/\delta_z$.
Under this interpretation, the closed fiber  $\mathrm{Spf}(\widehat{\mathcal{O}}_{\pi^{-1}(y),z})$ of $\pi_z$ corresponds to
the sub deformation $D'$ of $D_{V_z,\mathcal{T}_z}$ defined by
$$D'(A):=\{[(V_A,\psi_A,\mathcal{T}_A)]\in D_{V_z,\mathcal{T}_z}(A)| \delta_{1,A}|_{\mathcal{O}_K^{\times}}=\delta_z\otimes_{E(z)}\mathrm{id}_A, \,\,\delta_{2,A}|_{\mathcal{O}_K^{\times}}=\delta'_z\otimes_{E(z)}\mathrm{id}_A\}$$ for any $A\in\mathcal{C}_{E(z)}$.
 Because $V_z$ is crystalline, this is equivalent to that $V_A$ is crystalline by Lemma \ref{29}. Therefore we have $D'=D^{\mathrm{cris}}_{V_z}$, hence we have an isomorphism $R^{\mathrm{cris}}_{V_z}
\isom \widehat{\mathcal{O}}_{\pi^{-1}(y),z}$.
\end{proof}

In the situation of Lemma \ref{6.3}, for any $y:=(\prod_{\sigma\in \mathcal{P}}\sigma^{n_{\sigma}},\prod_{\sigma\in\mathcal{P}}
\sigma^{n_{\sigma}-m_{\sigma}})\in (\mathcal{W}_E\times_E\mathcal{W}_E)_{\mathrm{cl},x}$,
we set $U_y:=\pi^{-1}(y)\cap U$, which is smooth over $E(y)$ by the assumption on $U$, and define a subset
$$U_{y,b}:=\{z=([V_z],\delta_z,\lambda_z)\in U_y| V_z \,\text{is benign} \} .$$
\begin{prop}\label{6.5}
In the above situation, if $U_y$ is not empty, then $U_{y,b}$ is an admissible open which is scheme theoretically dense
in $U_y$, in particular $U_{y,b}$ is non-empty.
\end{prop}
\begin{proof}
We denote by $U_y:=\mathrm{Spm}(R')$.
First, by Lemma \ref{6.2}, any point $z\in U_y$ satisfies the condition (1) and (2) of Definition \ref{26}
and $V_z$ is crystalline
with Hodge-Tate weight
$\{n_{\sigma}, n_{\sigma}-m_{\sigma}\}_{\sigma\in\mathcal{P}}$. Because $U_y$ is smooth, so in particular  $U_y$ is reduced.
Hence, by Corollary 6.3.3 of \cite{Be-Co08} and by Corollary 3.19 of \cite{Ch09a},
$$D_{\mathrm{cris}}(V_{R'}(\delta_{R'}^{-1})):=\varinjlim_n(\frac{1}{t^n}B^+_{\mathrm{max}}\hat{\otimes}_{\mathbb{Q}_p}V_{R'}(\delta_{R'}^{-1}))^{G_K}$$ is a locally
free $K_0\otimes_{\mathbb{Q}_p}R'$-module of rank two and, for any $z\in U_y$, we have
an isomorphism $$D_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))\otimes_{R'}E(z)\isom D_{\mathrm{cris}}(V_z(\tilde{\delta}_z^{-1}))$$ and
$$K\otimes_{K_0}D_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))\isom (B_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}V_{R'}(\tilde{\delta}_{R'}^{-1}))^{G_K}
=(B^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}V_{R'}(\tilde{\delta}_{R'}^{-1}))^{G_K},$$
where the last equality follows from the assumption on the Hodge-Tate weight of $V_z$ for any $z\in U_y$.
  Because $U_y\subseteq U_Q$, we have an isomorphism $$(B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}V_{R}(\tilde{\delta}_R^{-1}))^{G_K}\otimes_{R}R'\isom
(B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}V_{R'}(\tilde{\delta}_{R'}^{-1}))^{G_K},$$
which is a locally free $K\otimes_{\mathbb{Q}_p}R'$-module of rank one by Corollary 2.6 of \cite{Ki03}.
Because the natural map $K\otimes_{K_0}(B^+_{\mathrm{max}}\hat{\otimes}_{\mathbb{Q}_p}R')^{\varphi^f=Y}
\hookrightarrow B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}R'$ is injection, so we have an isomorphism $$K\otimes_{K_0}D^+_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))^{\varphi^f=Y}\isom (B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}V_{R'}(\tilde{\delta}_{R'}^{-1}))^{G_K}.$$

From these, we can see that the natural map $$(B^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}V_{R'}(\tilde{\delta}_{R'}^{-1}))^{G_K}\rightarrow
(B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}V_{R'}(\tilde{\delta}_{R'}^{-1}))^{G_K}$$ is surjection. Hence,
we have a short exact sequence
$$0\rightarrow \mathrm{Fil}^kD_{\mathrm{dR}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))\rightarrow D^+_{\mathrm{dR}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))
\rightarrow (B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}V_{R'}(\tilde{\delta}_{R'}^{-1}))^{G_K}\rightarrow 0,$$
where we define $\mathrm{Fil}^kD_{\mathrm{dR}}(V_{R'}(\tilde{\delta}_{R'}^{-1})):=(t^kB^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}V_{R'}(\tilde{\delta}_{R'}^{-1}))^{G_K}$ which is a locally free $K\otimes_{\mathbb{Q}_p}R'$-module
of rank one. If we denote by
$$D_2:=D_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))/D^+_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))^{\varphi^f=Y},$$
 then
the above facts imply that
$D_2$ is also a locally free $K_0\otimes_{\mathbb{Q}_p}R'$-module of rank one. By taking a sufficiently fine affinoid covering of $\mathrm{Spm}(R')$, we may assume
that all these modules are free over $K_0\otimes_{\mathbb{Q}_p}R'$ or $K\otimes_{\mathbb{Q}_p}R'$.
If we decompose $D_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))=\oplus_{\tau:K_0\rightarrow K_0}D_{\tau}$ etc,then
we have
a short exact sequence
$$0\rightarrow D_{\tau}^{+, \varphi^f=Y}\rightarrow D_{\tau}\rightarrow D_{2,\tau}\rightarrow 0$$
of free $R'$-modules with an $R'$-linear $\varphi^f$-action for any $\tau$. We take the $Y_1\in R^{' \times}$ such that
$\varphi^f(e)=Y_1e$ where $e\in D_{2,\tau}$ is a $R'$-base of $D_{2,\tau}$. Because $Y_1$ is a lift of
the other Frobenius eigenvalue of $D_{\mathrm{cris}}(V_z(\tilde{\delta}_z^{-1}))$ (one is $\lambda_z$) for
any $z\in U_y=\mathrm{Spm}(R')$ and because $D_{\mathrm{cris}}(V_z(\tilde{\delta}_z^{-1}))$ is weakly admissible,
the condition $\sum_{\sigma\in \mathcal{P}}m_{\sigma}\geqq 2e_Kv_p(\lambda_z)+[K:\mathbb{Q}_p]+1$ for any $z\in U_y$ implies
that
$$Y-Y_1 (\text{ and } Y-p^{\pm f}Y_1)  \in R^{' \times}.$$
Then, an easy linear algebra implies that there exists a decomposition $D_{\tau}=R'e'_1\oplus  R'e'_2$ such that
$R'e'_1=D_{\tau}^{\varphi^f=Y}=D^{+,\varphi^f=Y}_{\tau}$ and $R'e'_2=D_{\tau}^{\varphi^f=Y_1}$.  Twisting these by $\varphi^i$ for
any $0\leqq i\leqq f-1$, we obtain a
decomposition $D_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))=D^+_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))^{\varphi^f=Y}\oplus D_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))^{\varphi^f=Y_1}$. We denote by $e_1$ (resp. $e_2$)  a $K_0\otimes_{\mathbb{Q}_p}R'$-base of $D^+_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))^{\varphi^f=Y}$ (resp. $D_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))^{\varphi^f=Y_1}$).
For any $\sigma\in \mathcal{P}$, we denote by $e_{1,\sigma}, e_{2,\sigma}$ the $R'$-basis of the $\sigma$-component of
$D^+_{\mathrm{dR}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))\isom K\otimes_{K_0}D_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))$ naturally
induced from $e_1, e_2$. Under this situation, we write the $\sigma$-component $\mathrm{Fil}^kD_{\mathrm{dR}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))_{\sigma}$ by using the basis
$e_{1,\sigma}, e_{2,\sigma}$ as follows. Because the natural map $D^+_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))^{\varphi^f=Y}\isom (B^+_{\mathrm{dR}}/t^kB^+_{\mathrm{dR}}\hat{\otimes}_{\mathbb{Q}_p}V_{R'}(\tilde{\delta}_{R'}^{-1}))^{G_K}$ is isomorphism, the natural map
$$\mathrm{Fil}^kD_{\mathrm{dR}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))\rightarrow K\otimes_{K_0}D_2,$$
 which is the composition
of the natural inclusion
$$\mathrm{Fil}^kD_{\mathrm{dR}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))\rightarrow D_{\mathrm{dR}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))=
K\otimes_{K_0}D_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))$$ with the natural projection
$K\otimes_{K_0}D_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))\rightarrow K\otimes_{K_0}D_2$, is an isomorphism. Hence, for any $\sigma\in\mathcal{P}$,
 we can take a $R'$-base of $\mathrm{Fil}^kD_{\mathrm{dR}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))_{\sigma}$ of the form
$e_{2,\sigma}+a_{\sigma}e_{1,\sigma}$ for some $a_{\sigma}\in R'$. Then, by the definition of benign representations,
for any $z\in U_y$, $V_z$ is benign if and only if $\prod_{\sigma\in \mathcal{P}}a_{\sigma}(z)\not=0 \in E(z)$ because we have
$D_{\mathrm{dR}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))\otimes_{R'}E(z)\isom D_{\mathrm{dR}}(V_z(\tilde{\delta}_z^{-1}))$ and
$D_{\mathrm{cris}}(V_{R'}(\delta_{R'}^{-1}))\otimes_{R'}E(z)\isom D_{\mathrm{cris}}(V_z(\tilde{\delta}_z^{-1}))$ etc.

Hence, to finish the proof of this proposition, it is enough to show that $\prod_{\sigma\in \mathcal{P}}a_{\sigma}$ is a
non-zero divisor in $R'$. To prove this claim, it is enough to show that $a_{\sigma}\in \widehat{\mathcal{O}}_{U_y,z}\isom
R_{V_z}^{\mathrm{cris}}$ is non-zero for any $\sigma\in \mathcal{P}$, $z\in U_y$  because $R_{V_z}^{\mathrm{cris}}$ is domain.
To prove this claim,
first we note that we have isomorphisms  $D_{\mathrm{cris}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))\otimes_{R'}R_{V_z}^{\mathrm{cris}}\isom
D_{\mathrm{cris}}(V_{R_{V_z}^{\mathrm{cris}}}(\tilde{\delta}_{R_{V_z}^{\mathrm{cris}}}^{-1})):=\varprojlim_n D_{\mathrm{cris}}(V_{R_{V_z}^{\mathrm{cris}}/\mathfrak{m}^n}(\tilde{\bar{\delta}}_{R_{V_z}^{\mathrm{cris}},n}^{-1}))$ and $D_{\mathrm{dR}}(V_{R'}(\tilde{\delta}_{R'}^{-1}))\otimes_{R'}R_{V_z}^{\mathrm{cris}}\isom D_{\mathrm{dR}}(V_{R_{V_z}^{\mathrm{cris}}}(\tilde{\delta}_{R_{V_z}^{\mathrm{cris}}}^{-1})):=\varprojlim_n D_{\mathrm{dR}}(V_{R_{V_z}^{\mathrm{cris}}/\mathfrak{m}^n}(\tilde{\bar{\delta}}_{R_{V_z}^{\mathrm{cris}},n}^{-1}))$ by construction and by Corollary 6.3.3 of \cite{Be-Co08}, where we denote by $\mathfrak{m}$ the maximal ideal of $R_{V_z}^{\mathrm{cris}}$ and denote
by $\delta_{R_{V_z}^{\mathrm{cris}}}:\mathcal{O}_K^{\times}\rightarrow (R^{\mathrm{cris}}_{V_z})^{\times}$ the homomorphism induced from
$\delta_{R'}$ and denote by $\bar{\delta}_{R_{V_z}^{\mathrm{cris}}, n}:\mathcal{O}_K^{\times}\rightarrow (R_{V_z}^{\mathrm{cris}}/\mathfrak{m}^n)^{\times}$ the reduction
of $\delta_{R_{V_z}^{\mathrm{cris}}}$. Hence, the claim follows from the following lemma.

\end{proof}

\begin{lemma}\label{6.6}
Let $V$ be a crystalline $E$-representation with Hodge-Tate weight $\{0, -k_{\sigma}\}_{\sigma\in \mathcal{P}}$ such
that $k_{\sigma}\in\mathbb{Z}_{\geqq 1}$ for any $\sigma\in\mathcal{P}$.
If we can write $D_{\mathrm{cris}}(V_{R_{V}^{\mathrm{cris}}}):=\varprojlim_n D_{\mathrm{cris}}(V_{R_{V}^{\mathrm{cris}}}/\mathfrak{m}^n)$
by $D_{\mathrm{cris}}(V_{R_V^{\mathrm{cris}}})=K_0\otimes_{\mathbb{Q}_p}R_{V}^{\mathrm{cris}}e_1\oplus K_0\otimes_{\mathbb{Q}_p}R_{V}^{\mathrm{cris}}e_2$ such that $\varphi^f(e_1)=\tilde{\lambda}_1e_1, \varphi^f(e_2)=\tilde{\lambda}_2e_2$ for some
$\tilde{\lambda}_1,\tilde{\lambda}_2\in (R_{V}^{\mathrm{cris}})^{\times}$ and that $\mathrm{Fil}^{k_{\sigma}}D_{\mathrm{dR}}(V_{R^{\mathrm{cris}}_V})_{\sigma}$ is generated by $e_{2,\sigma}+a_{\sigma}e_{1,\sigma}$ for $\sigma\in\mathcal{P}$.
Then, $a_{\sigma}\not= 0\in R_{V}^{\mathrm{cris}}$ for any $\sigma\in \mathcal{P}$.
\end{lemma}
\begin{proof}
If we denote by $\lambda_i:=\bar{\tilde{\lambda}}_i\in E^{\times}$ and $\bar{a}_{\sigma}\in E$ the reductions
of $\tilde{\lambda}_i$ and $a_{\sigma}$ by the natural quotient map
$R_{V}^{\mathrm{cris}}\rightarrow E$, then $D_{\mathrm{cris}}(V)=K_0\otimes_{\mathbb{Q}_p}E\bar{e}_1\oplus
K_0\otimes_{\mathbb{Q}_p}E\bar{e}_2$ such that $\varphi^f(\bar{e}_i)=\lambda_i\bar{e}_i$ and
$\mathrm{Fil}^{k_{\sigma}}D_{\mathrm{dR}}(V)_{\sigma}=E(\bar{e}_{2,\sigma}+\bar{a}_{\sigma}\bar{e}_{1,\sigma})$.
For any $b:=\{b_{\sigma}\}_{\sigma\in \mathcal{P}}\in \prod_{\sigma\in \mathcal{P}}E$, we construct a deformation $D(b)$ of
$D_{\mathrm{cris}}(V)$ over $E[\varepsilon]$ by $D(b):=D_{\mathrm{cris}}(V)\otimes_E E[\varepsilon]$ as a $\varphi$-module
and $\mathrm{Fil}^{0}(K\otimes_{K_0}D(b))=K\otimes_{K_0}D(b)$ and
$$\mathrm{Fil}^1(K\otimes_{K_0}D(b))_{\sigma}=
\mathrm{Fil}^{k_{\sigma}}(K\otimes_{K_0}D(b))_{\sigma}:=E[\varepsilon](\bar{e}_{2,\sigma}+ (\bar{a}_{\sigma}+b_{\sigma}\varepsilon)\bar{e}_{1,\sigma}),$$
$\mathrm{Fil}^{k_{\sigma}+1}(K\otimes_{K_0}D(b))_{\sigma}=0$. For any $b$ as above, $D(b)$ is a deformation of $D_{\mathrm{cris}}(V)$ over $E[\varepsilon]$. The existence of such deformations implies that $a_{\sigma}\not=0$ for any $\sigma\in\mathcal{P}$.

\end{proof}

Next, we will prove a proposition concerning  Zariski density of benign points in $\mathcal{E}(\bar{\rho})$.
Before proving this proposition, we first prove some lemmas concerning general (maybe well-known easy)
facts about rigid geometry.

\begin{lemma}\label{6.7}
Let $T_n$ be the $n$-dimensional closed unit disc defined over $E$. Then, for any admissible open $U$ of $T_n$
which contains the origin $0:=(0,\cdots,0)\in T_n$, there exists $m>>0$ such that
$\{(x_1,\cdots,x_n)\in T_n| |x_i|\leqq 1/p^m $ for any $1\leqq i\leqq n \} \subseteq U$.
\end{lemma}
\begin{proof}
Because $U$ is admissibly covered by rational sub-domains, we may assume that $U$ is itself
a rational sub-domain, namely, we may assume that there exist $f_1,\cdots,f_d, g\in E\{\{T_1,\cdots,T_n\}\}$ such that
$(f_1,\cdots,f_d,g)=E\{\{T_1,\cdots,T_n\}\}$ and $U=\{x=(x_1,\cdots,x_n)\in T_n| |f_i(x)|\leqq |g(x)| $ for any $1\leqq i\leqq d \}$.
Then, the condition $0\in U$ means that $|f_{i,0}|\leqq |g_0|$ for any $i$, where $f_{i,0}, g_0\in E$ are the constant terms of $f_i$  and $g$.
If $g_0=0$, then $f_{i,0}=0$ for any $i$, this means that $(f_1,\cdots,f_d,g)\subseteq (T_1,T_2\cdots,T_n)$, which is a contradiction.
Hence we have $g_0\not=0$ and then, because the norms of coefficients of $f_i$ and $g$ are bounded, there exits $m>>0$ large enough
such that $|f_i(x)|\leqq \mathrm{max}\{|f_{i,0}|, |g_0|\}=|g_0|$ and $ |g(x)|=|g_0|$ for any $x=(x_1,\cdots,x_n)\in T_n$ such that $|x_i|\leqq 1/p^m$ for any $i$, i.e. $\{x\in T_n| |x_i|\leqq 1/p^m $ for any $i \}\subseteq U$.
\end{proof}

\begin{lemma}\label{6.8}
Let $x:=([V_x],\delta_x,\lambda_x)\in \mathcal{E}(\bar{\rho})$ be an $E$-rational  point such that $V_x$ is crystalline trianguline
as in before Lemma $\mathrm{\ref{6.2}}$ and $U\subseteq \mathcal{E}(\bar{\rho})$ be an admissible open neighborhood of $x$. Then, there exists an admissible open neighborhood
$U'\subseteq U$ of $x$ such that $U'_{\mathrm{cl},x}:=U'\cap \pi^{-1}(\mathcal{W}_E\times_E\mathcal{W}_E)_{\mathrm{cl},x})$

is Zariski dense
in $U'$.
\end{lemma}
\begin{proof}
Re-taking smaller $U$, we may assume that  $U$ satisfies the properties as in before Lemma \ref{6.2} and
that the morphism $\pi|_U:U\rightarrow \mathcal{W}_E\times_E\mathcal{W}_E$ is smooth
and $U$ is irreducible smooth of dimension $3[K:\mathbb{Q}_p]+1$ by Theorem \ref{5.18} and Lemma \ref{6.1}. In particular, we may assume that
$\pi(U)\subseteq \mathcal{W}_E\times_E\mathcal{W}_E$
is an admissible open by Corollary 5.11 of \cite{BL93}. By definition of $\mathcal{W}_E\times_E\mathcal{W}_E$ and $(\mathcal{W}_E\times_E\mathcal{W}_E)_{\mathrm{cl},x}$ and by Lemma \ref{6.7}, if we re-take $U$ smaller, then we may assume that there exists an admissible open
neighborhood $V$ of $y:=\pi(x)$ which is isomorphic to $T_n\isom V$ where $n:=2[K:\mathbb{Q}_p]$ such that $y$ corresponds to the origin $0\in T_n$ and that, for any $m\geqq 1$,
the set $V_{\mathrm{cl},m}:=\{x\in T_n| |x_i|\leqq 1/p^m $ for any $1\leqq i\leqq n\}\cap (\mathcal{W}_E\times_E\mathcal{W}_E)_{\mathrm{cl},x}$ is Zariski
dense in $V$ and that $\pi(U)\subseteq V$ and that $\pi|_U:U\rightarrow V$ factors through an \'etale morphism $\pi':U\rightarrow V\times_E T_{n'}$,  where $n':=[K:\mathbb{Q}_p]+1$,
satisfying $\pi'(x)=(y,0)$. Because $V_{\mathrm{cl},m}$ is Zariski dense in $V$ for any $m$, the set $(V\times_ET_{n'})_{\mathrm{cl},m}
:=\{(y',z)\in V_{\mathrm{cl},m}\times_ET_{n'}| |z_i|\leqq 1/p^m $ for any $1\leqq i\leqq n' \}$ is also Zariski dense in
$V\times_E T_{n'}$. Because $\pi'(U)$ is admissible open neighborhood of $(y,0)\in V\times_ET_{n'}$, there exists
$m>>0$ such that $(V\times_ET_{n'})_{\mathrm{cl},m}$

is contained in $\pi'(U)$ by Lemma \ref{6.7}. Then, we have $\pi^{' -1}((V\times_E T_{n'})_{\mathrm{cl},m})\subseteq
\pi^{-1}(V_{\mathrm{cl},m})\subseteq U_{\mathrm{cl},x}$, then the lemma follows from the following lemma.

\end{proof}

\begin{lemma}\label{6.9}
Let $f:U:=\mathrm{Spm}(B)\rightarrow V:=\mathrm{Spm}(E\{\{T_1,\cdots,T_n\}\})$ be an \'etale morphism between $E$-affinoids for
some $n$.
We assume that $U$ is irreducible and reduced.
If $V_{\mathrm{cl}}\subseteq V$ is a Zariski dense sub set of $V$ such that $V_{\mathrm{cl}}\subseteq f(U)$, then
$f^{-1}(V_{\mathrm{cl}})$ is also Zariski dense in $U$.
\end{lemma}
\begin{proof}
By the assumption, the natural map $A:=E\{\{T_1,\cdots,T_n\}\}\rightarrow \prod_{x\in V_{\mathrm{cl}}}E(x)$ is injection.
For proving the lemma, it suffices to show that the kernel of the natural map
$B\rightarrow \prod_{y\in f^{-1}(V_{\mathrm{cl}})}E(y)$ is zero. If $I$ is the kernel of this map, then the map
$A\rightarrow B/I\hookrightarrow \prod_{y\in f^{-1}(V_{\mathrm{cl}})}E(y)$ is equal to the
map $A\hookrightarrow \prod_{x\in V_{\mathrm{cl}}}E(x)\rightarrow \prod_{y\in f^{-1}(V_{\mathrm{cl}})}E(y)$.
Because $V_{\mathrm{cl}}\subseteq f(U)$ by the assumption, the map $\prod_{x\in V_{\mathrm{cl}}}E(x)\rightarrow
\prod_{y\in f^{-1}(V_{\mathrm{cl}})}E(y)$ is injection. Therefore, the map $A\hookrightarrow B/I$ is also injection.
Then, we have $\mathrm{dim}(A)\leqq \mathrm{dim}(B/I)(\leqq \mathrm{dim}(B))$ by Lemma \ref{6.10} below.
From this, we have $\mathrm{dim}(B/I)=\mathrm{dim}(B)$ because $B$ is \'etale over $A$.
Because $U$ is irreducible and reduced, we have $I=0$.

\end{proof}

\begin{lemma}\label{6.10}
Let $f:Z:=\mathrm{Spm}(B')\rightarrow \mathrm{Spm}(E\{\{T_1,\cdots,T_n\}\})$ be a morphism of
affinoids over $E$.
We assume that the induced map $A:=E\{\{T_1,\cdots,T_n\}\}\rightarrow B'$ is injection.
Then, we have $\mathrm{dim}(A)\leqq \mathrm{dim}(B')$.
\end{lemma}
\begin{proof}
Because $A\hookrightarrow B'$ is injection,
the base change $\mathrm{Frac}(A)\hookrightarrow \mathrm{Frac}(A)\otimes_A B'$ is
also injection, in particular, the generic fiber of the morphism of schemes
 $f_0:\mathrm{Spec}(B')\rightarrow \mathrm{Spec}(A)$ induced from the injection $A\hookrightarrow B'$
 is not empty. We denote by $x$ the generic point of $\mathrm{Spec}(A)$ and take a point
 $y\in f_0^{-1}(x)$.

 Then, by Proposition 2.1.1 of \cite{Berk93}, if we denote by $\kappa(x)$ and
 $\kappa(y)$ the residue fields (in the sense of scheme) at $x$ and $y$, then the natural
 inclusion $\kappa(x)\hookrightarrow \kappa(y)$ is an inclusion of valuation fields which induces an
 inclusion $\tilde{\kappa}(x)\hookrightarrow \tilde{\kappa}(y)$, where $\tilde{\kappa}(-)$ is the residue
 field of the valuation field $\kappa(-)$. Form this inclusion, we have
 $(\mathrm{dim}(A)=n=)s(\tilde{\kappa}(x)/E)\leqq s(\tilde{\kappa}(y)/E)$, where $s(\tilde{\kappa}(-)/E)$ is the transcendence
  degree of $\tilde{\kappa}(-)$ over $E$. Then, by Lemma 2.5.2 of \cite{Berk93},
  we also have $s(\tilde{\kappa}(y)/E)\leqq \mathrm{dim}(B')$, hence we have $\mathrm{dim}(A)\leqq \mathrm{dim}(B')$.

\end{proof}

We set

$$\mathcal{E}(\bar{\rho})_{\mathrm{b}}:=\{x\in \mathcal{E}(\bar{\rho})| V_x\text{ is benign and crystalline }\}.$$

\begin{prop}\label{6.11}
Let $x$ be an $E$-rational  point  in $\mathcal{E}(\bar{\rho})$ as in Lemma $\mathrm{\ref{6.8}}$ and
$U$ be an admissible open neighborhood of $x$.
If we take an affinoid neighborhood $U':=\mathrm{Spm}(R)$ of $x$ as in Lemma $\mathrm{\ref{6.8}}$.
Then, $U'_{\mathrm{b}}:=\mathcal{E}(\bar{\rho})_{\mathrm{b}}\cap U'$ is also Zariski dense in $U'$.
\end{prop}
\begin{proof}
Consider any element $f\in R$ in the kernel of the natural map
$R\rightarrow \prod_{z\in U_{\mathrm{b}}}E(z)$. Then, for any
$y\in (\mathcal{W}_E\times_E\mathcal{W}_E)_{\mathrm{cl},x}\cap \pi(U)$, $f|_{\pi^{-1}(y)\cap U}\in \mathcal{O}_{\pi^{-1}(y)\cap U}$ is equal to zero by Proposition \ref{6.5} because $\mathcal{O}_{\pi^{-1}(y)\cap U}$ is reduced. Hence,  we obtain $f=0\in R$ by
Lemma \ref{6.8}.
\end{proof}

\begin{corollary}\label{6.11.1}
Let $Y$ be the Zariski closure of $\mathcal{E}(\bar{\rho})_{\mathrm{b}}$
in $\mathcal{E}(\bar{\rho})$. Then, $Y$ is a union of irreducible components
of $\mathcal{E}(\bar{\rho})$.
\end{corollary}
\begin{proof}
This follows from Proposition \ref{6.11}.
\end{proof}

We denote by
\[
\begin{array}{ll}
\mathfrak{X}(\bar{\rho})_{\mathrm{reg}-\mathrm{cris}}:=&\{x\in \mathfrak{X}(\bar{\rho})| V_x \text{ is crystalline and the Hodge-Tate weight }\\
& \text{ of }V_x\text{ is }\{k_{1,\sigma},k_{2,\sigma}\}_{\sigma\in \mathcal{P}}\text{ such that }k_{1,\sigma}\not= k_{2,\sigma}\text{ for any }\sigma\in\mathcal{P} \},
\end{array}
\]

\[
\begin{array}{ll}
\mathfrak{X}(\bar{\rho})_{\mathrm{b}}:=&\{x\in \mathfrak{X}(\bar{\rho})| V_x\text{ is benign and crystalline }\}.\\
\end{array}
\]

\begin{lemma}\label{6.12}
If $\mathfrak{X}(\bar{\rho})_{\mathrm{reg}-\mathrm{cris}}$ is not empty, then
$\mathfrak{X}(\bar{\rho})_{\mathrm{b}}$ is also not empty.
\end{lemma}
\begin{proof}
The proof is similar to that of Proposition \ref{6.5}.
Let $x\in \mathfrak{X}(\bar{\rho})_{\mathrm{reg}-\mathrm{cris}}$ be a point. Twisting by a suitable character, we may assume that
the Hodge-Tate weight of $V_x$ is $\tau:=\{0, -k_{\sigma}\}_{\sigma\in \mathcal{P}}$ such that
$k_{\sigma}\in \mathbb{Z}_{\geqq 1}$. Then, the subset of
$\mathfrak{X}(\bar{\rho})$ consisting of points corresponding crystalline representations
with Hodge-Tate weight $\tau:=\{0,-k_{\sigma}\}_{\sigma\in \mathcal{P}}$ forms a Zariski closed
rigid analytic subspace $\mathfrak{X}(\bar{\rho})_{\mathrm{cris}}^{\tau}$ of $\mathfrak{X}(\bar{\rho})$ by Corollary 2.7.7 of \cite{Ki08}. By the proof of
Theorem 3.3.8 of \cite{Ki08},
we have an isomorphism $\hat{\mathcal{O}}_{\mathfrak{X}(\bar{\rho})_{\mathrm{cris}}^{\tau},y}\isom
R_{V_y}^{\mathrm{cris}}$ for any point $y\in \mathfrak{X}(\bar{\rho})_{\mathrm{cris}}^{\tau}$. Then, by Corollary 6.3.3 of \cite{Be-Co08} and by Corollary 3.19 of \cite{Ch09a},
there exists an admissible open neighborhood $U$ of $y$ in $\mathfrak{X}(\bar{\rho})_{\mathrm{cris}}^{\tau}$,
such that $D_{\mathrm{cris}}(V_R):= ((R\hat{\otimes}_{\mathbb{Q}_p}B_{\mathrm{cris}})\otimes_{R}V_R)^{G_K}$
is a rank two finite free $K_0\otimes_{\mathbb{Q}_p}R$-module and $D_{\mathrm{dR}}(V_R)\isom K\otimes_{K_0}D_{\mathrm{cris}}(V_R)$,
where $V_R$ is the restriction to $U$ of the universal deformation of $\bar{\rho}$. For any $\sigma'\in \mathrm{Gal}(K_0/\mathbb{Q}_p)$, we denote
by $D_{\sigma'}$ the $\sigma'$-component of $D_{\mathrm{cris}}(V_R)$. We denote by $T^2-aT+b:=\mathrm{det}_R(T\mathrm{id}_{D_{\sigma'}}-\varphi^f|_{D_{\sigma'}})
\in R[T]$ the characteristic polynomial of relative Frobenius on $D_{\sigma'}$, which does not depend on the choice of $\sigma'$.
Then, we claim that $a^2-4b$ is a non zero divisor of $R$, i.e. the subset $U_{a^2-4b}\subseteq U$ consisting of points
$z$ such that $D_{\mathrm{cris}}(V_z)$ has two distinct relative Frobenius eigenvalues is scheme theoretically dense
in $U$. For proving this claim, it suffices to show that,
for any $z\in U$, $a^2-4b\not= 0$ in $\hat{\mathcal{O}}_{\mathfrak{X}(\bar{\rho})_{\mathrm{cris}}^{\tau},z}\isom R_{V_z}^{\mathrm{cris}}$ because
$R_{V_z}^{\mathrm{cris}}$ is domain.
But it is easy to see that $D_{\mathrm{cris}}(V_z)$ can be deformed to $E(z)[\varepsilon]$ such that with two distinct relative Frobenius
eigenvalues, hence $a^2-4b\not =0 $ in $R_{V_z}^{\mathrm{cris}}$. In the same way, we can show that the subset $U''\subseteq U$ consisting of points $z$ such that $D_{\mathrm{cris}}(V_z)$ has relative Frobenius eigenvalues $\alpha_1, \alpha_2$
satisfying $\alpha_1\not= p^{\pm f}\alpha_2$ is also scheme theoretically dense in $U$, hence their intersection
$U_{a^2-4b}\cap U''$ is also scheme theoretically dense in $U$.
Next, we take an element $z\in U_{a^2-4b}\cap U''\subseteq U$, then by extending scalar,
we may assume that $D_{\mathrm{cris}}(V_z)\isom K_0\otimes_{\mathbb{Q}_p}Ee_{1,z}\oplus K_0\otimes_{\mathbb{Q}_p}Ee_{2,z}$ such that
$\varphi^f(e_{i,z})=\alpha_{i,z}e_{i,z}$ for some $\alpha_{i,z}\in E^{\times}$ for $i=1,2$ such that $\alpha_{1,z}\not=\alpha_{2,z},p^{\pm f}\alpha_{2,z}$. Then, because
$\mathcal{O}_{\mathfrak{X}(\bar{\rho})^{\tau}_{\mathrm{cris}},z}$ is Henselian by Theorem 2.1.5 of \cite{Berk93}, for any sufficiently small affinoid open
neighborhood $U'=\mathrm{Spm}(R')$ of $z$ in $U_{a^2-4b}\cap U''$, we have $D_{\mathrm{cris}}(V_R)\otimes_RR'\isom K_0\otimes_{\mathbb{Q}_p}R'e_1\oplus K_0\otimes_{\mathbb{Q}_p}R'e_2$
such that $K_0\otimes_{\mathbb{Q}_p}R'e_i$ is $\varphi$-stable and $\varphi^f(e_i)=\tilde{\alpha}_ie_i$ for some $\tilde{\alpha}_i\in R^{' \times}$ for $i=1,2$
satisfying that $\tilde{\alpha}_1-\tilde{\alpha}_2, \tilde{\alpha}_1-p^{\pm f}\tilde{\alpha}_2\in R^{' \times}$. Then, for sufficiently small $U'$, we have $D_{\mathrm{dR}}(V_R)\otimes_{R}R'\isom D_{\mathrm{dR}}(V_{R'})$ and
$\mathrm{Fil}^1D_{\mathrm{dR}}(V_{R'}):=((R'\hat{\otimes}_{\mathbb{Q_p}}tB^+_{\mathrm{dR}})\otimes_{R'}V_{R'})^{G_K}\subseteq D_{\mathrm{dR}}(V_{R'})$ is a free $K\otimes_{\mathbb{Q}_p}R'$-module of rank one which is compatible with any base change by Proposition 2.5 and Corollary 2.6 of \cite{Ki03} and by the proof of Theorem 5.3.2 of
\cite{Be-Co08}. Then, for any $\sigma\in \mathcal{P}$, the $R'$-base of $\sigma$-component
of $\mathrm{Fil}^1D_{\mathrm{dR}}(V_{R'})$ can be written by $a_{\sigma}e_{1,\sigma}+b_{\sigma}e_{2,\sigma}$ for some
$a_{\sigma},b_{\sigma}\in R'$ where $e_{i,\sigma}$ is the $\sigma$-component of $e_{i}\in D_{\mathrm{cris}}(V_R)\otimes_R R'\subseteq
D_{\mathrm{dR}}(V_R)\otimes_R R'\isom D_{\mathrm{dR}}(V_{R'})$. Then, by the same argument as in the proof of Proposition \ref{6.5},
the subset of $U'$ consisting of benign representations is $U'_{(\prod_{\sigma\in \mathcal{P}}a_{\sigma}b_{\sigma})}$. Because
$\prod_{\sigma\in \mathcal{P}}a_{\sigma}b_{\sigma}$ is non zero divisor of $R'$, which can be proved in the same way as in the proof of Proposition \ref{6.5},
$U'_{(\prod_{\sigma\in \mathcal{P}}a_{\sigma}b_{\sigma})}\subseteq U'$ is scheme theoretically dense in $U'$, in particular, $\mathfrak{X}(\bar{\rho})_{\mathrm{b}}$ is non-empty.

\end{proof}

\begin{rem}\label{6.13}
In the proof of the above lemma, we prove that $\mathfrak{X}(\bar{\rho})_{\mathrm{cris},\mathrm{b}}^{\tau}:=\{z\in \mathfrak{X}(\bar{\rho})_{\mathrm{cris}}^{\tau}| V_z $ is benign $\}$ is scheme theoretically dense
in $\mathfrak{X}(\bar{\rho})_{\mathrm{cris}}^{\tau}$ for any $\tau:=\{k_{1,\sigma},k_{2,\sigma}\}_{\sigma\in \mathcal{P}}$ such that
$k_{1,\sigma}\not =k_{2,\sigma}$ for any $\sigma\in \mathcal{P}$.
\end{rem}

For a rigid analytic space $Y$ over $E$ and for a point $y\in Y$ , we denote by
$$t_{Y,x}:=\mathrm{Hom}_{E(y)}(\mathfrak{m}_y/\mathfrak{m}_y^2, E(y))$$
 the tangent space at $y$, where
$\mathfrak{m}_y$ is the maximal ideal of $\mathcal{O}_{Y,y}$. The following theorems are the
main theorems of this article.

We denote by $\overline{\mathfrak{X}(\bar{\rho})}_{\mathrm{b}}$ the Zariski closure of $\mathfrak{X}(\bar{\rho})_{\mathrm{b}}$ in $\mathfrak{X}(\bar{\rho})$.

\begin{thm}\label{6.14}
If $\mathfrak{X}(\bar{\rho})_{\mathrm{reg}-\mathrm{cris}}$ is non empty,
then $\overline{\mathfrak{X}(\bar{\rho})}_{\mathrm{b}}$ is non empty and a union of irreducible components
of $\mathfrak{X}(\bar{\rho})$.
\end{thm}
\begin{proof}
First, by Lemma \ref{6.12}, $Z:=\overline{\mathfrak{X}(\bar{\rho})}_{\mathrm{b}}$ is non empty.
Because any irreducible components have at most $4[K:\mathbb{Q}_p]+1$ dimension, so
it suffices to show that any irreducible components of $Z$ have $4[K:\mathbb{Q}_p]+1$ dimension.
Let $Z'$ be an irreducible component of $Z$.

Because the singular locus $Z'_{\mathrm{sing}}\subseteq Z'$ is a proper Zariski closed set in $Z'$, there exists a benign point
$x\in \mathfrak{X}(\bar{\rho})_{\mathrm{b}}\cap Z'$ such that $Z'$ ( and $\mathfrak{X}(\bar{\rho})$ ) is smooth at $x$.

By the definition of benign representation and by Theorem \ref{5.14}, there exists
the different two points
$$x_1:=([V_x],\delta_{x_1},\lambda_{x_1}), x_2:=([V_x],\delta_{x_2},\lambda_{x_2})\in \mathcal{E}(\bar{\rho})$$
such that $p_1(x_i)=x$ and with property (ii) in the Theorem \ref{5.14}.  We denote by $Y'_i$ an irreducible component of
$p_1^{-1}(Z)$ containing $x_i$ for $i=1,2$ respectively. By Corollary \ref{6.11.1}, these are also irreducible components
of $\mathcal{E}(\bar{\rho})$ and $Y'_i$ is unique by Theorem \ref{5.18}.

Because the natural morphisms $p_1|_{Y'_i}:Y'_i\rightarrow \mathfrak{X}(\bar{\rho})$ factor through $Z'$ for $i=1,2$,
we obtain maps, for $i=1,2$,
$$t_{\mathcal{E}(\bar{\rho}),x_i}=t_{Y'_i,x_i}\rightarrow t_{Z',x}\hookrightarrow t_{\mathfrak{X}(\bar{\rho}),x}.$$

Hence, we obtain a map
$$\bigoplus_{i=1,2}t_{\mathcal{E}(\bar{\rho}),x_i}\rightarrow t_{Z',x}\hookrightarrow t_{\mathfrak{X}(\bar{\rho}),x}.$$
By Theorem \ref{33} and Theorem \ref{5.18}, this map is surjective, hence we obtain an equality
$$t_{Z',x}=t_{\mathfrak{X}(\bar{\rho}),x}. $$ Because $x$ is smooth at $Z'$, then $Z'$ has dimension $4[K:\mathbb{Q}_p]+1$.

\end{proof}

Concerning the assumption that $\mathfrak{X}(\bar{\rho})_{\mathrm{reg}-\mathrm{cris}}$ is non empty, in this paper we prove the following (maybe well-known) lemma.

\begin{lemma}\label{4.16}
If $\bar{\rho}\otimes_{\mathbb{F}}\bar{\mathbb{F}}\not\sim\begin{pmatrix} \omega & \ast \\ 0 & 1 \end{pmatrix}\otimes \chi$ and
$\bar{\rho}\otimes_{\mathbb{F}}\bar{\mathbb{F}}\not\sim\begin{pmatrix} 1 & \ast \\ 0 & 1\end{pmatrix} \otimes \chi$ for any
character $\chi:G_K\rightarrow \bar{\mathbb{F}}^{\times}$, where $\omega$ is the mod $p$ cyclotomic
character. Then, $\mathfrak{X}(\bar{\rho})_{\mathrm{reg}-\mathrm{cris}}$ is non empty.
\end{lemma}
\begin{proof}
First, we prove the absolutely reducible cases. Extending $\mathbb{F}$, we may assume that $\bar{\rho}$ is reducible.
Because any character $\chi:G_K\rightarrow \mathbb{F}^{\times}$ has
a crystalline lift, we may assume that $\bar{\rho}=\begin{pmatrix}\chi & \ast \\ 0 & 1\end{pmatrix}$ for
a character $\chi:G_K\rightarrow \mathbb{F}^{\times}$ such that $\chi\not= 1$ and $\chi\not= \omega$. Using
twists of a Lubin-Tate character of $K$ by $\sigma\in\mathcal{P}$ and a unramified character, we can take a crystalline
lift $\tilde{\chi}:G_K\rightarrow \mathcal{O}^{\times}$ of $\chi$ whose Hodge-Tate weight  is
$\{k_{\sigma}\}_{\sigma\in \mathcal{P}}$ such that $k_{\sigma}\geqq 1$ for any $\sigma\in \mathcal{P}$.
Under the assumption $\chi\not= 1, \omega$, $\mathrm{H}^1(G_K,\mathcal{O}(\tilde{\chi}))$ is a free
$\mathcal{O}$-module of rank $[K:\mathbb{Q}_p]$ and the natural map
$\mathrm{H}^1(G_K,\mathcal{O}(\tilde{\chi}))\rightarrow \mathrm{H}^1(G_K,\mathbb{F}(\chi))$ is surjection.
By the choice of Hodge-Tate weight of $\tilde{\chi}$, we have an equality
$\mathrm{H}^1_f(G_K,E(\tilde{\chi}))=\mathrm{H}^1(G_K,E(\tilde{\chi}))$.
These imply that any extension in $\mathrm{H}^1(G_K,\mathbb{F}(\chi))$ lifts to
a extension in $\mathrm{H}^1(G_K,\mathcal{O}(\tilde{\chi}))$ which is crystalline.

Next, we prove the absolutely irreducible case. In this case, if we denote by $K_2$ the unramified extension
of $K$ such that $[K_2:K]=2$ and denote by $\chi_2:G^{\mathrm{ab}}_{K_2}\rightarrow \mathbb{F}^{\times}$
the mod $p$ reduction of the Lubin-Tate character $\chi_{2,\mathrm{LT}}$ of $K_2$ associated
to a uniformizer $\pi_{K_2}$
of $K_2$, then it is known that $\bar{\rho}\isom (\mathrm{Ind}^{G_{K}}_{G_{K_2}}\chi_2^i)\otimes \chi$ for a character
$\chi:G_K\rightarrow \mathbb{F}^{\times}$ and for some $i\in \mathbb{Z}$ such that $i\not\equiv 0 $ (mod $p^f+1$).
Hence, it suffices to show that $\mathrm{Ind}^{G_{K}}_{G_{K_2}}\chi_2^i$ has a crystalline lift. Because
$\chi_2$ is the mod $p$ reduction of $\chi_{2,\mathrm{LT}}$, we can take a lift of $\chi_2^i$ of the form
$\prod_{\sigma\in \mathcal{P}}\tilde{\sigma}(\chi_{2,\mathrm{LT}})^{k_{\sigma}}$ such that $k_{\sigma}\geqq 1$ for all
$\sigma\in \mathcal{P}$, where $\tilde{\sigma}:K_2\hookrightarrow \overline{K}$  is an extension of $\sigma$. Then,
$\mathrm{Ind}^{G_{K}}_{G_{K_2}}\chi_2^i$ has a crystalline lift $\mathrm{Ind}^{G_{K}}_{G_{K_2}}(\prod_{\sigma\in \mathcal{P}}
\tilde{\sigma}(\chi_{2,\mathrm{LT}})^{k_{\sigma}})$ whose Hodge-Tate weight is $\{0,k_{\sigma}\}_{\sigma\in \mathcal{P}}$.

\end{proof}
We can obtain the following theorem.
We remark that, from $\S$ 3.3, we assume $\mathrm{End}_{G_K}(\bar{\rho})=\mathbb{F}$ but, even if
$\mathrm{End}(\bar{\rho})\not=\mathbb{F}$,
we can prove Theorem \ref{5.14} and Theorem \ref{5.18} and Theorem \ref{6.14} without any additional
difficulties if we consider the universal framed deformations
instead of usual deformations.
But, up to now,  the author does not know the proof of the following theorem in the case
$\mathrm{End}_{G_K}(\bar{\rho})\not=\mathbb{F}$.

\begin{thm}\label{4.17}
We assume the following conditions.
\begin{itemize}
\item[(0)]$\mathrm{End}_{G_K}(\bar{\rho})=\mathbb{F}$.
\item[(1)]$\mathfrak{X}(\bar{\rho})_{\mathrm{reg}-\mathrm{cris}}$ is not empty.
\item[(2)]$\bar{\rho}\not\sim \begin{pmatrix} 1 & \ast \\ 0 & \omega \end{pmatrix}\otimes \chi$ for
any $\chi:G_K\rightarrow \mathbb{F}^{\times}$.

\item[(3)]$[K(\zeta_p):K]\not=2$ or, for any $\chi$, $\bar{\rho}|_{I_K}\not\sim \begin{pmatrix} \chi_2^i & 0 \\ 0 & \chi_2^{ip^f}\end{pmatrix}\otimes \chi$ such that $i=\frac{p^f+1}{2}$, where $\zeta_p$ is a primitive $p$-th root of $1$.
\end{itemize}
Then, we have an equality $\overline{\mathfrak{X}(\bar{\rho})}_{\mathrm{b}}=\mathfrak{X}(\bar{\rho})$.

\end{thm}
\begin{proof}
First, we prove in the case $\zeta_p\not\in K$, i.e. $\omega$ is not trivial.
In this case, under the conditions of (2) and (3), we have
$$\mathrm{H}^2(G_K,\mathrm{ad}(\bar{\rho}))\isom
\mathrm{H}^0(G_K, \mathrm{ad}(\bar{\rho})(\omega))^{\vee}=\mathrm{Hom}_{G_K}(\bar{\rho}, \bar{\rho}\otimes \omega)^{\vee}=0.$$
Hence, the universal deformation ring $R_{\bar{\rho}}$ is formally smooth over $\mathcal{O}$ and
$\mathfrak{X}(\bar{\rho})$ is isomorphic to $(4[K:\mathbb{Q}_p]+1)$-dimensional open unit disc, in particular irreducible.
Hence, by Theorem \ref{6.14}, $\mathfrak{X}(\bar{\rho})_{\mathrm{b}}$ is Zariski dense in $\mathfrak{X}(\bar{\rho})$.

Next, we prove in the case $\zeta_p\in K$. In this case $\mathfrak{X}(\bar{\rho})$ is not irreducible.
Let $P\subset \mathcal{O}_{K}^{\times}$ be the sub group of $\mathcal{O}_K^{\times}$ consisting of
all the $p$-th power roots of $1$ and let $p^n$ be the order of this sub group and let $\zeta_{p^n}\in \mathcal{O}_K^{\times}$ be
a primitive $p^n$-th root of $1$. For any $0\leqq i\leqq p^n-1$, we define a sub functor $D_{\bar{\rho},i}$ of $D_{\bar{\rho}}$
by, for any $A\in \mathcal{C}_{\mathcal{O}}$,
$$ D_{\bar{\rho},i}(A):=\{(V_A,\psi_A)\in D_{\bar{\rho}}(A)| \mathrm{det}(V_A)(\mathrm{rec}_K(\zeta_{p^n}))=\iota_A(\zeta_{p^n})^i \},$$
where $\iota_A:\mathcal{O}\rightarrow A$ is the morphism which gives an $\mathcal{O}$-algebra structure to $A$.
It is easy to see that the canonical inclusion $D_{\bar{\rho},i}\hookrightarrow D_{\bar{\rho}}$ is relatively representable, i.e.
this satisfies the conditions (1) and (2) and (3) in the proof of Proposition \ref{15}. For any $i$, let $R_{\bar{\rho},i}$ be the quotient of
$R_{\bar{\rho}}$ which represents $D_{\bar{\rho},i}$ and let $\mathfrak{X}(\bar{\rho})_i\subseteq \mathfrak{X}(\bar{\rho})$ be the
Zariski closed rigid analytic space associated to $R_{\bar{\rho},i}$, then it is easy to see that, as rigid analytic space,
$\mathfrak{X}(\bar{\rho})$ is the disjoint union of $\mathfrak{X}(\bar{\rho})_i$ for $0\leqq i\leqq p^n-1$,
$$ \mathfrak{X}(\bar{\rho})=\coprod_{0\leqq i\leqq p^n-1} \mathfrak{X}(\bar{\rho})_i.$$ Moreover, we claim that
each $\mathfrak{X}(\bar{\rho})_i$ is isomorphic to $(4[K:\mathbb{Q}_p]+1)$-dimensional open unit disc. For proving this claim,
it suffices to show that the functor $D_{\bar{\rho},i}$ is formally smooth.

We prove formally smoothness of $D_{\bar{\rho},i}$ as follows.
Let $A$ be an object of $\mathcal{C}_{\mathcal{O}}$ and $I\subseteq A$ be a non zero ideal such that $I\mathfrak{m}_A=0$.
Let $(V_{A/I},\psi_{A/I})$ be a deformation of $\bar{\rho}$ in $D_{\bar{\rho},i}(A/I)$, then  it suffices to show that
$(V_{A/I},\psi_{A/I})$ lifts to $D_{\bar{\rho},i}(A)$. Fixing $A/I$-base of $V_{A/I}$, we represent $V_{A/I}$ by
a continuous homomorphism $\rho_{A/I}:G_K\rightarrow \mathrm{GL}_2(A/I)$. Because the obstruction of the liftings
of $\mathrm{det}(\bar{\rho})$ comes only from that of $\mathrm{det}(\bar{\rho})|_{\mathrm{rec}_K(P)}$, we can take a continuous
 character
$c_A:G_K^{\mathrm{ab}}\rightarrow A^{\times}$ which is a lift of $\mathrm{det}(\rho_{A/I})$ and $c_{A}(\mathrm{rec}_K(\zeta_{p^n}))
=\iota_A(\zeta_{p^n})^i$. We take a continuous lift $\tilde{\rho}_A:G_K\rightarrow \mathrm{GL}_2(A)$ of
$\rho_{A/I}$ such that $\mathrm{det}(\tilde{\rho}_A(g))=c_A(g)$ for any $g\in G_K$ and then
we define a $2$-cocycle $f:G_K\times G_K\rightarrow I\otimes_{\mathbb{F}}\mathrm{ad}(\bar{\rho})$ by
$$\tilde{\rho}_A(g_1g_2)\tilde{\rho}_A(g_2)^{-1}\tilde{\rho}_A(g_1)^{-1}:=1+f(g_1,g_2)\in 1+I\otimes_A\mathrm{M}_2(A)=1+I\otimes_{\mathbb{F}}\mathrm{ad}(\bar{\rho}).$$
Because $\mathrm{det}(\tilde{\rho}_A)=c_A$ is a homomorphism, $f(g_1,g_2)$ is contained in $I\otimes_{\mathbb{F}}\mathrm{ad}^0(\bar{\rho})$, where we denote by $\mathrm{ad}^0(\bar{\rho}):=\{a\in \mathrm{ad}(\bar{\rho})| \mathrm{trace}(a)=0\}$.
Hence, we obtain
a class of $2$-cocycle $[f]\in \mathrm{H}^2(G_K,\mathrm{ad}^0(\bar{\rho}))$. But, under the assumption (0) and $\zeta_p\in K$, we have
$$\mathrm{H}^2(G_K, \mathrm{ad}^0(\bar{\rho}))\isom \mathrm{H}^0(G_K, \mathrm{ad}^0(\bar{\rho})(\omega))^{\vee}=
\mathrm{H}^0(G_K, \mathrm{ad}^0(\bar{\rho}))^{\vee}=0.$$ Hence, twisting $\tilde{\rho}_A$ by using a suitable continuous one cochain
$d:G_K\rightarrow I\otimes_{\mathbb{F}}\mathrm{ad}^0(\bar{\rho})$, we obtain a continuous homomorphism
 $\rho_A:G_K\rightarrow \mathrm{GL}_2(A)$ such that $\rho_A$ is a lift of $\rho_{A/I}$ and $\mathrm{det}(\rho_{A})=c_A$, in particular
 $(V_{A/I,\psi_{A/I}})$ lifts to $D_{\bar{\rho},i}(A)$, which proves the claim.

 By this claim and by Theorem \ref{6.14}, it suffices to show that, under the assumption (1), $\mathfrak{X}(\bar{\rho})_i\cap
 \mathfrak{X}(\bar{\rho})_{\mathrm{b}}$ is non empty for any $i$. We prove this claim as follows.
 First, by the assumption (1), there exists $i$ such that $\mathfrak{X}(\bar{\rho})_i\cap \mathfrak{X}(\bar{\rho})_{\mathrm{b}}$ is
 non empty. We take a point $x=[V_x]\in \mathfrak{X}(\bar{\rho})_i\cap \mathfrak{X}(\bar{\rho})_{\mathrm{b}}$.

When $p\not= 2$,  the twist $V_x(\chi_{\mathrm{LT}}^{j(p^f-1)})$ of $V_x$ for any $j\in \mathbb{Z}$ is contained in $\mathfrak{X}(\bar{\rho})_{\mathrm{b}}\cap \mathfrak{X}(\bar{\rho})_{i_j}$, where
we define $i_j$ such that $0\leqq i_j\leqq p^n-1$ and $i_j\equiv i+2j(p^f-1)$ ( mod $p^n$). When $p\not=2$, $i_j$ runs through all
$0\leqq i'\leqq p^n-1$, hence $\mathfrak{X}(\bar{\rho})_{\mathrm{b}}\cap \mathfrak{X}(\bar{\rho})_{i'}$ is non empty
for any $i'$. Hence, $\mathfrak{X}(\bar{\rho})_{\mathrm{b}}$ is Zariski dense in $\mathfrak{X}(\bar{\rho})$ in this case.

Finally, we consider $p=2$ case. By the same argument as above, $\mathfrak{X}(\bar{\rho})_{\mathrm{b}}\cap
\mathfrak{X}(\bar{\rho})_{i'}$ is non empty for any $0\leqq i'\leqq 2^n-1$ such that $i'\equiv i$ (mod $2$). Hence, it suffices to show
that there exists $i_1$ and $i_2$ such that $0\leqq i_1,i_2\leqq 2^n-1$ and $i_1\not\equiv i_2$ (mod $2$) and
$\mathfrak{X}(\bar{\rho})_{i_j}\cap \mathfrak{X}(\bar{\rho})_{\mathrm{b}}$ is non empty for $j=1,2$.
We prove this claim as follows.

If $\rho$ is absolutely reducible, then under the assumption of  this Theorem,
$\rho\sim \begin{pmatrix} \chi & \ast \\ 0 & 1\end{pmatrix}\otimes \eta$ for some $\chi$ and $\eta$ such that
$\chi\not= 1$.  We may assume that $\eta=1$. By the proof of Lemma \ref{4.16}, there exists a crystalline lift $\tilde{\chi}:G_K
\rightarrow \mathcal{O}^{\times}$ such that there exists an $x\in \mathfrak{X}(\bar{\rho})_{\mathrm{reg}-\mathrm{cris}}\cap
\mathfrak{X}(\bar{\rho})_i$ such that
$[V_x]\in \mathrm{H}^1(G_K, E(\tilde{\chi}))$ for some $i$. There also exists an $x'\in \mathfrak{X}(\bar{\rho})_{\mathrm{reg}-\mathrm{cris}}\cap\mathfrak{X}(\bar{\rho})_{i'}$ such that
$[V_{x'}]\in \mathrm{H}^1(G_K, E(\tilde{\chi}\chi_{\mathrm{LT}}^{2^f-1}))$ and $i\not\equiv i'$ (mod $2$).
By the same proof as in Lemma \ref{6.12}, we can show that $\mathfrak{X}(\bar{\rho})_{\mathrm{b}}\cap \mathfrak{X}(\bar{\rho})_{i}$
and $\mathfrak{X}(\bar{\rho})_{\mathrm{b}}\cap \mathfrak{X}(\bar{\rho})_{i'}$ are non empty, which proves the above claim.
In absolutely irreducible case, we can prove in the same way. Hence, when $p=2$, $\mathfrak{X}(\bar{\rho})_{\mathrm{b}}$ is
Zariski dense in $\mathfrak{X}(\bar{\rho})$. We finish the proof of this theorem.

\end{proof}

\section{Appendix : Continuous cohomology of $B$-pairs}
In \cite{Na09}, we defined a cohomology $\mathrm{H}^i(G_K,W)$ by using continuous cochains which
we review below. On the other hand,  Liu \cite{Li08} defined another cohomology which we write by $\mathrm{H}^i_{\mathrm{Liu}}(G_K, W):=
\mathrm{H}^i(D(W))$
by using a complex defined from the $(\varphi,\Gamma)$-module $D(W)$ associated to $W$, see 2.1 of \cite{Li08} for the definition. Moreover, he proved that this cohomology
satisfies Euler-Poincar\`e formula and Tate duality. In this appendix, we prove  that  $\mathrm{H}^i(G_K, W)$ also
satisfies Euler-Poincar\`e  formula and Tate duality and that $\mathrm{H}^i(G_K, W)$ is canonically isomorphic to  $\mathrm{H}^i_{\mathrm{Liu}}(G_K, W)$.

We recall the definition of $\mathrm{H}^i(G_K, W)$. Let $G$ be a topological group. For a continuous $G$-module $M$, we define the group of
 $i$-th continuous cochains by $$\mathrm{C}^i(G, M):=\{c:G^{\times i}\rightarrow M| c\text{ is a continuous map }\}.$$
 As usual, we define the boundary map
  $$\delta:\mathrm{C}^i(G, M)\rightarrow \mathrm{C}^{i+1}(G, M)$$ by
  \[
\begin{array}{ll}
\delta(c)(g_1,g_2,\cdots, g_{i+1}):=& g_1c(g_2,\cdots, g_{i+1}) +(-1)^{i+1}c(g_1,g_2,\cdots, g_i)\\
 &+\sum_{s=1}^{i}(-1)^i c(g_1,\cdots, g_{s-1}, g_s g_{s+1},g_{s+2},\cdots, g_{i+1}) .
\end{array}
\]
 Let $W=(W_e, W^+_{\mathrm{dR}})$ be a $B$-pair and let $W_{\mathrm{dR}}:=W_e\otimes_{B_e}B_{\mathrm{dR}}$. For $W$, we define a complex $\mathrm{C}^{\bullet}(G_K, W)$ of $\mathbb{Q}_p$-vector spaces as
 the mapping cone of the map $$\mathrm{C}^{\bullet}(G_K, W_e)\oplus \mathrm{C}^{\bullet}(G_K, W^+_{\mathrm{dR}})\rightarrow
 \mathrm{C}^{\bullet}(G_K, W_{\mathrm{dR}}): (c_e, c_{\mathrm{dR}})\mapsto c_e-c_{\mathrm{dR}},$$ i.e.
 we define by
 $$\mathrm{C}^0(G_K, W):=\mathrm{C}^0(G_K, W_e)\oplus \mathrm{C}^0(G_K, W^+_{\mathrm{dR}})$$
 and by
 $$\mathrm{C}^i(G_K, W):=\mathrm{C}^{i}(G_K, W_e)\oplus \mathrm{C}^i(G_K, W^+_{\mathrm{dR}})\oplus \mathrm{C}^{i-1}(G_K, W_{\mathrm{dR}})$$ for $i\geqq 1$ and the differentials $\delta$ are defined by $\delta:\mathrm{C}^0(G_K, W_e)\oplus \mathrm{C}^0(G_K,  W^+_{\mathrm{dR}}) \rightarrow \mathrm{C}^1(G_K, W_e)\oplus \mathrm{C}^1(G_K, W^+_{\mathrm{dR}})\oplus \mathrm{C}^0(G_K, W_{\mathrm{dR}})(c_e, c_{\mathrm{dR}})  \mapsto (\delta(c_e),  \delta(c_{\mathrm{dR}}), c_e-c_{\mathrm{dR}})$ and
 $\delta:\mathrm{C}^i(G_K, W_e)\oplus \mathrm{C}^i(G_K, W^+_{\mathrm{dR}})\oplus \mathrm{C}^{i-1}(G_K, W_{\mathrm{dR}})
 \rightarrow \mathrm{C}^{i+1}(G_K, W_e) \oplus \mathrm{C}^{i+1}(G_K, W^+_{\mathrm{dR}})\oplus \mathrm{C}^i(G_K, W_{\mathrm{dR}})(c_e, c_{\mathrm{dR}}, c)\mapsto (\delta(c_e), \delta(c_{\mathrm{dR}}), c_e-c_{\mathrm{dR}}-\delta(c))$
  for $i\geqq 1.$ We define  the cohomology of $W$ by
 $$\mathrm{H}^i(G_K, W):=\mathrm{H}^i(\mathrm{C}^{\bullet}(G_K, W)) ,$$ and
 $$\mathrm{H}^i(G_K, W_{e}):=\mathrm{H}^i(\mathrm{C}^{\bullet}(G_K, W_e))$$ and
 $$\mathrm{H}^i(G_K, W^+_{\mathrm{dR}}):=
 \mathrm{H}^i(\mathrm{C}^{\bullet}(G_K, W^+_{\mathrm{dR}})), \,\,\mathrm{H}^i(G_K, \allowbreak W_{\mathrm{dR}}):=\mathrm{H}^i(\mathrm{C}^{\bullet}(G_K, W_{\mathrm{dR}})).$$
 By these definitions, we have the following long exact sequence,
 $$ \cdots \rightarrow \mathrm{H}^{i-1}(K, W_{\mathrm{dR}})\rightarrow \mathrm{H}^i(G_K, W)\rightarrow \mathrm{H}^i(G_K, W_e)\oplus
 \mathrm{H}^i(G_K, W^+_{\mathrm{dR}})\rightarrow \cdots .$$

 Before proving Euler-Poincar\`e formula, we recall some results of \cite{Be09}.
 Let $W$ be an almost $\mathbb{C}_p$-representation and $V_1$ and $V_2$ be
 $\mathbb{Q}_p$-representations of $G_K$ of dimension $d_1$ and $d_2$ respectively
 and $d\geqq 0$ be an integer
 such that $V_1\subseteq W$ and $V_2\subseteq \mathbb{C}_p^{\oplus d}$ and
 $W/V_1\isom \mathbb{C}_p^{\oplus d}/V_2$, then we define
 the dimension of $W$ by $\mathrm{dim}_{\mathcal{C}(G_K)}(W):=d$ and the height of $W$ by $\mathrm{ht}(W):=d_1-d_2$.
 Let $W:=(W_e, W^+_{\mathrm{dR}})$ be a $B$-pair.
 We define $X_0(W):=W_e\cap W^+_{\mathrm{dR}}$ and $X_1(W):=W_{\mathrm{dR}}/(W_e+W^+_{\mathrm{dR}})$.
 In \cite{Be09}, Berger proved the following theorem.
 \begin{thm}\label{5-0}
 Let $W$ be a $B$-pair of rank $d$, then :
 \begin{itemize}
 \item[(1)]$X_0(W)$ and $X_1(W)$ are almost $\mathbb{C}_p$-representations,
 \item[(2)]If $W$ is pure of slope $s\leqq 0$, then $\mathrm{dim}_{\mathcal{C}(G_K)}(X_0(W))=-
 sd$ and $\mathrm{ht}(X_0(W))=d$ and $X_1(W)=0$,
 \item[(3)]If $W$ is pure of slope $s>0$, then $X_0(W)=0$ and $\mathrm{dim}_{\mathcal{C}(G_K)}(X_1(W))=sd$ and
 $\mathrm{ht}(X_1(W))=-d$.

 \end{itemize}
 \end{thm}
 \begin{proof}
 See Theorem 3.1 of \cite{Be09}.
 \end{proof}
 \begin{lemma}\label{5-0-5}
 Let $W$ be an almost $\mathbb{C}_p$-representation, then $\mathrm{H}^i(G_K W)$ is finite dimensional over $\mathbb{Q}_p$ for $i=0,1,2$ and zero for $i\geqq 3$.
 \end{lemma}
 \begin{proof}
 This follows from the definition of almost $\mathbb{C}_p$-representations and
 the facts that $\mathrm{H}^i(G_K, V)=0$ for $i\geqq 3$ for any $\mathbb{Q}_p$-representation $V$
 of $G_K$ and that $\mathrm{H}^{i}(G_K, \mathbb{C}_p)=0$ for $i\geqq 2$ and that $\mathrm{H}^i(G_K, V)$ and
 $\mathrm{H}^i(G_K, \mathbb{C}_p)$ are finite dimensional over $\mathbb{Q}_p$.
 \end{proof}
 For an almost $\mathbb{C}_p$-representation $W$, we write $\chi(W):=\sum_{i=0}^2(-1)^i\mathrm{dim}_{\mathbb{Q}_p}
 \mathrm{H}^i(G_K, W)$.
 \begin{lemma}\label{5-0-6}
 $\chi(W)=-[K:\mathbb{Q}_p]\mathrm{ht}(W)$.
 \end{lemma}
 \begin{proof}
 This follows from the definition of almost $\mathbb{C}_p$-representations and Euler-Poincar\`e formula for
 $\mathbb{Q}_p$-representations of $G_K$ and the fact that $\chi(\mathbb{C}_p)=0$.
 \end{proof}

 \begin{lemma}\label{5-1}
 The following equalities hold.
 \begin{itemize}
 \item[(1)]$\mathrm{C}^{\bullet}(W_e)=\varinjlim_n \mathrm{C}^{\bullet}(W_e\cap \frac{1}{t^n}W^+_{\mathrm{dR}}).$
 \item[(2)]$\mathrm{C}^{\bullet}(W_{\mathrm{dR}})=\varinjlim_n\mathrm{C}^{\bullet}(G_K, \frac{1}{t^n}W^+_{\mathrm{dR}}).$
 \end{itemize}

 \end{lemma}
 \begin{proof}
 For any $n$, $\frac{1}{t^n}W^+_{\mathrm{dR}}$ is closed in $\frac{1}{t^{n+1}}W^+_{\mathrm{dR}}$ and
 the topology on $\frac{1}{t^n}W^+_{\mathrm{dR}}$ is the topology induced from $\frac{1}{t^{n+1}}W^+_{\mathrm{dR}}$.
 Hence, by Proposition 5.6 of \cite{Schn01}, we obtain the equality (2).
 For $W_e$, if we fix an isomorphism $W_e\isom B_e^{\oplus d}$ as $B_e$-module, the topology on $W_e$
 is defined by the direct sum topology of $B_e$. Because we have an equality
 $tB_{\mathrm{max}}^{+, \varphi=p^n}=
 \cap_{m\geqq 0} \mathrm{Ker}(\theta(\varphi^{m}): B_{\mathrm{max}}^{+,\varphi=p^{n+1}}\rightarrow \mathbb{C}_p)$ by
 Proposition 8.10 (2) of \cite{Co02},
 $\frac{1}{t^n}B_{\mathrm{max}}^{+,\varphi=p^n}$ is closed in $\frac{1}{t^{n+1}}B_{\mathrm{max}}^{+,\varphi=p^{n+1}}$ and the
 topology on $\frac{1}{t^n}B_{\mathrm{max}}^{+,\varphi=p^n}$ is the topology induced from $\frac{1}{t^{n+1}}B_{\mathrm{max}}^{+,\varphi=p^{n+1}}$.
 Hence, by Proposition 5.6 of \cite{Schn01}, we have $\mathrm{C}^{\bullet}(G_K, W_e)
 =\varinjlim_n\mathrm{C}^{\bullet}(G_K, (\frac{1}{t^n}B_{\mathrm{max}}^{+,\varphi=p^n})^{\oplus d})=\varinjlim_n
 \mathrm{C}^{\bullet}(G_K, W_e\cap \frac{1}{t^n}W^+_{\mathrm{dR}})$.
 \end{proof}

\begin{lemma}\label{5-2}
Let $W^+_{\mathrm{dR}}$ be a finite free $B^+_{\mathrm{dR}}$-module with
a continuous semi-linear $G_K$-action. Then the canonical map
$\mathrm{H}^i(G_K,W^+_{\mathrm{dR}})\rightarrow \varprojlim_{n}\mathrm{H}^i(G_K, W^+_{\mathrm{dR}}/t^n W^+_{\mathrm{dR}})$
is isomorphism.
\end{lemma}
\begin{proof}
Because we have $\mathrm{C}^{\bullet}(G_K, W^+_{\mathrm{dR}})\isom \varprojlim_n \mathrm{C}^{\bullet}(G_K, W^+_{\mathrm{dR}}/t^nW^+_{\mathrm{dR}})$, for any $i\geqq 0$, we have the following short exact sequence,
$$0\rightarrow \mathbb{R}^1\varprojlim_n\mathrm{H}^{i-1}(G_K, W^+_{\mathrm{dR}}/t^nW^+_{\mathrm{dR}})\rightarrow
\mathrm{H}^i(G_K, W^+_{\mathrm{dR}})\rightarrow \varprojlim_n\mathrm{H}^i(G_K, W^+_{\mathrm{dR}}/t^nW^+_{\mathrm{dR}}))\rightarrow
0.$$
Because $\mathrm{H}^{i-1}(G_K, W^+_{\mathrm{dR}}/t^nW^+_{\mathrm{dR}})$ is finite dimensional over $\mathbb{Q}_p$, Mittag-Leffler condition implies that $\mathbb{R}^1\varprojlim_n\mathrm{H}^{i-1}(K, W^+_{\mathrm{dR}}/t^nW^+_{\mathrm{dR}})=0$.
The lemma follows from this.
\end{proof}
\begin{corollary}\label{5-3}
Let $W^+_{\mathrm{dR}}$ be as above. Let $\{h_1,h_2,\cdots,h_d\}$ be the generalized Hodge-Tate weight of $W^+_{\mathrm{dR}}/tW^+_{\mathrm{dR}}$. Let $k\geqq 1$ be any integer such that $k+h_j\geqq0$ for any $h_j\in \mathbb{Z}$.
Then the natural map $\mathrm{H}^i(G_K, W^+_{\mathrm{dR}})\rightarrow \mathrm{H}^i(G_K, W^+_{\mathrm{dR}}/t^kW^+_{\mathrm{dR}})$ is
isomorphism and $\mathrm{H}^i(G_K, t^{k+1}W^+_{\mathrm{dR}})=0.$

\end{corollary}
\begin{proof}
By the assumption on $k$, we have $\mathrm{H}^i(G_K, t^lW^+_{\mathrm{dR}}/t^{l+1}W^+_{\mathrm{dR}})=0$ for any $l\geqq k+1$.
The corollary follows from this and Lemma \ref{5-2}.
\end{proof}

\begin{corollary}\label{5-4}
Let $W^+_{\mathrm{dR}}$ be as above, then
$\mathrm{H}^i(G_K, W^+_{\mathrm{dR}})=\mathrm{H}^i(G_K, W_{\mathrm{dR}})= 0$ for $i\geqq 2$ and $\mathrm{H}^i(G_K, W^+_{\mathrm{dR}})$ and $\mathrm{H}^i(G_K, W_{\mathrm{dR}})$ are finite dimensional
over $\mathbb{Q}_p$ for $i=0,1$.
\end{corollary}
\begin{proof}
Because $\mathrm{H}^i(G_K, W^+_{\mathrm{dR}}/t^nW^+_{\mathrm{dR}})=0$ for $i\geqq 2$ and
$\mathrm{H}^i(G_K, W^+_{\mathrm{dR}}/t^n W^+_{\mathrm{dR}})$ is finite dimensional for $i=0,1$, we obtain the  corollary
for $W^+_{\mathrm{dR}}$ by Lemma \ref{5-3}. We prove the corollary for $W_{\mathrm{dR}}$. Because
$\mathrm{C}^{\bullet}(G_K, W_{\mathrm{dR}})\isom \varinjlim_n \mathrm{C}^{\bullet}(G_K, \frac{1}{t^n} W^+_{\mathrm{dR}})$
by Lemma \ref{5-1} (2),
we have an isomorphism $\mathrm{H}^i(G_K, W_{\mathrm{dR}})\isom \varinjlim_n \mathrm{H}^i(G_K, \frac{1}{t^n}W^+_{\mathrm{dR}})$.
Then we can show that for $n$ large enough the natural map $\mathrm{H}^i(G_K, \frac{1}{t^{n+j}}W^+_{\mathrm{dR}})\rightarrow
\mathrm{H}^i(G_K, \frac{1}{t^{n+j+1}}W^+_{\mathrm{dR}})$ is isomorphism for any $j\geqq 0$,  then the natural map
$\mathrm{H}^i(G_K, \allowbreak \frac{1}{t^n} W^+_{\mathrm{dR}})\rightarrow \mathrm{H}^i(G_K, W_{\mathrm{dR}})$ is isomorphism, the corollary for $W_{\mathrm{dR}}$ follows
from this.

\end{proof}

\begin{lemma}\label{5-5}
Let $W=(W_e,W^+_{\mathrm{dR}})$ be a $B$-pair.
Then we have $\mathrm{H}^i(G_K, W_e)=0$ for $i\geqq 3$.
\end{lemma}
\begin{proof}
Because we have $\mathrm{C}^{\bullet}(G_K, W_e)=\varinjlim_n \mathrm{C}^{\bullet}(G_K, W_e\cap \frac{1}{t^n}W^+_{\mathrm{dR}})$
by Lemma \ref{5-1} (1),
we have an isomorphism $\mathrm{H}^i(G_K,W_e)\isom \varinjlim_n \mathrm{H}^i(G_K,W_e\cap \frac{1}{t^n}W^+_{\mathrm{dR}})$.
For any $n$, because $W_e\cap\frac{1}{t^n}W^+_{\mathrm{dR}}$ is an almost $\mathbb{C}_p$-representation by Theorem \ref{5-0},
we have $\mathrm{H}^i(G_K, W_e\cap \frac{1}{t^n}W^+_{\mathrm{dR}})=0$ for $i\geqq 3$ by Lemma \ref{5-0-5}.
The lemma follows from these facts.
\end{proof}

\begin{thm}\label{5-6}
Let $W$ be a $B$-pair,  then the following hold.
\begin{itemize}
\item[(1)]$\mathrm{H}^i(G_K,W)$ is zero for $i\geqq 3$ and
$\mathrm{H}^i(G_K,W)$ is finite dimensional over $\mathbb{Q}_p$ for
$i=0,1,2$.
\item[(2)](Euler-Poincar\'e characteristic formula)
$$\sum_{i=0}^2 \mathrm{dim}_{\mathbb{Q}_p}(-1)^i\mathrm{H}^i(G_K, W)=-[K:\mathbb{Q}_p]\mathrm{rank}(W).$$
\end{itemize}
\end{thm}
\begin{proof}
We first prove that $\mathrm{H}^i(G_K,W)=0$ for $i\geqq 3$. Because there is an exact sequence
$$\cdots \rightarrow \mathrm{H}^{i-1}(K, W_{dR})\rightarrow \mathrm{H}^i(G_K, W)\rightarrow
\mathrm{H}^i(G_K, W_e)\oplus \mathrm{H}^i(G_K, W^+_{\mathrm{dR}})\rightarrow \cdots ,$$
the claim follows from Corollary \ref{5-4} and Lemma \ref{5-5}.
Next we prove that $\mathrm{H}^i(G_K, W)$ is finite dimensional over $\mathbb{Q}_p$.
By slope filtration theorem, it suffices to show this claim when $W$ is pure. Let
$W$ be a $B$-pair pure of slope $s$. When $s\leqq 0$ we have the following short exact sequence,
$$0\rightarrow W_e\cap W^+_{\mathrm{dR}}\rightarrow W_e\oplus W^+_{\mathrm{dR}}\rightarrow W_{\mathrm{dR}}
\rightarrow 0$$ by Theorem \ref{5-0} (2). Hence the natural map $\mathrm{H}^i(G_K, W_e\cap W^+_{\mathrm{dR}})\rightarrow
\mathrm{H}^i(G_K, W)$ is isomorphism. Because $W_e\cap W^+_{\mathrm{dR}}$ is an almost $\mathbb{C}_p$-representation by
Theorem \ref{5-0}, $\mathrm{H}^i(G_K, W_e\cap W^+_{\mathrm{dR}})$ is finite dimensional by Lemma \ref{5-0-5}, this prove the claim
for $s\leqq 0$. When $s>0$, then we have the following short exact sequence
$$0\rightarrow W_e\oplus W^+_{\mathrm{dR}}\rightarrow W_{\mathrm{dR}}\rightarrow W_{\mathrm{dR}}/(W_e + W^+_{\mathrm{dR}})
\rightarrow 0 $$ by Theorem \ref{5-0} (3). Hence we obtain a natural isomorphism $\mathrm{H}^{i}(K, W)\isom
\mathrm{H}^{i-1}(K, \allowbreak W_{\mathrm{dR}}/(W_e+W^+_{\mathrm{dR}}))$. Because $W_{\mathrm{dR}}/(W_e+W^+_{\mathrm{dR}})$ is
an almost $\mathbb{C}_p$-representation by Theorem \ref{5-0}, $\mathrm{H}^{i-1}(K, W_{\mathrm{dR}}/(W_e+W^+_{\mathrm{dR}}))$ is finite dimensional, the claim for $s>0$ follows from this.

Next we prove (2). For $W$ a $B$-pair or an almost $\mathbb{C}_p$-representation, we write
$\chi(W):=\sum_{i=0}^2 (-1)^i\mathrm{dim}_{\mathbb{Q}_p}\mathrm{H}^i(G_K, W)$.
It suffices to show (2) when $W$ is pure of slope $s$.
When $s\leqq 0$, then by the above proof $\chi(W)=\chi(X_0(W))$.
By Lemma \ref{5-0-6} and by Theorem \ref{5-0} (2),  $\chi(X^0(W))=-[K:\mathbb{Q}_p]\mathrm{ht}(W)
=-[K:\mathbb{Q}_p]\mathrm{rank}(W)$.  When $s>0$, then by the above proof we have $\chi(W)=-\chi(X^1(W))$.
By Lemma \ref{5-0-6} and by Theorem \ref{5-0} (3), we have $\chi(X^1(W))=-[K;\mathbb{Q}_p]\allowbreak \mathrm{ht}(X^1(W))=
[K:\mathbb{Q}_p]\mathrm{rank}(W)$.  (2) follows from these.

\end{proof}

Next, we define the cup product pairing for $B$-pairs $W:=(W_e, W^+_{\mathrm{dR}})$ and
$W':=(W'_e, W^{' +}_{\mathrm{dR}})$ as follows.
First, for two continuous cochains $c\in \mathrm{C}^{i}(G_K, W_{?})$ and $c'\in \mathrm{C}^{j}(G_K, W'_{?})$ where
$W_{?}$ is $W_e$ or $W^+_{\mathrm{dR}}$ or $W_{\mathrm{dR}}$, we define a continuous cochain
$$c\cup c'\in \mathrm{C}^{i+j}(G_K, W_{?}\otimes_{B_{?}}W'_{?})$$ by $$c\cup c'(g_1, \cdots, g_{i+j}):=
c(g_1,\cdots, g_i)\otimes g_1g_2\cdots g_i c'(g_{i+1},\cdots, g_{i+j})$$
where $B_{?}$ is $B_e$ or $B^+_{\mathrm{dR}}$ or $B_{\mathrm{dR}}$ when
$W_{?}$ is $W_e$ or $W^+_{\mathrm{dR}}$ or $W_{\mathrm{dR}}$ respectively.
Then, $c\cup c'$ satisfies
$$\delta(c\cup c')= \delta(c)\cup c' + (-1)^i c\cup \delta(c').$$
For $c=(c_e, c^+_{\mathrm{dR}}, c_{\mathrm{dR}}) \in \mathrm{C}^i(G_K, W_e)\oplus
\mathrm{C}^i(G_K, W^+_{\mathrm{dR}})\oplus \mathrm{C}^{i-1}(G_K, W_{\mathrm{dR}})$ and
$c'=(c'_e, c^{' +}_{\mathrm{dR}}, c'_{\mathrm{dR}})\in \mathrm{C}^{j}(G_K, W'_e)\oplus
\mathrm{C}^{j}(G_K, W^{' +}_{\mathrm{dR}})\oplus \mathrm{C}^{j-1}(G_K, W'_{\mathrm{dR}})$ and for
a parameter $\gamma\in \mathbb{Q}_p$,
we define
$$c\cup_{\gamma} c'\in \mathrm{C}^{i+j}(G_K, W_{e}\otimes_{B_e}W'_e)\oplus
\mathrm{C}^{i+j}(G_K, W^+_{\mathrm{dR}}\otimes_{B^+_{\mathrm{dR}}}W^{' +}_{\mathrm{dR}})
\oplus \mathrm{C}^{i+j-1}(G_K, W_{\mathrm{dR}}\otimes_{B_{\mathrm{dR}}}W'_{\mathrm{dR}})$$
by $$c\cup_{\gamma} c':=( c_e\cup c'_e, c^+_{\mathrm{dR}}\cup c^{' +}_{\mathrm{dR}}, c_{\mathrm{dR}}\cup(\gamma c'_e+(1-\gamma)c^{'+}_{\mathrm{dR}})+ (-1)^i((1-\gamma)c_e+\gamma c^+_{\mathrm{dR}})\cup c'_{\mathrm{dR}})).$$
Then, we can check that if $\delta(c)=\delta(c')=0$ then $\delta(c\cup_{\gamma} c')=0$ and
if $\delta{c}=0$ and $c'=\delta(c'')$ ( or $c=\delta(c'')$ and $\delta(c')=0$) then
$c\cup_{\gamma}c'\in \mathrm{Im}(\delta)$. Therefore, this paring induces a $\mathbb{Q}_p$-bi-linear paring
$$\cup_{\gamma}:\mathrm{H}^i(G_K, W)\times \mathrm{H}^j(G_K, W')\rightarrow
\mathrm{H}^{i+j}(G_K, W\otimes W').$$
Moreover, we can check that $\cup_{\gamma}$ doesn't depend on the choice of a parameter
$\gamma$, so we just write $\cup$ instead of $\cup_{\gamma}$.

We define the paring
$$\cup: \mathrm{H}^i(G_K, W)\times \mathrm{H}^{2-i}(G_K, W^{\vee}(\chi_p))\rightarrow \mathbb{Q}_p$$
by composing
$$\cup: \mathrm{H}^i(G_K, W)\times \mathrm{H}^{2-i}(G_K, W^{\vee}(\chi_p))\rightarrow
\mathrm{H}^2(G_K, W\otimes W^{\vee}(\chi_p))$$ with the map
$\mathrm{H}^2(G_K, W\otimes W^{\vee}(\chi_p))\rightarrow \mathrm{H}^2(G_K, W(\mathbb{Q}_p)(\chi_p))$ which is
induced from the evaluation map $W\otimes W^{\vee}(\chi_p)\rightarrow W(\mathbb{Q}_p)(\chi_p)$ and with
the natural isomorphism $\mathrm{H}^2(G_K, W(\mathbb{Q}_p)(\chi_p))\isom \mathrm{H}^2(G_K, \mathbb{Q}_p(\chi_p))$
and with Tate's trace map $\mathrm{H}^2(G_K, \mathbb{Q}_p(\chi_p))\isom \mathbb{Q}_p$ where $W(\mathbb{Q}_p)$ is the $B$-pair
associated to the trivial representation $\mathbb{Q}_p$.
Tate  duality  theorem for $B$-pairs is following.
\begin{thm}\label{5-7}
For $i=0,1,2$,the above paring
$$\cup: \mathrm{H}^i(G_K, W)\times \mathrm{H}^{2-i}(G_K, W^{\vee}(\chi_p))\rightarrow \mathbb{Q}_p$$
is a perfect paring.
\end{thm}
\begin{proof}
We can prove this theorem in the same way as in the proof of Theorem 4.7 of \cite{Li08} if we use the Euler-Poincar\`e formula
and the facts  that
$\mathrm{H}^0(G_K, W(\prod_{\sigma\in \mathcal{P}}\sigma))\allowbreak =0$ and $\mathrm{H}^0(G_K, W(|\prod_{\sigma\in \mathcal{P}}\sigma|))=0$ which
have been already proved in Proposition \ref{i}.
\end{proof}
Finally, we prove that our continuous cohomology is canonically isomorphic to Liu's cohomology.
We first define an isomorphism between $\mathrm{H}^0$ by
$\mathrm{H}^0_{\mathrm{Liu}}(G_K, W)\isom \mathrm{Hom}_{\varphi,\Gamma}(R, D(W))\isom \mathrm{Hom}(W(\mathbb{Q}_p), W)
\isom \mathrm{H}^0(G_K, W)$ where $D(W)$ is the $(\varphi,\Gamma)$-module associated to $D$ and $R$ is the trivial
$(\varphi,\Gamma)$-module, where the second isomorphism follows from the equivalence of categories between
$B$-pairs and $(\varphi,\Gamma)$-modules.
\begin{thm}
The above isomorphism $\mathrm{H}^0_{\mathrm{Liu}}(G_K, W)\isom \mathrm{H}^0(G_K, W)$ extends uniquely to
an isomorphism of $\delta$-functors $\mathrm{H}^i_{\mathrm{Liu}}(G_K, W)\isom \mathrm{H}^i(G_K, W)$.
\end{thm}
\begin{proof}
This follows from weakly effaceabilities of functors $\mathrm{H}^i_{\mathrm{Liu}}(G_K, -)$
and $\mathrm{H}^i(G_K, -)$.
For $\mathrm{H}^i_{\mathrm{Liu}}$, these facts  are proved in the proof of Theorem 8.1 of \cite{Ke09}.
For $\mathrm{H}^i(G_K,- )$, we can also prove in the same way as in Theorem 8.1 of \cite{Ke09} because
we have already proved Euler-Poincar\`e formula and Tate duality for $\mathrm{H^i}(G_K, -)$ and
we have an isomorphism $\mathrm{H}^1(G_K, W)\isom \mathrm{Ext}^1( W(\mathbb{Q}_p), W)$ which is proved in Proposition 2.2 of \cite{Na09}.

\end{proof}