text:
\begin{document}
\bibliographystyle{alpha}
\newtheorem{theorem}[subsubsection]{Theorem}
\newtheorem*{theorem*}{Theorem}
\newtheorem{conjecture}[subsubsection]{Conjecture}
\newtheorem{proposition}[subsubsection]{Proposition}
\newtheorem{lemma}[subsubsection]{Lemma}
\newtheorem*{lemma*}{Lemma}
\newtheorem{corollary}[subsubsection]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[subsubsection]{Definition}
\newtheorem*{definition*}{Definition}
\newtheorem{property}[subsubsection]{Property}

\newtheorem{rema}[subsubsection]{Remark}
\newtheorem{example}[subsubsection]{Example}
\newtheorem{notation}[subsubsection]{Notation}
\newtheorem{construction}[subsubsection]{Construction}

\setcounter{tocdepth}{2}

\title{Some remarks on the companions conjecture for normal varieties}

\author{Marco D'Addezio}
\date{\today}

\address{Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany}
\email{daddezio@mpim-bonn.mpg.de}

\begin{abstract}
Drinfeld in 2010 proved the companions conjecture for smooth varieties over a finite field, generalizing L. Lafforgue's result for smooth curves. We study the obstruction to prove the conjecture for arbitrary normal varieties. To do this, we introduce a new property of morphisms. We verify this property in some cases, showing thereby the companions conjecture for some singular normal varieties.
\end{abstract}
\maketitle
\tableofcontents

\section{Introduction}
\subsection{The companions conjecture}
Let $\Fq$ be a finite field of characteristic $p$ and $X_0$ a connected normal variety over $\Fq$. Let $\ell$ be a prime different from $p$ and $\calV_0$ an irreducible Weil lisse $\Qlbar$-sheaf over $X_0$ with finite order determinant. Write $E$ for the subfield of $\Qlbar$ generated by the coefficients of all these Frobenius polynomials at closed points. Deligne proved that $E$ is a finite extension of $\Q$, \cite{Del}. He showed this finiteness reducing geometrically to the case of curves, which was already proven by L. Lafforgue as a consequence of the Langlands correspondence, \cite{Laf}. This property of the field $E$ was conjectured by Deligne in \cite[Conjecture 1.2.10]{Weil2} together with other properties for $\calV_0$. The principal part of the conjecture is the following one.

\begin{conjecture}[Companions conjecture]\label{intro-companions-c}After possibly replacing $E$ with a finite extension, for every finite place $\lambda$ not dividing $p$ there exists a Weil lisse $E_{\lambda}$-sheaf $E$-compatible\footnote{Cf. \cite[Definition 3.1.15]{Dad}.} with $\calV_0$.
\end{conjecture}

When $X_0$ has dimension $1$, the conjecture is again a consequence of the Langlands correspondence. For higher dimensional varieties, Drinfeld proved Conjecture \ref{intro-companions-c} when $X_0$ is smooth, \cite{Dri}. Unluckily, his method cannot be applied directly to prove the full conjecture, [\textit{ibid.}, §6].

\subsection{The obstruction}\label{obst:ss}
Suppose for simplicity that the singular locus of $X_0$ consists of one closed point and that we can solve that singularity. In other words, suppose that there exists a smooth variety $Y_0$ and a proper morphism $h_0:Y_0\to X_0$ sending $i_0:Z_0\hookrightarrow Y_0$ to a closed point $x_0\in |X_0|$ and such that $h_0$ is an isomorphism outside $Z_0$. Write $\F$ for an algebraic closure of $\Fq$ and suppose that $Z:=Z_0\otimes_{\Fq}\F$ is connected.
\begin{lemma}[Corollaire 6.11, SGA I, Exposé XI]

\label{i-contraction-fundamental-group-lemma} For every geometric point $z$ of $Z_0$ there exists an exact sequence $$\pi_1^{\et}(Z,z)\xrightarrow{i_*} \pi_1^{\et}(Y_0,z)\xrightarrow{h_{0*}} \pi_1^{\et}(X_0,h(z))\to 1$$ in the sense that the smallest normal closed subgroup containing the image of $i_*$ is the kernel of $h_{0*}$.
\end{lemma}
By the lemma, every étale Weil lisse sheaf $\calV_0$ over $Y_0$ which is trivial over $Z$ is the inverse image of an étale Weil lisse sheaf defined over $X_0$.
Since we know the companions conjecture for $Y_0$, in order to deduce it for $X_0$ we have to verify the following property.

\begin{itemize}
	\item[$\mathcal{P}(Z_0)$ :] For every pair $(\calV_0,\calW_0)$ of compatible absolutely irreducible Weil lisse sheaves with finite order determinant over $Y_0$, the sheaf $\calV_0$ is trivial over $Z$ if and only if the same is true for $\calW_0$.
\end{itemize}
If $Z_0\subseteq Y_0$ satisfies $\calP(Z_0)$ we say that $Z_0$ is a \textit{$\lambda$-uniform subvariety}. Thanks to Lemma \ref{i-contraction-fundamental-group-lemma} and the companions conjecture for smooth varieties, we have the following result.

\begin{theorem}\label{lamd-unif-comp:t}
If $Z_0\subseteq Y_0$ is $\lambda$-uniform, $X_0$ satisfies the companions conjecture.
\end{theorem}
\subsection{Main results}

The aim of this text is to shed some new lights on the companions conjecture for normal varieties. For this scope we focus on $\lambda$-uniformity. We extend the notion of $\lambda$-uniform subvarieties of §\ref{obst:ss} with the one of \textit{$\lambda$-uniform morphisms} of varieties (Definition \ref{uniform-morphisms:d}). We intend to investigate the following conjecture.
\begin{conjecture}[Conjecture \ref{unif:c}]\label{i-unif:c}
Let $Y_0$ and $Z_0$ be varieties over $\Fq$. If $Y_0$ is normal, every morphism $f_0:Z_0\to Y_0$ is $\lambda$-uniform (cf. Definition \ref{uniform-morphisms:d}).
\end{conjecture}

We shall verify Conjecture \ref{i-unif:c} in some particular cases.

\begin{theorem}[Theorem \ref{normal-uniform:t}, Theorem \ref{tree:t}, and
	Theorem \ref{finite:t}]\label{i-main:t}
Let $f_0: Z_0 \to Y_0$ be a morphism of geometrically connected varieties over $\Fq$ with $Y_0$ normal. Let $z$ be a geometric point of $Z_0$. The morphism $f_0$ is $\lambda$-uniform in the following cases.
\begin{itemize}
\item[{\rm (i)}] If $Z_0$ is a normal variety.
\item[{\rm (ii)}] If $Z_0$ is a semi-stable curve with simply connected dual graph.
\item[{\rm (iii)}]  If the smallest closed normal subgroup of $\pi_1^{\et}(Y_0,f(z))$ containing the image of $\pi_1^{\et}(Z,z)$ is open inside $\pi_1^{\et}(Y,f(z))$.
\item[{\rm (iv)}] If $\pi_1^{\et}(Y,f(z))$ contains an open solvable profinite subgroup.
\end{itemize}
\end{theorem}

Combining the previous results we get the following.
\begin{theorem}Let $Y_0$ be a smooth geometrically connected variety over $\Fq$. If $X_0$ be a normal variety that can be written as a contraction of a geometrically connected subvariety $Z_0\subseteq Y_0$ satisfying one of the conditions of Theorem \ref{i-main:t}, then $X_0$ verifies Conjecture \ref{intro-companions-c}.
\end{theorem}

An independent property we prove in this text is a property of invariance of $\lambda$-uniformity “under deformations” (Theorem \ref{homotopy:t}). This might be useful for further developments in the direction of Conjecture \ref{i-unif:c}. Besides, we present in §\ref{fina-comm:ss} a concrete example, proposed by de Jong, where we think it might be interesting to study Conjecture \ref{i-unif:c}.

\subsection{Acknowledgements}
I am grateful to my advisor Hélène Esnault for introducing me to this topic and for all the time we spent talking about this problem. I thank Emiliano Ambrosi, Raju Krishnamoorthy, and Jacob Stix for discussions and suggestions. Finally, I thank Piotr Achinger and Daniel Litt for sharing with me de Jong's Example \ref{dJ:ex}.

\subsection{Notation and conventions}
\subsubsection{}For us, a variety over a field $k$ is a separated scheme of finite type over $k$. We write $X_0,Y_0,Z_0,...$ for varieties over $\F_q$ and $X,Y,Z,...$ for the base change to $\F$. Further, we put a subscript $_0$ to indicate objects and morphisms defined over $\F_q$ and the suppression of this subscript shall mean that we are extending the scalars to $\F$. If $E$ is a number field we write $|E|_{\neq p}$ for the set of finite places of $E$ which do not divide $p$. For every $\lambda\in |E|_{\neq p}$, we denote by $E_{\lambda}$ the completion of $E$ with respect to $\lambda$.

\subsubsection{}
We use the notation for Weil lisse sheaves as in \cite[§2.2]{Dad}. We say that a Weil lisse $E_\lambda$-sheaf $\calV_0$ is \textit{split untwisted} if every subquotient of $\calV_0$ is absolutely irreducible and has finite order determinant. We say instead that $\calV_0$ is \textit{untwisted} if it is split untwisted after possibly extending $E_\lambda$. Recall that if $Y_0$ is a normal variety over $\Fq$, every untwisted Weil lisse sheaf over $Y_0$ is pure of weight $0$, \cite[Théorème 1.6]{Del}. Besides, by \cite[Proposition 1.3.14]{Weil2} and \cite[Proposition 3.1.16]{Dad}, every untwisted Weil lisse sheaf is étale.

\subsubsection{}

An \textit{$E$-compatible system} over $X_0$, denoted by $\underline{\calV_0}$, is a family $\{\calV_{\lambda,0}\}_{\lambda\in |E|_{\neq p}}$ where each $\calV_{\lambda,0}$ is an $E$-rational Weil lisse $E_{\lambda}$-sheaf and such that all sheaves are pairwise $E$-compatible. Each $\calV_{\lambda,0}$ is called the \textit{$\lambda$-component} of $\underline{\calV_0}$. We say that a compatible system is \textit{semi-simple, untwisted, split untwisted}, ... if each $\lambda$-component has the respective property.

\section{\texorpdfstring{$\lambda$}{}-uniform morphisms}

\label{the-conj:ss}
\subsection{General properties}
\begin{definition}
\label{uniform-systems:d}
Let $Z_0$ be a connected variety. A compatible system $\underline{\calV_0}$ over $Z_0$ is \textit{$\lambda$-uniform} if one of the following disjoint conditions is verified.

\begin{itemize}
\item [(i)] For every $\lambda\nmid p$, the lisse sheaf $\calV_{\lambda,0}$ is geometrically trivial.
\item [(ii)]For every $\lambda\nmid p$, the lisse sheaf $\calV_{\lambda,0}$ is geometrically non-trivial.
\end{itemize}
We say that $\underline{\calV_0}$ is \textit{strongly $\lambda$-uniform} if the dimension of $H^0(Z,\calV_{\lambda})$ does not depend on $\lambda$. Strongly $\lambda$-uniform compatible systems are clearly $\lambda$-uniform. If $Z_0$ is not connected we say that a compatible system is \textit{$\lambda$-uniform} (resp. \textit{strongly $\lambda$-uniform}) if it is $\lambda$-uniform (resp. strongly $\lambda$-uniform) over every connected component.
\end{definition}

\begin{theorem}
\label{normal-uniform:t}
Let $Z_0$ be a normal variety over $\F_q$. Every untwisted $E$-compatible system $\ucalVz$ over $Z_0$ is strongly $\lambda$-uniform.
\end{theorem}

\begin{proof}
After extending the base field, we may assume that $Z_0$ is geometrically connected. As said before, since $Z_0$ is a normal variety, each $\lambda$-component of $\ucalVz$ is pure of weight $0$, \cite[Théorème 1.6]{Del}. Also, if $U_0$ is the smooth locus of $Z_0$, the \'etale fundamental group of $U$ maps surjectively onto the \'etale fundamental group of $Z$. Therefore, we have a canonical isomorphism $$H^0(Z,\calV_{\lambda})=H^0(U,\calV_{\lambda}|_U)$$ for every $\lambda$.  Applying \cite[Cor. VI.3]{Laf} to the $E$-compatible system $\{\calV_{\lambda,0}|_{U_0}\}_{\lambda\in |E|}$, we obtain the desired result.
\end{proof}

If we do not assume $Z_0$ normal, Theorem \ref{normal-uniform:t} becomes false in general (Example \ref{noda-curv:ex}). The issue is the lack of a Chebotarev density theorem for non-normal varieties. In what follows we want to understand if a weaker variant of Theorem \ref{normal-uniform:t} is still true for singular varieties.

\begin{definition}
\label{uniform-morphisms:d}
Let $f_0:Z_0\to Y_0$ be a morphism of varieties over $\Fq$. We say that $f_0$ is a \textit{$\lambda$-uniform morphism} if for every untwisted compatible system $\underline{\calV_0}$ over $Y_0$, the pullback $f_0^*\underline{\calV_0}$ is $\lambda$-uniform. If $f_0$ is a closed immersion we say that $Z_0$ is a \textit{$\lambda$-uniform subvariety} of $Y_0$.
\end{definition}
\begin{conjecture}\label{unif:c}
	Let $Y_0$ and $Z_0$ be varieties over $\Fq$. If $Y_0$ is normal, every morphism $f_0:Z_0\to Y_0$ is $\lambda$-uniform.
\end{conjecture}

If $Y_0$ is smooth we have an equivalent definition of a $\lambda$-uniform morphism. This other definition uses the \textit{Grothendieck semirings} of Weil lisse sheaves (cf. \cite[§4.3.3]{Dad}).
\begin{notation}
As in [\textit{ibid.}], we write $\KK(\calV_{\lambda,0})$ for the Grothendieck semiring of $\langle\calV_{\lambda,0} \rangle$, the Tannakian category spanned by $\calV_{\lambda,0}$. We also use an analogous notation for the lisse sheaf $\calV_\lambda$ over $Y$. Besides, we write $\KK^+(\calV_{\lambda})^0\subseteq \KK^+(\calV_{\lambda})$ for the sub-semiring of (geometrically) trivial lisse sheaves in $\langle\calV_{\lambda} \rangle$.
\end{notation}
If $\ucalVz$ is a split untwisted $E$-compatible system over a smooth variety, thanks to the companions conjecture, for every $\lambda,\lambda'\in |E|_{\neq p}$ we have a natural isomorphism of semirings $\KK^+(\calV_{\lambda,0})\iso\KK^+(\calV_{\lambda',0}),$ \cite[Proposition 4.3.6]{Dad}.
\begin{proposition}
	If $Y_0$ is a smooth variety, a morphism $f_0:Z_0\to Y_0$ is $\lambda$-uniform if and only if for every split untwisted $E$-compatible system $\underline{\calV_0}$ over $Y_0$ the preimage of $\KK^+(f^*\calV_{\lambda})^0$ via $\KK^+(\calV_{\lambda,0})\to \KK^+(f^*\calV_{\lambda})$ does not depend on $\lambda$.
\end{proposition}
\begin{proof}
	Note that by \cite[Corollary 3.5.2.(ii)]{Dad}, every $\lambda$-component of $\underline{\calV_0}$ is geometrically semi-simple. Therefore, if we take the semi-simplification of each $\lambda$-component we do not change the isomorphism classes of the lisse sheaves over $Y$. This shows that we may assume that each $\lambda$-component of $\underline{\calV_0}$ is semi-simple.
	Next, we observe that the preimage of $\KK^+(f^*\calV_{\lambda})^0$ is precisely the sub-semiring of $\KK^+(\calV_{\lambda,0})$ given by those semi-simple Weil lisse sheaves in $\langle\calV_{\lambda,0} \rangle$ which are geometrically trivial over $Z_0$.

	Suppose that the preimage of $\KK^+(f^*\calV_{\lambda})^0$ does not depend on $\lambda$. Looking at the class of $[\calV_{\lambda,0}]\in \KK^+(\calV_{\lambda,0})$ when $\lambda$ varies we deduce that $f_0$ is $\lambda$-uniform. For the converse, if $\calV_{\lambda,0}$ is untwisted, every object in $\langle\calV_{\lambda,0} \rangle$ is untwisted as well, \cite[Theorem 3.4.7]{Dad}. Therefore, every $\calW_{\lambda,0}\in \langle\calV_{\lambda,0} \rangle$ sits in an untwisted compatible system $\underline{\calW_0}$ over $Y_0$ which by assumption is $\lambda$-uniform when restricted to $Z_0$. This concludes the proof.
\end{proof}

\subsection{Homotopic invariance}

Let us look more closely at $\lambda$-uniform morphisms by analysing the relation with the induced morphism on fundamental groups.

\subsubsection{}

Let $f_0:Z_0\to Y_0$ be a morphism of geometrically connected varieties over $\Fq$ with $Y_0$ normal. If we choose a geometric point $z$ of $Z_0$ we have a morphism $$\pi_1^{\et}(Z,z)\xrightarrow{f_{*}} \pi_1^{\et}(Y_0,f(z)).$$
	For every étale compatible system $\underline{\calV_0}$ over $Y_0$ we denote by $\{\rho_{\lambda,0}\}_{\lambda \in |E|_{\neq p}}$ the associated family of $\ell$-adic representations of $\pi_1^{\et}(Y_0,f(z))$. Let $\overline{\Image(f_*)}$ be the smallest normal closed subgroup of $\pi_1^{\et}(Y_0,f(z))$ containing the image of $f_*$. The following lemma is a direct consequence of the definition of a $\lambda$-uniform morphism.

\begin{lemma}\label{uniform-group-property:l}A morphism $f_0$ is $\lambda$-uniform if and only if for every untwisted compatible system over $Y_0$, if $\overline{\Image(f_*)}\subseteq \Ker(\rho_{\lambda,0})$ for one $\lambda$ then the same is true for every other $\lambda\in |E|$. In particular, the property of a morphism of being $\lambda$-uniform depends only on the inclusion $\overline{\Image(f_*)}\subseteq \pi_1^{\et}(Y_0,y)$ as topological groups together with the assignment of the conjugacy classes of the Frobenii at closed points of $\pi_1^{\et}(Y_0,y)$ and their degrees.
\end{lemma}

As a consequence of the previous lemma, we prove an “homotopic invariance” of $\lambda$-uniformity. Let $T_0$ and $S_0$ be geometrically connected varieties over $\F_q$ and $h_0: T_0\to S_0$ a proper and flat morphism with connected and reduced geometric fibres. Let $s_0$ and $s'_0$ be closed points of $S_0$ and write $\iota_0:Z_0\hookrightarrow T_0$ and $\iota'_0:Z'_0\hookrightarrow T_0$ for the closed immersions of the fibres of $h_0$ above $s_0$ and $s'_0$ respectively.
\begin{theorem}
\label{homotopy:t}For every morphism $\widetilde{f}_0: T_0\to Y_0$, the restriction $f_0:=\widetilde{f}_0|_{Z_0}$ is $\lambda$-uniform if and only if $f'_0:=\widetilde{f}_0|_{Z'_0}$ is $\lambda$-uniform.
\end{theorem}

\begin{proof}
Let $z$ and $z'$ be geometric points of $Z_0$ and $Z'_0$ respectively. By \cite[Tag 0C0J]{Stacks}, we have exact sequences $$\pie(Z,z)\xrightarrow{\iota_{*}} \pie(T,z)\xrightarrow{h_{*}} \pie(S,h(z))\to 1$$
$$\pie(Z',z')\xrightarrow{\iota'_{*}} \pie(T,z')\xrightarrow{h_{*}} \pie(S,h(z'))\to 1.$$
The choice of an étale path $\gamma$ joining $z$ with $z'$ induces isomorphisms $\gamma:\pie(T,z)\iso \pie(T',z')$ and $h_*(\gamma):\pie(S,h(z))\iso \pie(S',h(z'))$. Thanks to the two exact sequences this implies that $\gamma$ restricts to an isomorphism $\overline{\Image(\iota_*)}\iso \overline{\Image(\iota'_*)}$. In turn, this implies that the induced isomorphism $f_*(\gamma):\pi_1^{\et}(Y_0,f(z))\iso \pi_1^{\et}(Y_0,f'(z'))$ restricts to an isomorphism $\overline{\Image(f_*)}\iso \overline{\Image(f'_*)}$. By construction, $f_*(\gamma)$ respects the conjugacy classes of Frobenii at closed points and their degrees. We conclude applying Lemma \ref{uniform-group-property:l}.
\end{proof}

\section{Some examples}
In this section, we verify Conjecture \ref{unif:c} in some cases. Note that by virtue of Theorem \ref{normal-uniform:t} we already know the conjecture when $Z_0$ is normal.
\subsection{Semi-stable curves}
\label{semi-stab-curv:ss}
Let $Z_0$ be a connected semi-stable curve over $\F_q$. Write $Z^{(i)}$ where ${1\leq i \leq n}$ for the irreducible components of $Z$. Suppose that for every $i$, the irreducible component $Z^{(i)}$ is smooth. Let $z$ be a geometric point of $Z$ and for every $1\leq i \leq n$, let $z^{(i)}$ be a generic geometric point of $Z^{(i)}$. We denote by $\Gamma$ the dual graph of $Z$ and by $P$ the point of $\Gamma$ associated to the connected component where $z$ lies.
\begin{lemma}[\cite{Stix}]
	\label{Stix:l}
The choice of étale paths $\{\gamma^{(i)}\}_{1\leq i \leq n}$ joining $z$ to $z^{(i)}$ for every $i$ determines an isomorphism $$\pie(Z,z)\simeq \pie(Z^{(1)},z^{(1)})*\dots*\pie(Z^{(n)},z^{(n)})* \pie(\Gamma,P)^\wedge,$$ where ${\pi}_1(\Gamma,P)^\wedge$ is the profinite completion of the topological fundamental group of $\Gamma$.
\end{lemma}

\begin{theorem}\label{tree:t}
If $\Gamma$ is a tree, every untwisted compatible system over $Z_0$ is $\lambda$-uniform.
\end{theorem}
\begin{proof}
Let $\ucalVz$ be an untwisted compatible system over $Z_0$. If $\calV_{\lambda,0}$ is geometrically trivial for one $\lambda$, then it remains geometrically trivial when restricted to every irreducible component of $Z_0$. By Theorem \ref{normal-uniform:t}, for every other $\lambda'\in |E|_{\neq p}$, the restriction of $\calV_{\lambda',0}$ to every irreducible component of $Z_0$ is geometrically trivial as well. By Lemma \ref{Stix:l}, since $\Gamma$ is a tree, the geometric étale fundamental group of $Z_0$ is generated by the étale fundamental groups of the irreducible components $Z^{(i)}$. This shows that for every $\lambda$ the Weil lisse sheaf $\calV_{\lambda',0}$ is geometrically trivial over $Z_0$ and this yields the desired result.
\end{proof}

\begin{example}
	\label{noda-curv:ex}
If $Z_0$ is an irreducible split nodal cubic curve over $\Fq$ with nodal point $z_0$, then $\pie(Z,z)$ is isomorphic to $\hZ$. In addition, the action of $\Gal(\F/\Fq)$ on $\pie(Z,z)$ is trivial. Therefore, $\pie(Z_0,z)$ is isomorphic to $\hZ\times \Gal(\F/\Fq)$, where the embedding $\Gal(\F/\Fq)\subseteq \pie(Z_0,z)$ is induced by the closed immersion of $z_0\hookrightarrow Z_0$. The Frobenius elements of $\pie(Z_0,z)$ correspond via this isomorphism to elements of $\hZ\times \Gal(\F/\Fq)$ of the form $(0,F^d)$, where $F$ is the geometric Frobenius of $\Fq$ and $d$ is some positive integer. This implies that every pair of étale lisse sheaves over $Z_0$ which are trivial at $z_0$ are $\Q$-compatible with all the eigenvalues at closed points equal to $1$.

On the other hand, for every prime number $\ell\neq p$ and every continuous automorphism $\alpha$ of $\overline{\Z}_\ell^{\oplus r}$, where $\overline{\Z}_\ell$ is the ring of integers of $\Qlbar$, there exists an étale lisse $\Qlbar$-sheaf of this type such that the induced $\ell$-adic representation sends $(1,\mathrm{id})$ to $\alpha$. In particular, we may take $r=1$ and as $\alpha$ the multiplication by a root of unit. This construction produces lots of examples of untwisted compatible systems over $Z_0$ which are not $\lambda$-uniform.
\end{example}
\subsection{Finite monodromy}

As in Theorem \ref{normal-uniform:t}, we may use the theory of weights in order to prove that certain morphisms are $\lambda$-uniform. This strategy needs strong finiteness conditions.
\begin{theorem}
\label{finite:t}
Let $f_0: Z_0 \to Y_0$ be a morphism of geometrically connected varieties over $\Fq$ with $Y_0$ normal. Let $z$ be a geometric point of $Z_0$. The morphism $f_0$ is $\lambda$-uniform in the following cases.
\begin{itemize}
\item [{\rm (i)}] If the smallest closed normal subgroup of $\pi_1^{\et}(Y_0,f(z))$ containing the image of $\pi_1^{\et}(Z,z)$ is an open subgroup of $\pi_1^{\et}(Y,f(z))$.
\item [{\rm (ii)}] If $\pi_1^{\et}(Y,f(z))$ contains an open solvable profinite subgroup.
\end{itemize}

\end{theorem}
\begin{proof}Let $\underline{\calV_0}$ be an untwisted compatible system over $Y_0$. For every $\lambda\in |E|_{\neq p}$, write $G_{\lambda}$ for the geometric monodromy group of $\calV_{\lambda,0}$.\\

\begin{itemize}
\item [(i)]
If $f_0^*(\calV_{\lambda,0})$ is geometrically trivial over $Z_0$ for one $\lambda$, then $\rho_{\lambda}$ is trivial when restricted to the image of $\pi_1^{\et}(Z,z)$ in $\pi_1^{\et}(Y,f(z))$. Thanks to the hypothesis, we deduce that $\rho_{\lambda}$ factors through a finite quotient of $\pi_1^{\et}(Y,f(z))$. This implies that $G_{\lambda}$ is a finite algebraic group. By \cite[Proposition 2.2]{LP5}, the subgroups $$\Ker(\rho_{\lambda'})\cap \pi_1^{\et}(Y,f(z))\subseteq \pi_1^{\et}(Y,f(z))$$ are all equal when $\lambda'$ varies in $|E|_{\neq p}$. Since the image of $\pi_1^{\et}(Z,z)$ in $\pi_1^{\et}(Y,f(z))$ is contained in $\Ker(\rho_{\lambda})$, it is also contained in $\Ker(\rho_{\lambda'})$ for every $\lambda'\in |E|_{\neq p}$. Therefore, the lisse sheaf $f_0^*(\calV_{\lambda',0})$ is geometrically trivial for every $\lambda'\nmid p$, as we wanted.\\

\item [(ii)]
By \cite[Corollary 3.4.10]{Dad}, the algebraic groups $G_\lambda$ are all semi-simple because $\underline{\calV_0}$ is pure. On the other hand, thanks to the assumption that $\pi_1^{\et}(Y,f(z))$ is solvable, we know that all the algebraic groups $G_\lambda$ are solvable as well. Combining these two properties we deduce that each algebraic group $G_\lambda$ is finite and we can proceed as in the previous case.
\end{itemize}

\end{proof}
\begin{corollary}
A dominant morphism $f_0:Z_0\to Y_0$ is $\lambda$-uniform. In particular, if $Y_0$ is a smooth curve, every morphism to $Y_0$ is $\lambda$-uniform.
\end{corollary}
\begin{proof} Let $\eta\hookrightarrow Y$ be the generic point of $Y$ and write $Z_{\eta}$ the preimage of $\eta$ via $f$. Since $f$ is dominant, the scheme $Z_{\eta}$ is a non-empty variety over the function field of $Y$. Fix a closed point $\eta'$ of $Z_{\eta}$ and choose a geometric point $\overline{\eta}$ over $\eta'$. We have the following commutative diagram

			\begin{center}
		\begin{tikzcd}
		\pie(\eta',\overline{\eta}) \arrow[hook,r] \arrow[d] &  \pie(\eta,f(\overline{\eta}))  \arrow[d, two heads]\\
		\pi_1^{\et}(Z,\overline{\eta}) \arrow[r] & \pi_1^{\et}(Y,f(\overline{\eta})).
		\end{tikzcd}
	\end{center}
Note that $\pie(\eta',\overline{\eta})$ maps to a finite index subgroup of $\pie(\eta,f(\overline{\eta}))$. Therefore, the image of $\pi_1^{\et}(Z,\overline{\eta})$ in $\pi_1^{\et}(Y,f(\overline{\eta}))$ has finite index as well. Thanks to this, we may apply Theorem \ref{finite:t}.(i) to conclude.
\end{proof}

\subsection{Final comments}\label{fina-comm:ss}
Apart from the cases presented above, it seems difficult to prove in general that a closed subvariety of a smooth variety is $\lambda$-uniform. Note that the property of a Weil lisse sheaf of being geometrically trivial can be thought as the combination of two properties: the property of being geometrically unipotent and the one of being geometrically finite\footnote{For us, a geometrically unipotent (resp. finite) Weil lisse sheaf is a Weil lisse sheaf with unipotent (resp. finite) geometric monodromy group.}. Let us focus on the second one.
\begin{definition}
We say that a subvariety $Z_0\subseteq Y_0$ is \textit{pseudo-$\lambda$-uniform} if it satisfies the analogous condition as a $\lambda$-uniform subvariety where “geometrically \textit{trivial} Weil lisse sheaves” are replaced by “geometrically \textit{finite} Weil lisse sheaves”.
\end{definition}
 In the following example, we present a concrete class of subvarieties where the pseudo-$\lambda$-uniformity is not immediate and, at the best of our knowledge, it is not known.

\begin{example}[de Jong]
	\label{dJ:ex}
	For an integer $n\geq 3$ such that $(n,p)=1$, we write $Y^{(n)}$ for the moduli scheme of principally polarized abelian surfaces over $\F$ with a symplectic level-$n$-structure. Let $Z^{(n)}\subseteq Y^{(n)}$ be the supersingular locus of $Y^{(n)}$. Since we are working with abelian surfaces, this coincides with the Moret--Bailly locus of $Y^{(n)}$, which is a union of copies of $\PP^1_{\F}$. Therefore, by \cite[Proposition 7.3]{Oor01}, the variety $Z^{(n)}$ is connected for every choice of $n$. Let $N$ be a positive multiple of $n$ which is also prime to $p$. The preimage of the natural finite étale Galois cover $Y^{(N)}\to Y^{(n)}$ is $Z^{(N)}$. As $Z^{(N)}$ is connected, the restriction $Z^{(N)}\to Z^{(n)}$ is a finite étale Galois cover with the same Galois group as $\Gal(Y^{(N)}/Y^{(n)})$. If $z$ is a geometric point of $Z^{(n)}$, we have the following commutative diagram.
	\begin{center}
		\begin{tikzcd}
		\pi_1^{\et}(Z^{(n)},z) \arrow{r} \arrow[d,two heads] &  \pi_1^{\et}(Y^{(n)},z)\arrow[d,two heads]\\
		\Gal(Z^{(N)}/Z^{(n)})   & \arrow[l,"="'] \Gal(Y^{(N)}/Y^{(n)}).\
		\end{tikzcd}
	\end{center}
	When $N$ goes to infinity, the cardinality of $\Gal(Y^{(N)}/Y^{(n)})$ goes to infinity as well. This implies that the image of $\pi_1^{\et}(Z^{(n)},z)\to \pi_1^{\et}(Y^{(n)},z)$ is infinite. The varieties $Z^{(n)}$ and $Y^{(n)}$ descend to geometrically connected varieties $Z_0^{(n)}$ and $Y_0^{(n)}$ over some finite field $\F_q$. The restriction of a Weil lisse sheaf over $Y_0^{(n)}$ is not in general geometrically finite over $Z_0^{(n)}$. We cannot prove in this case that $Z_0^{(n)}\subseteq Y_0^{(n)}$ is pseudo-$\lambda$-uniform. Nonetheless, we think this might be a nice concrete example to analyse.

	\end{example}

\bibliographystyle{ams-alpha}
\input{companions.bbl}
\end{document}


TITLE: How do I find :$b a^2+ c b^2+ a c^2$ according to below given conditions?
QUESTION [0 upvotes]: Let $a,b,c>0$ such that:
$a+b+c=5$ and
$ \frac 1a +b+\frac 1b +c+ \frac 1c +a=9/10.$
I have tried to find the value of :$b a^2+ c b^2+ a c^2$ using the above  gaven but i didn't succeed however i have added a third equality which is :$(a+b+c)^2=25$ but no result
My question is to find :
$b a^2+ c b^2+ a c^2$ ?

REPLY [1 votes]: Your two equations imply:
$\frac{9}{10} = \frac{1}{a}+b+\frac{1}{b}+c+\frac{1}{c}+a = \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+ (a+b+c) = \frac{1}{a}+\frac{1}{b}+\frac{1}{c} + 5.$
Thus $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} = \frac{9}{10}-5 = \frac{9}{10}-\frac{50}{10}<0$.  Since you also state that $a,b,c>0$, there can be no numbers $a,b,c$ meeting your initial conditions.


text:
\section{Conclusion}\label{sec:conclusion}

We proposed a constructive method, based on the synthesis of stochastic barrier certificates, for computing an exponentially decreasing upper bound, if existent, on the tail probability that an SDE system violates a given safety specification. We showed that such an upper bound facilitates a reduction of the verification problem over an unbounded temporal horizon to that over a bounded one. Preliminary experimental results on a set of interesting examples from the literature demonstrated the effectiveness of the reduction and that our method often produces tighter bounds on the failure probability.

For future work, we plan to investigate a possible convergence result in the sense that the derived failure probability bound may converge to the exact one as increasing the degree of the barrier certificates. Extending our technique to tackle SDEs with control inputs will also be of interest. Moreover, checking whether a given parametric (polynomial) formula keeps probabilistic invariance plays a central in the verification of SDEs. Several kinds of sufficient conditions on probabilistic barrier certificates were proposed, including the ones given in this paper. It consequently deserves to investigate a necessary and sufficient condition for checking the probabilistic invariance of a given template, like for ODEs in~\cite{LZZ11}. Apart from that, we are interested in carrying our results to the verification of probabilistic programs without conditioning, which can be viewed as discrete-time stochastic dynamics.

config: arxiv