text:
Regularization of inverse problems formulated in Banach spaces have been of recent interest.
On the one hand there are a several theoretical regularization results such as convergence rates in a general Banach spaces setting,
see e.g.~\cite{burger2004convarreg,resmerita2005regbanspaces,resmerita2006nonquadreg,
burger2007errorinvscalespace,hofmann2007convtikban,
hein2008convratesbanach},
or convergence rates for special sequence spaces such as $\ell^p$, $1 \leq p <2$,
see e.g.~\cite{lorenz2008reglp,ramlau2008regproptikhonov}.
On the other hand there are results which deal with solving inverse problems formulated in Banach spaces,
such as Landweber-like iterations or minimization methods for Tikhonov functionals,
see e.g.~\cite{schoepfer2006illposedbanach} and~\cite{daubechies2003iteratethresh,bredies2008harditer,bredies2008itersoftconvlinear,bredies2008forwardbackwardbanach,bonesky2007gencondgradnonlin,ramlau2006tikhproject,griesse2008ssnsparsity,schuster2008tikbanach},
respectively.

The interest in Banach spaces is due to the fact that in many situations a Banach space is better suited to model the data under consideration than a Hilbert space.
In the context of image processing, for example, the Banach space $BV$ of functions of bounded variation is used to model images with discontinuities along lines~\cite{rudin1992tv,acar1994analysis,burger2004convarreg}.
Moreover,~\cite{hofmann2007convtikban} presents two examples in which the use of Banach spaces is necessary for a thorough formulation of the problem.
Another class of Banach spaces are the Besov spaces $B^s_{p,q}$ which play an important role in inverse problems related to image processing, see e.g.~\cite{chambolle1998shrink,chan2006imageprocessing,lorenz2007wavproj}.

In this paper we make a first attempt to analyze inverse problems in scales of Banach spaces generalizing classical Hilbert scales~\cite{natterer1984hilbertscales}.
The easiest scale of Banach spaces is the scale of Sobolev spaces $W^s_p$.
However, we are going to use Besov spaces $B^s_{p,p}$ since they coincide with the Sobolev scale in most cases if the integrability indices coincide.
Moreover, they come with a characterization in terms of wavelet coefficients which make them easy to use for our purposes.
We apply previous convergence rates regularization results from~\cite{burger2004convarreg,hofmann2007convtikban} in the scale of Besov spaces and develop optimal convergence rates.
To this end, we derive the source conditions that lead to a convergence rate of $\bigO(\sqrt{\delta})$ in a certain Sobolev space.

Consider the equation
\begin{equation}\label{op_eq}
  F \; u^\dagger = v,
\end{equation}
where $F$ is a linear continuous operator
\[
  F: \besov_D \to L_2
\]
between the Besov space $\besov_D:=B_{p_D,p_D}^{s_D}$,
$s_D \in \R$, $p_D > 1$,
and Lebesgue space $L_2$.
In general these function spaces contain functions or distributions defined on the subset $\Omega \subset \R^d$.
Due to clarity we omit $\Omega$ in the following.
The Besov spaces $B_{p,p}^s$ are subspaces of the
space of tempered distributions $\mathcal{S}'$ and, in contrast to $\mathcal{S}'$,
they are Banach spaces for $p\geq 1$~\cite{runst1996spaces}.
Different from classical approaches we use the domain $\besov_D$
---often a superset of $L_2$---and not $L_2$ itself.
That may be of interest in some applications, e.g. mass-spectrometry where the data consists of delta peaks
(see \cite{klann2007shrink_vs_deconv,dahlke200multiscale}) which are not elements of $L_2$.

If we assume that only noisy data $v^\delta$ with noise level $\norm{v-v^\delta} \leq \delta$ are available,
the solution of (\ref{op_eq})
could be unstable and has to be stabilized by regularization methods.
We use regularization with a Besov constraint, i.e.~we regularize by minimizing a not necessarily quadratic functional $T_\alpha:\besov_D\to [0,\infty]$ defined by
\begin{equation}\label{tikhonov_eq}
  T_\alpha(u) := \norm{Fu - v^\delta}_{L_2}^2 + \alpha \norm{u}_{\besov_R}^{p_R},
\end{equation}
where $\besov_R:=B_{p_R,p_R}^{s_R}$ is a Besov space,
not necessarily equal to $\besov_D$.
Since $T_\alpha$ shall be defined on $\besov_D$ we define
\[
  \norm{u}_{\besov_R} = \infty
\]
for $u\notin \besov_R$.

In this paper we will investigate regularization properties and convergence rates
of the regularization method consisting of the minimization of (\ref{tikhonov_eq}),
i.e. $u^{\alpha,\delta}\in\argmin T_\alpha(u)$.
The proceeding is as follows.
\begin{enumerate}[i)]
  \item In section~\ref{sec_notation} we introduce the notation and collect preliminary results.
  \item In section~\ref{sec_convergence} we apply convergence rates results for Banach
    spaces~~\cite{burger2004convarreg,hofmann2007convtikban}.
    With the constraints on $p_R$ and $s_R$ in mind and the
    parameter rule $\alpha = \delta$ we will
    get a stable approximation, i.e.
    \[
      \norm{u^{\alpha,\delta} - u^\dagger}_{\besov_R} \to 0, \quad \delta \to 0,
    \]
    and a convergence rate in the Sobolev space $H^{\sigma}$
    \[
      \norm{u^{\alpha,\delta} - u^\dagger}_{H^{\sigma}}
        =
        \bigO(\sqrt{\delta}),
    \]
    with $\sigma$ depending on $s_R$ and $p_R$
    (theorems~\ref{convergence} and~\ref{convergence_rate}).
    These results restrict the choice of possible regularization spaces $\besov_R$.
    Using Besov space embeddings in section~\ref{sec_optimization} we will get a generalization of the first result.
    We find a convergence rate---also formulated in a Sobolev space---which
    holds for a larger set of Besov space penalties $\norm{\cdot}_{\besov_R}^{p_R}$
    (theorem~\ref{convergence_rate_weaker_source}).
  \item The convergence result gets stronger as $\sigma$ increases since for $\theta >0$ it holds $H^{\sigma+\theta} \subset H^{\sigma}$.
    Since $\sigma$ depends on $s_R$ and $p_R$, we address the question
    how to choose $\besov_R$ in a way such that $\sigma$ is maximal.
    We will find the regularization penalty $\norm{\cdot}_{\besov_R}^{p_R}$,
    which gives the best estimate with respect to $\sigma$.
  \item In section~\ref{sec_examples} we apply these results to some operators defined in
    Sobolev and Besov spaces to demonstrate the differences.
\end{enumerate}

\section{Notation and Basic Besov Space Properties}
 \label{sec_notation}
As already mentioned, Besov spaces $B_{p,q}^s$ are subspaces of the space of tempered distributions $\mathcal{S}'$.
They coincide with special cases of traditional smoothness function spaces
such as H\"older and Sobolev spaces.
Note e.g., that $B^s_{p,p} = W^s_p$ for $s\notin\Z$ and $p \geq 1$ and even
$B^s_{2,2} = H^s := W^s_2$ for all $s\in\R$.
As from now we use the term Sobolev space only for the Hilbert spaces $H^s$.
This clarifies the characterization that the Besov space $B_{p,q}^s$ contains functions
having $s$ derivatives in $L_p$ norm. The second integrability index $q$ declares a finer nuance of smoothness.
In the following we omit the second integrability index $q$ of the Besov spaces
which is always equal to the corresponding first one $p$.

There are a several ways of defining Besov spaces. Most commonly they are defined via the modulus of smoothness,
a way to model differential properties.
For a detailed introduction of Besov spaces via moduli of smoothness
in conjunction with other smoothness spaces see e.g. \cite[section 4.5]{devore1998nonapprox}.

Another way defining Besov spaces is based on wavelet coefficients.
According to \cite{cohen2003numwav}
for all $s\in\R$,$\ p>0$,
there exists a wavelet basis
$\{\psi_\lambda\}_{\lambda\in\Lambda}$
such that
\begin{equation}
  \label{eq_equiv_besov_norm}
\norm{u}_{B^s_p}^{p} \asymp
\sum\limits_{\lambda\in\Lambda}
2^{p(s+d(\frac{1}{2}-\frac{1}{p})) \; |\lambda|}
|u_\lambda|^{p},
\end{equation}
where $u_\lambda = \ip{u}{\psi_\lambda} = \int u\, \psi_\lambda dx$ are the wavelet coefficients of $u$.
The notation $A\asymp B$ means that there exist constants $c,C>0$ such that $cA\leq B\leq CA$.
We will use this equivalent norm throughout the paper.

An important ingredient in the analysis of the regularization method (\ref{tikhonov_eq}) is the embedding result (cf.~\cite{runst1996spaces}):

\begin{proposition}\label{embed_besov_spaces}
  Let $B_{p_1}^{s_1}$, $B_{p_2}^{s_2}$ be Besov spaces. If
  \begin{equation}\label{eq-embed_besov_spaces}
    s_1 - \frac{d}{p_1} > s_2 - \frac{d}{p_2}
    \quad \mbox{and} \quad
    p_1 \leq p_2,
  \end{equation}
  then $B_{p_1}^{s_1} \subset B_{p_2}^{s_2}$ continuously.
  The term $s - \frac{d}{p}$ is called \emph{differential dimension} of $B_p^s$.
\end{proposition}

The embedding of Besov spaces is often visualized with the help of the DeVore diagram~\cite{devore1998nonapprox} where one plots the smoothness $s$ against $1/p$, see figure~\ref{bild_embed_besov_spaces}.
By $B_{p_1}^{s_1} \subset B_{p_2}^{s_2}$, in the following,
we denote not only the set-theoretical embedding but also the continuous embedding.

\begin{figure}[h]
  \[
  \beginpicture
    \setcoordinatesystem units <2cm,1cm>
    \setplotarea x from -0.5 to 4, y from -1.5 to 3
    \inboundscheckon
    \axis left shiftedto x=0
          ticks long in unlabeled at 1 /
                long out unlabeled at 1 /
    /
    \axis bottom shiftedto y=0
          ticks long in unlabeled at 2 /
                long out unlabeled at 2 /
    /
    \put{$\scriptstyle \blacktriangle$} at 0 3
    \put{$s$} at -0.25 3
    \put{$\scriptstyle \blacktriangleright$} at 4 0
    \put{$\frac{1}{p}$} at 4 -0.5

    \put{$\bullet$} at 2 1
    \put{$B_{p_0}^{s_0}$} at  1.75 1.25

    \put{$s_0$} at -0.25 1
    \put{$\frac{1}{p_0}$} at 2.25 -0.5

    \setdots <2pt>
    \plot 0 -1  4 3 /
    \put{$s=\frac{d}{p} + s_0 - \frac{d}{p_0}$} at 4 2
    \setsolid
    \plot 2 -1.5  2 3 /

    \setshadegrid span <1.5pt>
    \vshade 0 -1.5 -1
            <z,z,z,z>
            2 -1.5  1
    /
    \put{\colorbox{white}{$\set{\besov}{\besov \supset B_{p_0}^{s_0}}$}} at 1.25 -1.25
    \setshadegrid span <2pt>
    \vshade 2 1 3
            <z,z,z,z>
            4 3 3
    /
    \put{\colorbox{white}{$\set{\besov}{\besov \subset B_{p_0}^{s_0}}$}} at 2.75 2.75
  \endpicture
  \]
  \caption{DeVore diagram for the embedding of Besov spaces.}
  \label{bild_embed_besov_spaces}
\end{figure}

We are going to use the following Besov spaces:
\begin{itemize}
\item $\besov_D := B^{s_D}_{p_D}$ for the domain of $F$.
\item $\besov_R := B^{s_R}_{p_R}$ for the space in which we regularize.
\item $\besov_S := B^{s_S}_{p_S}$ for the source condition.
\item $\besov_G := B^{s_G}_{p_G}$ for the range of $F^*$ which models the smoothing properties of $F$.
\end{itemize}

As stated above, the smoothing properties of the operator
\[
  F: \besov_D \to L_2
\]
are modeled by assuming that the range of its adjoint is small, namely
\[
\rg F^* = \besov_G \subset \besov_D^\ast = B_{p_D^\ast}^{-s_D},
\]
where $p^\ast$ is defined via $\frac{1}{p} + \frac{1}{p^\ast} = 1$,
hence $p^\ast = \frac{p}{p-1}$.
Consequently, we have
\[
  s_G - \frac{d}{p_G} > -s_D - \frac{d}{p_D^\ast},
  \quad
  \frac{1}{p_G} \geq \frac{1}{p_D^\ast}.
\]

config: arxiv


TITLE: What type of pseudoprime does the largest known pseudoprime tend to be?
QUESTION [6 upvotes]: It is a well-known fact that the largest known prime number for several decades now has been a Mersenne prime, even though more and more of them have been found over the years and there have also been efforts to find other kinds of primes, like the one that proves that $10223$ is not a Sierpinski number.
But what about pseudoprimes? Does the largest known pseudoprime tend to be a Fermat pseudoprime to say, base $2$? Does the search for large prime numbers help reveal larger pseudoprimes?

REPLY [1 votes]: One can generate arbitrarily large pseudoprimes. For example, if the exponent of a Mersenne number, $2^x-1$, is a Fermat probable prime base $2$, then so is the Mersenne number itself.
If we take $x$ not equal to a prime ($x$ a composite pseudoprime), then we can guarantee that $2^x-1$ is also a composite pseudoprime. In this manner,
$2^{2047}-1$, $2^{2^{2047}-1}-1$, $2^{2^{2^{2047}-1}-1}-1$, and so on are all composite pseudoprimes base $2$.


TITLE: Question About the Logic of my Proof
QUESTION [4 upvotes]: Okay, I am working on the following relatively simple problem:

Let $f(x) = |x-3| + |x-1|$ for all $x \in \mathbb{R}$. Find all $t$ for which $f(t+2) = f(t)$.

So, if $f(t+2)=f(t)$, the is equivalent to $|t+1| = |t-3|$. Thus, if this holds, one can square both sides and arrive at $t=1$. So, this value of $t$ is a necessary condition, but prima facie it isn't the only value. To show sufficiency, could I let $t = 1 + \epsilon$, plug it into the above equation, deduce that $\epsilon = 0$, and conclude that $t=1$ is the only value? Would that go into showing this is the only value?

REPLY [0 votes]: Noting wrong with breaking into cases.
|t+1|=|t-3|
Case 1: t+1 and t-3  both have same parity.
Then t+1=t-3 which is impossible.
Case 2: they have opposite parity.
The t+1=-(t-3)=3-t.  So t=1.
That's the only possible solution.


TITLE: Multiplying by an irrational number in combinatorial problems
QUESTION [14 upvotes]: Everybody knows that the number of derangements of a set of size $n$ is the nearest integer to $n!/e$.
It is also widely known that the $(n+1)$th Fibonacci number $F_{n+1}$ is the nearest integer to $(1+\sqrt{5})F_n/2$ where $F_n$ is the $n$th Fibonacci number (with the lone exception that $F_2=F_1$).
It had escaped my attention until today, when I wrote this answer, that the number of sequences of distinct elements of a set of size $n$ (including those of length $0$) is the nearest integer to $n!e$.  (Later note: Provided $n\ge2$.)
How widespread is this operation of mulitplying by an irrational number and then rounding, in combinatorial problems?  Are there other standard examples?  Is there some general theory accounting for this?
Postscript some hours later: If I'm not mistaken, the sequence whose $n$th term is the nearest integer to $n!e$ satifies the recurrence $x_{n+1} = (n+1)x_n + 1$.

REPLY [3 votes]: The number of paths from vertex $v_1$ to $v_2$ in a complete graph on $n$ vertices is $$\sum_{k=0}^{n-2}\dfrac{(n-2)!}{k!}=\lfloor(n-2)!e\rfloor$$
To see why note that $e=1+1+\frac{1}{2}+\frac1{3!}+\cdots+\frac{1}{(n-1!)}+\cdots$
So $\displaystyle e(n-2)!=\left(\sum_{k=0}^{n-2}\dfrac{(n-2)!}{k!}\right)+\frac{(n-2)!}{n-1!}+\frac{(n-2)!}{n!}+\cdots$
But the sum at the end is less than $1$ for $n\geq3$. (to see why see that the sum at the end is strictly less than the geometrc series $(n-1)+(n-1)^2+(n-1)^3+\cdots$ which is exactly $1$ when $n$ is $3$.


TITLE: How can I approximate $\sum_{k=0}^\infty \exp(-[(ak)^3+ibk])$ as an integral?
QUESTION [0 upvotes]: As the title says I would like to know the solution of the following series,
$$\sum_{k=0}^\infty \exp(-[(ak)^3+ibk]),$$
for $a\ll1$. I think this is too difficult to solve directly. Can we find an approximate solution by writing the sum as an integral? If so, how?

REPLY [1 votes]: Assume $b \in \mathbb R$ is fixed. The sum and the integral tend to two different constants:
$$\lim_{a \downarrow 0} \int_0^\infty e^{-a k^3 - i b k} dk =
-\frac i b, \\
\lim_{a \downarrow 0} \sum_{k \geq 0} e^{-a k^3 - i b k} =
\frac 1 2 - \frac i 2 \cot \frac b 2.$$
The limit of the integral can be found by applying the steepest descent method. For the sum, suppose $\cot$ is the distribution understood in the sense of the principal value. The Fourier coefficients are
$$\frac 1 {2 \pi} \int_{-\pi}^\pi e^{-i k b} \cot \frac b 2 \, db =
-i \operatorname{sgn} k, \\
\frac 1 {2 \pi} \int_{-\pi}^\pi e^{-i k b} \delta(b) db =
\frac 1 {2 \pi}.$$
Then, in the distributional sense,
$$\lim_{a \downarrow 0} \sum_{k \leq 0} e^{a k^3 + i b k} =
\sum_{k \leq 0} e^{i k b} =
\frac 1 2 -
 \frac 1 2 \sum_{k \in \mathbb Z} e^{i k b} \operatorname{sgn} k +
 \frac 1 2 \sum_{k \in \mathbb Z} e^{i k b} = \\
\frac 1 2 - \frac i 2 \cot \frac b 2 +
 \pi \sum_{k \in \mathbb Z} \delta(b - 2 \pi k).$$
When $b \neq 2 \pi k$, we have only the regular part.


text:
\begin{document}
   \baselineskip   .7cm
\newtheorem{defn}{Definition}[section]
\newtheorem{thm}{Theorem}
\newtheorem{propo}[defn]{Proposition}
\newtheorem{cor}[defn]{Corollary}
\newtheorem{lem}[defn]{Lemma}
\theoremstyle{remark}
\newtheorem{rem}{Remark}[section]
 \newtheorem{ex}[defn]{Example}

\renewcommand\o{{{\mathcal O}}}
\newcommand\s{\sigma}
\newcommand\w{\widehat}

\newcommand\cal{\mathcal}

\title [Annihilators for cusp forms of weight $2$ and level $4p^m$]
{Annihilators for cusp forms of weight $2$ and level $4p^m$}

\author[A. \'Alvarez]{A. \'Alvarez${}^*$     }
\address{Departamento de Matem\'aticas  \\
Universidad de Salamanca \\ Plaza de la Merced 1-4. Salamanca
(37008). Spain.}

\thanks{MSC: 11F11, 11F33, 11G20 \\$*$  Departamento de Matem\'aticas.
Universidad de Salamanca. Spain.}

\maketitle

  \begin{abstract} We obtain   operators,  given essentially by formal sums of Hecke operators,  that annihilate
  spaces of cusp forms of weight $2$ for $\Gamma_1( p^m)\cap
  \Gamma(4)$, whose dimensions will be specified.
  Moreover, we obtain      the principal part  ($\mathrm{mod} \, p$), over the
  cusps,
  of certain meromorphic modular functions of level $4p^m$.

  \end{abstract}

\tableofcontents

\section{Introduction}

{\bf{Previous notation:}}
In this this work $p$ is a prime integer $\neq 2,3$. We fix $\zeta_{4p^m}$,   a primitive $ {4p^m}$-root of unity.
 Let $\mathfrak p$ be a place   over $p$  in ${\Bbb Z} [\zeta_{4p^m}]$ and $R_{\mathfrak p}$  is the local ring ${\Bbb Z} [\zeta_{4p^m}]_{(\mathfrak p)}$; $k$ and $K$ are its residual field and fraction field, respectively.
The cardinality of $k$ is $p^\delta$, where $\delta=1$ when $4\vert
(p-1) $, and $\delta=2$ in  other cases.
If $X$ is a curve over $\mathrm{Spec}(R_{\mathfrak p})$, then
$ X_k$ denotes $X\otimes_{R_{\mathfrak p}}k$. Let us denote by
$\mathrm{Fr}$   the $p^\delta$-Frobenius morphism. We denote by $A^0_m$   the $p^m$-cyclic subgroup of $({\Bbb Z}/p^m)^{\oplus 2}$ generated by $(1,0)$ and  ${A^0_i}\subset A_m^0 $ its $p^i$-cyclic subgroup. Let $H/L$ be a Galois extension; the group $Aut_L(H)$ is called the Galois group of $H/L$.

\bigskip

Part of this  section is  extracted from \cite{DR}, \cite{L} and \cite{M}.
Let us denote by ${\mathfrak H}$  the upper plane of complex
numbers  with $\mathrm{Im}\, z>0$. Let us denote by ${\mathfrak H}^*$   the union
of the  upper plane with $\infty$ and ${\Bbb Q}$. The modular
congruence  subgroups $\Gamma(n)$  and $\Gamma_1(n)$ consist  of the elements  $\gamma \in Sl_2({\Bbb Z})$ such that
$$\gamma\equiv   \left(\begin{matrix} 1   & 0
  \\0 & 1
  \end{matrix}\right)\,\mathrm{mod} \, n \text{,
   }\gamma\equiv   \left(\begin{matrix} 1   & b
  \\0 & 1
  \end{matrix}\right)\,\mathrm{mod} \, n,$$
(where $b$ is arbitrary), respectively. The quotients  ${\mathfrak
H}^*/\Gamma(n)$ and ${\mathfrak H}^*/\Gamma_1(n)$ give   smooth,
compact modular curves over ${\Bbb C}$, usually denoted by $X(n)$
and $X_1(n)$, respectively. In the same way,  if one considers
commensurable subgroups  $\Gamma\subset Sl_2({\Bbb
 Z})$,   one obtains  modular curves $X_\Gamma$.

A  modular form of weight $2$  for $\Gamma(n)$,
 (respectively $\Gamma_1(n)$), is a  holomorphic function
 $g(z):{\mathfrak H}^*\to {\Bbb C} $
 such that $g(\frac{az+b}{cz+d})=(cz+d)^{ 2}g(z)$  for each $\gamma \in \Gamma(n)$, (respectively $ \Gamma_1(n)$), with
$$\gamma=   \left(\begin{matrix} a  & b
  \\c & d
  \end{matrix}\right). $$
  Therefore, if
  $g(z)$ is a modular form for $\Gamma(n)$,
 (respectively $\Gamma_1(n)$) then $g(z+n)=g(z)$, (respectively, $g(z+1)=g(z)$).
 If we denote $q:=e^{2\pi iz}$, then $g(z)$ has
  a $q$-expansion: $g(z)=\sum_{i=r}^\infty a_iq^{i/n}$, (respectively, $g(z)=\sum_{i=r}^\infty a_iq^{i }$), for some $r\in {\Bbb Z}$.

  The open subset    $X^0(n):={\mathfrak H}/\Gamma(n)$   of $X(n)$   has the
 following modu\-lar interpretation: its points are identified with isomorphism classes of elliptic
 curves  $E$  over ${\Bbb C} $, together with  an isomorphism of the
   $n$-torsion group of $E$ with $({\Bbb Z}/n)^{\oplus 2}$.  The points of the closed subset  $X(n)\setminus
 X^0(n)$, (respectively $X_1(n)\setminus
 X_1^0(n)$)
   are called  the cusps   of $X(n)$, (respectively $X_1(n)$).

A cusp form of weight $2$ for  $\Gamma_1(n)$
   is a modular form $g(z)$ for  $\Gamma_1(n)$
   of weight
$2$    that has a zero at each cusp point. Via the identification
$g(z)\to g(z)dz$,  the space of cusp forms for $\Gamma_1(n)$
  is isomorphic to
the space of $1$-holomorphic differentials on $X_1(n)$.

The meromorphic functions on $X(n)$ are called  meromorphic
modular functions of level $n$.

Let us set
$ R_i   \in \Gamma(1)/\Gamma_1(  p^m)$  as
$$R_i\equiv   \left(\begin{matrix} i   & 0
  \\0 & i^{-1}
  \end{matrix}\right) \mathrm{mod} \,\Gamma_1(  p^m) \text{ and } R_i\equiv   \mathrm{Id} \,\mathrm{mod}\,\Gamma (  4)$$
  with $1\leq i < p^m$ and $(i,p)=1$.
  We use the symbol $T_p$ for the  Hecke operator $T_p(\sum_{i=r}^\infty a_iq^{i } )=\sum_{i=r}^\infty a_{ip}q^{i } $ and $T_{p^h}=T_p^h$. Moreover, $R_i$ and $T_p$ operate   over the cusp forms of level $\Gamma_1( p^m)\cap \Gamma (4 )$.

The Deligne-Rapoport model $M(4p^m )$, (c.f. \cite{DR}
 IV, Corollary 3.9.2.) provides us with   proper models of $X(4p^m)$ over ${\Bbb Z}[1/4, \zeta_{p^m}]$.
  There exists an  open  subscheme $M^0(4p^m )\subset M (4p^m )$    with   $$M^0 (4p^m )\otimes_{{\Bbb Z}[1/4, \zeta_{p^m}]} {\Bbb C}= X^0(4p^m ),$$

   Moreover, there exits a moduli scheme    $M_1(4p^n )$   which is proper and flat over ${\Bbb Z}[1/4 ]$ and   is a model for $X(4p^m)/\Gamma_1(p^m)$, (c.f. \cite{DR}
 IV, Proposition 3.10).

 In this work,  we consider  the moduli  $M (4p^n )  $ and $M_1 (4p^m ) $ defined over the category of schemes over $R_{\mathfrak p}$.
We consider the open subscheme $M^0_1(4p^n )\subset  M_1(4p^n )$ such that $M^0_1(4p^n )\otimes_{R_{\mathfrak p}}{\Bbb C}=X_1^0(4p^m )$.

In this article for each $0\leq i \leq d-1$ we shall define a set $H_i$ and an operator $R_h$ for each $h\in H_i$ such that:

 The operator  $\sum _{i=0}^{d-1}
 (\sum_{h \in H_i}
R_{h }   \cdot T_{p^{\delta i}})$   annihilates  a space of cusp forms, of weight $2$ for the modular group  $\Gamma_1( p^m)\cap \Gamma ( 4)$, of dimension $N$, where $N=\mathrm{dim} (\mathrm{Pic}^0_{M_1(4p^m)_k/k})^{ab}$, (c.f. Theorem \ref{oper}). Let $G$ be an algebraic group, smooth and connected over $k$. There exists a  smallest linear algebraic subgroup $L\subset G$ with $G/L$ an abelian variety. $G^{ab}=:G/L$ is called the abelian part of $G$, c.f. \cite{Ro}.

In the second result of this article, we obtain the principal parts
of certain meromorphic modular functions of level $4p^m$  ($ \mathrm{mod}\,  p$) over  $M(4p^m)$. Let    $\mathrm{Spec}(B)=M^0(4p^m)$  and  let $\overline q^{1/4p^m}$  be the reduction  $ \mathrm{mod}\, \mathfrak p$ of the local parameter $q$ for the $\infty$.  We prove in  Theorem \ref{prin} that:

 For each $m\in {\Bbb N}$, there exists an element of $B_k[p(\sigma)^{-1}]$
  such that its principal is
 $\sum _{i=0}^{d-1}
 (\sum_{h \in H_i}
  \frac{1}{ ({\overline
q^{mp^{\delta i}} )^{R^{-1}_{h }}} }).$

 The function field
 of
${ M}_4[1/4]\otimes_{{\Bbb Z}[i]}{k}$ is $k(\sigma)$, (c.f \cite{H} 4.3), and $p(\sigma)\in k[\sigma]$ is   defined in section 4.1.

To prove the above two results, we use    the correspondences  found by
Anderson and Coleman, (c.f.\cite{An},\cite{C}),  which are  trivial
  (up to vertical and horizontal ones) and that are defined over
abelian extensions of   rational function fields.  These
correspondences are given by   Stickelberger's elements for
function fields and they provide proof  of the Brumer-Stark
theorem in the function field case.
To prove  the first result  of this article, we need    the Eichler-Shimura Theorem for $T_p$ over $M_1(4p^m)$.

The results of the first part  of this work is useful for obtaining,  in an algorithmic   way, the $p$-term in the Euler product of the Hasse-Weil $L$-function of $M_1(4p^m)$ (section 4.3), while the second result could be considered as an additive version of the arithmetic problem of explicitly obtaining  generators for principal ideals of global fields.

\section{Preliminaries}

\subsection{Models  for modular curves}\label{mo}

   We consider $M(4p^m )$  and $M_1(4p^m )$ defined over $R_{\mathfrak p}$. Note that $M(n)$ for  small   $n $ is not a scheme.
For the modular interpretation  of these models we use the Drinfeld level structures for elliptic curves, c.f. \cite{KM}.
\begin{defn}(c.f. \cite{KM}, chapter 3) Let $E$ be an elliptic curve over a scheme $S$. A $\Gamma(p^m)$ (respectively $\Gamma_1(p^m)$)-Drinfeld level structure  is an homomorphism of groups $\phi_{p^m}:({\Bbb Z}/p^m)^{\oplus 2}\to E_{p^m}(S) $, (respectively $\phi^1_{p^m}:{\Bbb Z}/p^m\to E_{p^m}(S) $), such that the divisor $\sum_{a\in ({\Bbb Z}/p^m)^{\oplus 2} } \phi_{p^m}(a)$, (respectively $\sum_{a\in {\Bbb Z}/p^m } \phi^1_{p^m}(a)$) gives a direct sum of two cyclic subgroup subschemes of $E$ of order $p^m$ (respectively a cyclic subgroup subscheme  of $E$ of order $p^m$). Note that in the case of $S=\mathrm{Spec}(k)$ with  $char(k)=p$, a  $p^m$-cyclic subgroup subscheme of $E_{p^m}$ could be
  $  \mathrm{Spec}(k[t]/t^{p^m})$  and hence the divisor given by $\phi^1_{p^m} $ could be  $p^m\cdot 0$. When $E_{n}$ is etale over $S$ and $\phi_{n}$ is a  usual $n$-level structure then  $\phi_{n}$   will be denoted by $\iota_{n}$.
\end{defn}

\begin{defn}(c.f. \cite{KM}, chapter 3) A  $\Gamma_1(p^m)$-level structure over an elliptic curve $E$ over $S$ is an object  $(E, C_{p^m}, \phi^1_{p^m} )$, with   $\phi^1_{p^m}$ a   $\Gamma_1 (p^m)$-Drinfeld level structure and $C_{p^m}=\phi^1_{p^m}({\Bbb Z}/p^m)$. The objects  $(E,    \phi_{p^m} )$ are said to be $\Gamma  (p^m)$-level structures.
\end{defn}

Bearing in mind that $M(4)$ is a fine moduli over $R_{\mathfrak p}$ for the $4$-level structures  (c.f. \cite{DR} IV, Corollaire 2.9), by \cite{KM} 5.5.1, and \cite{DR} IV, we have that
 $M^0(4p^m)$ (respectively  $M^0_1(4p^m)$) are schemes  and they have the following  modular  interpretation:
Let $S$ be a   scheme over $R_{\mathfrak p}$, $M^0(4p^m)(S) $ (respectively  $M^0_1(4p^m)(S) $)  are the sets    $$\{(E,\iota_4 , \phi_{p^m})\} \text{ (respectively  }\{(E,\iota_4, C_{p^m}, \phi^1_{p^m})\}),$$   where $E$ is an elliptic curve over $S$ and $\iota_4$ is a  usual $4$-level structure.

  Let ${A_m}$ be a cyclic subgroup of $({\Bbb
Z}/p^m)^{\oplus 2} $  of cardinality $p^m$. We define  the functor ${\mathcal C}^0_{A_m}(S)$  as the set of objects $(E,\iota_4,C_{p^m}, \phi_{p^m} )$
  such that   $\phi_{p^m}(A_m)=C_{p^m}$  is an etale $p^m$-cyclic subgroup of $E$ over the points of $S$ where $E$ is not supersingular. By \cite{DR} V, 4.8.
  ${\mathcal
C}^0_{A_m}\subset M^0(4p^m)$ gives a  curve   over $R_{\mathfrak p}$.

Let $\{A_m^i\}_{0\leq i\leq l-1  }$ be  the set of $p^m$-cyclic subgroups of $({\Bbb Z}/p^m)^{\oplus 2}$ and $l$  the cardinality of this set. According to    \cite{DR} V, Theorem 4.12,  the decomposition into the reduced
and irreducible components of the scheme $  M(4p^m )_k  $  takes the form
$\cup^{l-1}_{i=0}   ({\mathcal C}_{A^i_m})_k$.  The curves $({\mathcal C}_{A^i_m})_k$ are  smooth over $k$. We denote $ ({\mathcal C}^0_{A^i_m})_k=({\mathcal C}_{Aî_m})_k\cap M^0(4p^m )_k$.

\begin{rem}\label{psi}
Let $A_m^0   $  and ${A^0_i}\subset A_m^0 $ be as in the Introduction. We have an injective morphism  of functors $\Psi_i:({\mathcal C}^0_{A^0_i} )_k\hookrightarrow  M^0_1(4p^m)$.
Let us consider   an object  $(E,\iota_4 , \phi_{p^i} )\in ({\mathcal C}^0_{A^0_i} )_k$ and  the morphism of elliptic curves
$$h_i:E\to E/\phi_{p^i}(A^0_i)\overset{\mathrm{Fr}^{m-i}} \longrightarrow (\mathrm{Fr}^{m-i} )^*(E/\phi_{p^i}(A^0_i)),$$
 where, $\mathrm{Fr}$ denotes the $p$-Frobenius morphism. Thus, we define
 $$\Psi_i  (E,\iota_4  , \phi_{p^i})=(E, \iota_4, \mathrm{Ker }(h_i) , \phi^1_{p^m} ),$$  where  $\phi^1_{p^i}:{\Bbb Z}/p^m\simeq A_m^0 \to E_{p^m}(S)$
  with $\sum_{a=0}^{p^{m-i}}\phi^1_{p^m}(a)=p^{m-i}\cdot 0$ and  ${\phi^1_{p^m}}_{\vert {A^0_i}}={\phi_{p^i}}_{\vert {A^0_i}}$. We have that $\mathrm{Ker }(h_i)=\phi^1_{p^m}({\Bbb Z}/p^m)$.
\end{rem}

 Let $A^0_m,\cdots, A_m^m$ be   elements  for
 the equivalence classes of the quotient set for the action of $\Gamma_1(p^m)$ on   the set of $p^m$-cyclic subgroups  ${A^j_m}$  of $({\Bbb
Z}/p^m)^{\oplus 2} $, where  $A^0_i=A_m^0\cap A^{m-i}_m$.
The decomposition into reduced and irreducible components of $ {M^0_1(4p^m )}_k$ is $\cup^{m}_{i=0} \Psi_i(({\mathcal
C}^0_{A^0_i})_ k) $. This result is  deduced   from both \cite{DR} V, 4.8, 4.11 and 4.12,  and
  \cite{KM} 13.5.6.

\subsection{N\'eron models}
Let $R$ be a discrete valuation ring, $K$ its fraction field,  and $k$ the residual field, which we assume to be perfect. The N\'eron model  for an abelian variety $J_K$ defined over $K$  is a  group scheme ${\mathcal J}$  smooth over $R$,  such that ${\mathcal J}_K\simeq J_K$  and for each scheme ${\mathcal H}$ over $R$ the natural morphism
   $$(*)Hom_R({\mathcal H}, {\mathcal J})\to Hom_K({\mathcal H}_K, {\mathcal J}_K)$$
   is bijective.

   Let $X$ be a   flat, proper   curve over $R$ with $X_K$ smooth and geometrically irreducible over $K$.       We assume that $X_k$ has an irreducible component of  geometric multiplicity $1$.  In the above conditions, the identity component for
   the  N\'eron model ${\mathcal J}$ for $\mathrm{Pic}^0_{X_K/K}$  is $ \mathrm{Pic}^0_{\overline X/R}$, where $\overline X\to X$ is a desingularization for $X$. (C.f. \cite {BLR}, 9.5 Theorem 4 and 9.7 Theorem 1).

\begin{defn}Let  $G$, $H$ be two abelian group schemes  over $R$. $f:G\to H$ is said to be a quasi-isogeny if there exists a   homomorphism   $g:H\to G$, with $g\cdot f=[l\cdot]$, for some $l\in {\Bbb N}$.\end{defn}

   \begin{lem}Let ${\mathcal J}$, $ {\mathcal L}$ be the N\'eron models for the abelian varieties   $J_K$ and $ L_K$, respectively. Let $f:J_K\to L_K$ be an  isogeny and ${\mathfrak f}: {\mathcal J}\to {\mathcal L}$ the morphism associated with $f$ by the   bijection (*). We have that
   ${\mathfrak f}_k:{\mathcal J}_k\to {\mathcal L}_k$ is a  quasi-isogeny and ${\mathfrak f}^{ab}_k:{\mathcal J}^{ab}_k\to {\mathcal L}^{ab}_k$ is an isogeny. We denote by  ${\mathfrak f}_k^{ab}$   the homomorphism given by ${\mathfrak f}_k$ between the abelian parts of ${\mathcal J}_k$ and ${\mathcal L}_k$.
   \end{lem}
   \begin{proof} Since $f$ is an isogeny  ${\mathfrak f} $ is a quasi-isogeny  and therefore  ${\mathfrak f}_k$ is also a quasi-isogeny. The remaining assertion is proved because over abelian varieties quasi-isogenies are isogenies.
   \end{proof}

\begin{lem}\label{ab}Let $g$ be an endomorphism of an abelian variety $J_K$.   Let $ {\mathfrak g}_k^{ab}: \mathcal J^{ab}_k\to \mathcal J^{ab}_k$ be the endomorphism given by ${\mathfrak g}_k$. If $\mathrm{dim}\,\mathrm{Ker }({\mathfrak g}_k^{ab})\geq r$ then $dim \mathrm{Ker } (   g  )\geq r$. Here, we assume that $char(K)=0$.
\end{lem}
\begin{proof} By \cite{BLR} 7.1, Corollary 6, $\mathrm{Ker } ({ g} )$ and $ \mathrm{Im} (  g   )$ admit N\'eron models ${\mathcal N}$ and ${\mathcal I}$, respectively.
Because ${\mathcal N}$ is flat over $R$ we have
$ \mathrm{dim}\, \mathrm{Ker } (  g) =\mathrm{dim}\,{\mathcal N}_k$. Thus, to prove the Lemma it suffices to prove that $  {\mathcal N}^{ab}_k$ is  isogenic to $\mathrm{Ker }({\mathfrak g}_k^{ab})$.

We have the complex ${\mathcal N}\to {\mathcal J}\overset {\mathfrak g} \to {\mathcal I}.  $
Moreover, $J_K$ is   isogenous to $\mathrm{Ker } (  g  ) \times \mathrm{Im}\, ( g  )$ so by the above Lemma ${\mathcal J}$ and ${\mathcal J}_k$ are quasi-isogenous to
${\mathcal N}\times {\mathcal I} $  and ${\mathcal N}_k\times {\mathcal I}_k $, respectively and
${\mathcal J}^{ab}_k$ is isogenous to ${\mathcal N}^{ab}_k\times {\mathcal I}^{ab}_k $. Thus,  we deduce
an exact sequence,  up to isogenies
$$0\to {\mathcal N}^{ab}_k\to{\mathcal J}^{ab}_k\overset{{\mathfrak g}^{ab}_k}\to {\mathcal I}^{ab}_k\to 0$$
and therefore $\mathrm{Ker } ({\mathfrak g}_k^{ab})$ is isogenic to ${\mathcal N}^{ab}_k $.
\end{proof}

\section{The Eichler-Shimura Theorem}

\subsection{Convention for correspondences}\label{conv}
  Let us
consider a proper smooth and geometrically irreducible  curve   $Y$ over a field $k$.  A
correspondence  $C\subset
 Y\times Y$,    given by a Weil divisor,   defines an endomorphism, denoted by $\tilde C$, over the Jacobian  $J_Y$
 in the following way and convention: Let $R$ be an $k$-algebra  and $L(D)$
a line bundle associated  with a  locally principal
 ideal    $I_D\subseteq \o_{Y_R }$  (i.e: $I_D$ is given by an effective Cartier divisor
 $D$  on $Y_R$),   then, $\tilde C(L(D))$ is the line bundle on $Y_{R }$ with  the Cartier divisor
 given
 (locally over $Y_ R$)  by
 the line bundle obtained from the kernel ideal of the ring morphism:
 $$\phi: \o_{Y_R }\to \frac{  \o_{Y_R }  \otimes_R\o_{Y_R }}{I_D\otimes_R \o_{Y_R }
  +I_C},$$
 with  $\phi(a ):= 1\otimes a $, and $I_C$  is the ideal  associated with  $C$.
 We denote by $C(D)$ the Cartier divisor given by $ \mathrm{Ker} (\phi)$. By linear
 extension,
 one can obtain  $\tilde C(L(D))$ and  $C(D)$, with $D$  not
 effective.

Let $\tau:Y\to Y$ be  a scheme morphism. We denote  $\Gamma(\tau)=\{(x,\tau(x)):x\in Y\}$ and its transpose $^t\Gamma(\tau)=\{( \tau(x),x):x\in Y\}$.

\begin{lem}\label{Fr} Let   $k$ be a finite field of $q$ elements. If we denote by
 $D$ a  Cartier divisor  over $ Y_R  $, then $
 ^t{\Gamma(Fr)}(D) =(\mathrm{Fr}\times \mathrm{Id})^*D$ and $ {\Gamma(\mathrm{Fr})}(D) =(\mathrm{Id}\times \mathrm{Fr})^*D$. Here, $  \mathrm{Fr}$ denotes the $q$-Frobenius morphism and $\gamma^*D$ denotes
 the pullback of $D$ by  a morphism $\gamma$.
 \end{lem}
 \begin{proof} Let $I\subset  \o_{Y_R }$  be an ideal   and $I_{\Delta}\subset  \o_{Y_R }\otimes_R
  \o_{Y_R }$ the diagonal ideal. The ideal $I_\Gamma$  given by  $
 ^t{\Gamma(\mathrm{Fr})}$ is $(\mathrm{Id}_Y\otimes \mathrm{Id}_R)\otimes (\mathrm{Fr}\otimes \mathrm{Id}_R))( I_{\Delta})$. Thus,
 $  I\otimes_R  \o_{Y_R } +I_\Gamma = (\mathrm{Fr}\otimes \mathrm{Id}_R)I  \otimes_R \o_{Y_R }+I_\Gamma.$
 Thus,  $^t{\Gamma(\mathrm{Fr})}(D) = (\mathrm{Fr}\times \mathrm{Id}_R)^*D$ for any Cartier divisor $D$.

To prove the remaining equality we consider
 the composition of the
correspondences $
 ^t\Gamma(\mathrm{Fr}) * {\Gamma(\mathrm{Fr})}=q\Delta$, where $\Delta\subset Y\times Y$  is
 the diagonal subscheme. Moreover,   $(\mathrm{Fr}\times \mathrm{Fr})^*D=D^q$  for each  Cartier divisor  $D$  on $Y_R $ and
  $  ^t{\Gamma(\mathrm{Fr})}[  {\Gamma(Fr)} (D)- (\mathrm{Id}\times \mathrm{Fr})^*D]=0$  and
we conclude, since    $  {\Gamma(\mathrm{Fr})}$ is injective acting on the
Cartier group of divisors of  $ {Y_R }$.
 \end{proof}

\subsection{The Eichler-Shimura Theorem for $T_p$}\label{sh}
We fix an isomorphism of groups  $p/p^{m+1}\simeq {\Bbb Z}/p^m$ and we follow the notation of section \ref{mo}.

  First, we shall define the Hecke correspondence  $T_p$ on  $M^0_1( 4p^n)$.

  Following \cite{R}, we consider the morphisms,
  $$\beta,   \alpha:M^0_1(4p^{m+1}) \to M^0_1(4 p^{m }) $$
   $$\beta(E,\iota_4,C_{p^{m+1}}, \phi^1_{p^{m+1}})=(E/C_p,\iota'_4, C_{p^{m+1}}/C_p,  \overline \phi^1_{p^{m+1}}  ),$$
   $$\alpha(E,\iota_4 ,C_{p^{m+1}}, \phi^1_{p^{m+1}} )=(E ,\iota_4, C_{p^{m } } ,  {\phi^1_{p^{m+1}}}_{\vert p/p^{m+1}}),   $$

   $\iota'_4$ is deduced from the isomorphism  $ E_4\simeq E_4/C_p$,
   $\overline \phi^1_{p^{m+1}}$ is $\phi^1_{p^{m+1 }}$ $ \mathrm{mod} \, p^m$ and $C_{p^{i}}=\phi^1_{p^{m+1}}(p^{m+1-i}/p^{m+1 })$.

  Because $M^0_1(4p^m )$ is an affine scheme it is a separated scheme over $R_{\mathfrak p}$. Moreover,
  $\alpha$ is a finite morphism, and therefore proper, because the finite morphism of forgetting the $p^m$-level structures,  $M ^0(4p^m )\to M ^0(4 )$, factorizes through $\alpha$. Thus, we have the closed subscheme   $$\Gamma^0_p=\{ (\beta(x),\alpha(x))\text{ with } x\in M ^0_1(4p^{m+1} )\}\subset M^0_1(4p^m )\times_{R_{\mathfrak p}} M^0_1(4p^m ).$$
Let $S$ be an $  R_{\mathfrak p} $-scheme. Then, $\Gamma^0_p$ has the following modular interpretation
$$ \Gamma^0_p(S)=\{ (E,\iota_4 ,C_{p^{m }}, \phi^1_{p^{m }}  ),  (\overline E,\overline \iota_4 ,\overline C_{p^{m }}, \overline \phi^1_{p^{m }} )  \}$$
such that there exists an isogeny $\psi:E \to \overline E$ of degree $p$ with $ \overline \iota_4=\psi\cdot   \iota_4$,
$ \overline \phi^1_{p^{m }}=\psi\cdot   \phi^1_{p^{m }}$ and $\psi( C_{p^{m }})=\overline C_{p^{m }}$.

According to \cite{DR}
 IV, Proposition 3.10,  we consider $M_1(4p^m):=(M(4p^m)/G)_{ R_{\mathfrak p}}  $, which is a  proper and flat  scheme   over $ R_{\mathfrak p}$,   where  $$G:=\{\gamma \in Gl_2({\Bbb Z})\text{ such that }
 \gamma\equiv   \left(\begin{matrix} 1   & b
  \\0 & d
  \end{matrix}\right)\,\mathrm{mod} \, p^m\}.$$
 This scheme  is a compactification   of $M^0_1(4p^m)$.

  Now, let $f:\overline{M}\to M_1(4p^m)$ be a desingularization of  $M_1(4p^m)$.
  By \cite{KM}, Theorem 10.12.2,     $\overline{M}$  can be obtained from the  $\Gamma_1(p^m)$-balanced moduli problem over $M(4)$. Moreover, for each automorphism $R_j$, there exists an automorphism  $R_j:\overline{M}\to \overline{M}$, (which we also denote  by $R_j$), such that $R_j\cdot f=f\cdot R_j$.

  The   schematic closure of $\Gamma^0 _p\subset  M_1(4p^{m })\times_{R_{\mathfrak p}} M_1(4p^{m })$ will be denoted by $\Gamma_p$ and  $\overline \Gamma_p:=(f\times f)^{-1}( \Gamma_p)$.

\begin{lem}   $\overline \Gamma_p$  provides
an   endomorphism, which we denote  by $T_p$, of the scheme $\mathrm{Pic}^0_{\overline M/R_{\mathfrak p}}$.\end{lem}
\begin{proof}According to \cite{KM}, 13.5.6,  we are in the conditions of \cite{BLR}, Theorem 1, 9.7 , and hence
$\mathrm{Pic}^0_{\overline M/R_{\mathfrak p}}$ is a smooth scheme over $ \mathrm{Spec}(R_{\mathfrak p})$. It is the identity component of the  N\'eron model for $\mathrm{Pic}^0_{M_1(4p^m)_K/K}=\mathrm{Pic}^0_{\overline M_K/K}$.

To find the endomorphism $T_p$  it suffices to give for each smooth morphism
     $W\to \mathrm{Spec}(R_{\mathfrak p})$  an   endomorphism
   $$(T_p)_W: \mathrm{Pic}^0 _{\overline M/R_{\mathfrak p}}(W)\to \mathrm{Pic}^0_{\overline M/R_{\mathfrak p}}(W).$$

   Let $\pi_i$, $i=1,2$, be the natural projections $\pi_i:\overline M_W\times_W \overline M_W\to \overline M_W$, and $ \overline \pi_i   $ its restrictions to $(\overline \Gamma_p)_W$. Following the convention fixed in the previous section, let $L$ be a line bundle over $\overline M_W $,    $\overline \pi_1^*(L)$ is a line bundle on  $(\overline\Gamma_p)_W$ with Weil divisor $Z$. Therefore,  the pushforward $(\overline \pi_2)_*(Z)$ is a Weil divisor on the regular scheme $\overline M_W $. We define  $(T_p)_W(L)$ as the line bundle on $\overline M_W$ associated with this Weil divisor. Note that $\overline M_W$ is a regular scheme because $\overline M_W\to \overline M$ is smooth and $\overline M$ is regular.
\end{proof}

 For a  morphism  $R_{\mathfrak p}\to {\Bbb C}$,  one can write  down  $T_p$ explicitly in terms of the modular parameter $\tau$.  Let $E_\tau$ be the elliptic curve given by the torus obtained from the lattice $1\cdot {\Bbb Z}+\tau \cdot{\Bbb Z}$, $\tau \in {\Bbb C}$. Let $(E,C_{p^m},\phi_{p^m} )$  be given by the pair $(E_\tau, \frac{   p^{-m}\Bbb Z } {\Bbb Z}, \phi^1_{p^m})$. Here,  $\phi^1_{p^m}$ is given by the natural isomorphism
   ${\Bbb Z}/p^m{\Bbb Z}\simeq p^{-m}{\Bbb Z}/{\Bbb Z} $.  We have:
   $$T_p (E,p^{-m}{\Bbb Z}/{\Bbb Z}, \phi^1_{p^m})=\sum_{i=0}^{p-1}(E_{\frac{i+\tau }{p}}, \frac{   p^{-m}\Bbb Z } {\Bbb Z} , \phi^1_{p^m}).$$

If $f(\tau)$ is a  cusp form of weight $2$  and level $p^m$, with $q$-expansion $\sum_{i=r}^\infty a_iq^i$, then $T_p(f(z))= \sum_{i=r}^\infty a_{ip}q^i$.

{\bf{Notation}} We denote $M:= M_1(4p^{m })$.
We consider    the desingularization   morphisms   $h:(\widetilde  {M_k})_{\mathrm{red}}  \to M_k$ and $\overline h: (\widetilde  {\overline M_k})_{\mathrm{red}}  \to \overline M_k$.

Let us consider $\widetilde f_k:\widetilde {(\overline M_k)}_{\mathrm{red}}\to (\widetilde M_k)_{\mathrm{red}}$, the mosphism induced by $f_k:\overline M_k\to  {  M_k}$. The morphism $\widetilde f_k$ is surjective because $f$ is surjective. Moreover,
we have  $h\cdot \widetilde f_k=f_k\cdot \overline h$.

We denote $(\widetilde {\Gamma}_p)_k :=
 (h\times h)^{-1}({\Gamma_p})_k  $  and $(\widetilde  {\overline \Gamma _p})_k:=(\overline h\times \overline h)^{-1}(   {  \overline \Gamma _p})_k$.

  Note that $ (\Gamma_p^0)_k=(\widetilde \Gamma_p)_k\cap (h\times h)^{-1}( M^0_k\times M^0_k)$.

We have  that $(\widetilde {M}_k)_{\mathrm{red}}=\sqcup_{i=0}^m \Psi_i(({\mathcal C}_{A^0_i})_k)$ and therefore
  $\mathrm{Pic}^0_{(\widetilde {M}_k)_{\mathrm{red}}/k}=\prod_{i=0}^m\mathrm{Pic}^0_{({\mathcal C}_{A^0_i})_k/k}.$ Moreover, $\mathrm{Pic}^0_{({\mathcal C}_{A^0_0})_k/k}=\{0\}$, c.f \cite{KM} 13.5.6.
Now, we prove      the Eichler-Shimura Theorem for $T_p$ and $M_1(4p^m)$.
\begin{lem}\label{es} We have that $ (\widetilde {\Gamma ^0_p})_k(x)= {\Gamma}(\mathrm{Fr})(x)$, where $x$ is a geometric point of $  \Psi_i(({\mathcal C}_{A^0_i})_k)$, ($i>0$),  given by $\Psi_i(  E,  \iota_4 ,     \phi_{p^{i }} )$ with $E$  an ordinary elliptic curve. Moreover, $(\widetilde  {\overline \Gamma _p})_k( (\widetilde{f_k})^{-1} (x))= (\widetilde{f_k})^{-1}({\Gamma}(\mathrm{Fr})(  (x)))$. \end{lem}
  \begin{proof} Let us consider $\Psi_i(  E,  \iota_4 ,     \phi_{p^{i }} )=(  E,  \iota_4 ,  C_{p^{m }},   \phi^1_{p^{m }} )=x$. Since $i>0$ the subgroup $C_p\subset C_{p^{m }}$ is etale because $ \phi_{p^i}(A^0_i)$ is an etale subgroup of $C_{p^{m }}$ of cardinality $p^i$, (see Remark \ref{psi}).
Let us denote  $ (\widetilde {\Gamma ^0_p})_k(x)=\overline x$ with
 $\overline x=( \overline E,  \overline\iota_4 ,  \overline C_{p^{m }},   \overline\phi^1_{p^{m }} )$ and we have an isogeny of degree $p$, $\psi:E\to \overline E$, such that   $ \overline \iota_4=\psi\cdot   \iota_4$,
$ \overline \phi^1_{p^{m }}=\psi\cdot   \phi^1_{p^{m }}$ and $\psi( C_{p^{m }})=\overline C_{p^{m }}$. Since $\psi( C_{p^{m }})=\overline C_{p^{m }}$   we have $\mathrm{Ker}(\psi)\cap C_p=\{0\}$  and thus we have that $\psi$ is the $p$-Frobenius morphism $E\to \mathrm{Fr}^* E=\overline E$, (recall that $C_p$ is etale),    and therefore $\overline x=\mathrm{Fr}(x)$.
The remaining assertion is deduced bearing in mind the above result and
$(\widetilde  {\overline \Gamma _p})_k:=(  h\cdot \widetilde f_k\times h\cdot \widetilde f_k)^{-1}(   {   \Gamma _p})_k$. Recall that, $h\cdot \widetilde f_k=f_k\cdot \overline h$.

 \end{proof}

\begin{lem}\label{T} Let $(T_p)_K:\mathrm{Pic}^0_{M_K/K}\to \mathrm{Pic}^0_{M_K/K}$ be the endomorphism given by $T_p$ and ${\mathcal T}_p:{\mathcal J}\to {\mathcal J}$  the endomorphism given by  $(T_p)_K$ for the N\'eron model ${\mathcal J}$ of $\mathrm{Pic}^0_{M_K/K}$.  We have that  the endomorphism  ${\mathcal T}^0_p:{\mathcal J}^0 \to {\mathcal J}^0$ between the identity component of ${\mathcal J}$ is $T_p$. Recall that $M:=M_1(4p^m)$ and that $k$ is the function field of $R_{\mathfrak p}$.
\end{lem}
\begin{proof} This is deduced from the equalities
$$Hom_R(\mathrm{Pic}^0_{\overline M/R}, {\mathcal J})= Hom_K(\mathrm{Pic}^0_{M_K/K}, \mathrm{Pic}^0_{M_K/K})=Hom_R({\mathcal J}, {\mathcal J}). $$
 These equalities are obtained  because ${\mathcal J}$ is the N\'eron model for $ \mathrm{Pic}^0_{M_K/K}$.
\end{proof}

\section{Annihilators for cusp forms and a  calculation }

\subsection{Annihilators for cusp forms }\label{an}

An elliptic curve  over ${\Bbb F}_p$  in the Legendre form $y^2=x(x-1)(x-\lambda)$ is supersingular if and only if $H(\lambda)=0$, with $$H(\lambda)=(-1)^m\sum^m_{i=0}\left(\begin{matrix}    m
 \\ i  \end{matrix}\right)^2\lambda^i$$  the Deuring polynomial ($m=(p-1)/2$). For more details    see
\cite{H} 13.3.  The function field
 of
${ M}_4[1/4]\otimes_{{\Bbb Z}[i]}{k}$ is $k(\sigma)$, with
 $\lambda=(\sigma+1/2\sigma)^2  $. C.f \cite{H} 4.3.
We denote $p(\sigma):=\sigma^{p-1}
H((\sigma+1/2\sigma)^2) $. Here, $k$ denotes the residual field of $R_{\mathfrak p}$.

Let us consider  $q(\sigma)\in k[\sigma]$. We denote by $K_{q(\sigma)}$  the $q(\sigma){\mathfrak m}_\infty$-ray class field for $k(\sigma)$ that  is the maximal abelian extension, totally ramified, of $k(\sigma)$ and with ideal of ramification given by $q(\sigma){\mathfrak m}_\infty$. Here,  ${\mathfrak m}_\infty$ is the maximal ideal associated with   $\infty, ( 1/\sigma =0)$.

\begin{lem}
We have that the Galois group
of $K_{q(\sigma)}/k(\sigma)$ is isomorphic to  $  (k[\sigma]/q(\sigma))^\times $.
\end{lem}\begin{proof}
From class field theory,  the
abelian extension   $K_{q(\sigma)}/k(\sigma)$ has as Galois group
  $ I  /t_\infty^{\Bbb Z} \cdot {k(\sigma)}^\times \cdot U ({q(\sigma)}),$
  where $I  $ is the idele group of   $k(\sigma)  $, $U ({q(\sigma)} )$ denotes
  the idele subgroup   $\{\mu;  \mu\equiv 1\, mod\, q(\sigma){\mathfrak m}_\infty \}$ and $t_\infty$ is the idele whose entries are $\sigma^{-1}$ at $\infty$ and $1$ elsewhere.

Let $T$ be the zero locus of $q(\sigma){\mathfrak m}_\infty$. Let $I^T $ and  ${U}^T $  the ideles   of $I $
 and ${U}$  outside $T$,  respectively, and
$K^\times(q(\sigma)):={k(\sigma)}^\times \cap U ({q(\sigma)}). $
    We have an isomorphism of groups
$$\alpha: {I^T}/{   K^\times(q(\sigma))\cdot U^T}\to
 {I}/{t_\infty^{\Bbb Z}\cdot k(\sigma)^\times \cdot U(q(\sigma))},$$
where $\alpha(\mu_T)$ is the class of the idele in $ I$, whose
entries are given by $\mu_T$ outside $T  $ and $1$ over the places
of $T$. Moreover, we have an isomorphism
$$\delta: {I^T}/{  K^\times(q(\sigma))\cdot
U^T}\to  (k [\sigma]/p(\sigma))^\times $$
 defined as follows. Let  $\mathfrak h$ be a place on $k(\sigma)$ with $\mathfrak
 h\notin
T$ and    $h(\sigma) $   the irreducible polynomial associated
with $\mathfrak h$. Let $t_\mathfrak h\in I^T$ be   the idele whose entries are $h(\sigma)$ at $\mathfrak h$ and $1$ elsewhere. We define $\delta(t_\mathfrak h)$ as the class of
$h(\sigma)^{-1}$ within $(k
[\sigma]/p(\sigma))^\times $.
\end{proof}

We have that
$\Sigma$ is a totally ramified abelian extension of  $k(\sigma)$ whose ideal of ramification  divides $q(\sigma){\mathfrak m}_\infty$ if and only if $\Sigma$ is a  subextension of $K_{q(\sigma)}$ (see   {\cite{Ha}} Chapter 9).

Let $Y_{q(\sigma)}$ be the smooth, proper and geometrically irreducible
curve over $k$  with function field $K_{q(\sigma)}$. We have the following
Anderson-Coleman result, (c.f. \cite{An}, \cite{C}): there exists
a trivial correspondence, up to vertical and horizontal
correspondences  on $Y_{q(\sigma)}\times Y_{q(\sigma)}$
$$D_k:=\sum _{i=0}^{d-1}
 (  \underset { \mathrm{deg}(r(\sigma))=d-1- i}  {\sum_{r(\sigma)\text{(monic)} } }
\Gamma(\beta_{r(\sigma)}) * \Gamma(\mathrm{Fr}^{ i}))
,$$
with $\mathrm{deg}(q(\sigma))=d$,
  $\beta_{r(\sigma)}:=(\delta^{-1}\cdot \alpha)(r(\sigma))\in Aut_{k(\sigma)}(K_{q(\sigma)})$ and  $r(\sigma)$ are monic polynomials in $\sigma$.
 We   denote  the
composition of correspondences by $*$ and $\mathrm{Fr}$ denotes the $p^\delta$-Frobenius morphism.

\begin{lem} The   extension $\Sigma_{A^0_m}/k(\sigma)$,  given by
 $( {\mathcal C}_{A^0_m})_k\to
M(4 )_k$,   is an abelian extension of group
$({\Bbb Z}/p^m)^\times $. Moreover, $\Sigma_{A^0_m}$ is a
subextension of $K_{q(\sigma)}$, where  $q(\sigma)=p(\sigma)^rk[\sigma]$  for certain
$r\in{\Bbb N}$.
\end{lem}
\begin{proof}
By \cite{DR}  V, Lemma 4.16,   the morphism $  ({\mathcal C}_{A^0_m})_k\to  M( 4)_k$ is     a ramified abelian covering of    group $({\Bbb
Z}/p^m)^{\times}$ and it is
totally ramified  over the pairs $(E, \iota_4) $,  with $E$ a supersingular elliptic curve.
This  means that it  is totally ramified for the
values of $\sigma$ where  $E$ is supersingular. These values are given by the roots of the
Deuring polynomial $\sigma^{p-1}  H(\lambda)$, $\lambda=(\sigma+1/2\sigma)^2$,
(see for example, \cite{H} Chapter 13, {\S}3).
\end{proof}

We have that
$\Gamma(1)$ operates transitively over the irreducible components of
$ M(4p^m)_k=\cup^{l-1}_{i=0}({\mathcal C}^0_{A^i_m})_k $. By fixing $A^0_m$ as in the introduction, there exists $g_i\in \Gamma(1)$, with
$ g_i(({\mathcal C}^0_{A^0_m})_k)=({\mathcal C}^0_{A^i_m})_k  $. The action of the elements $R_l$ on $g_i(({\mathcal C}^0_{A^0_m})_k)$ is given by $g_i\cdot R_l\cdot g_i^{-1}$.

The Galois group of   $\Sigma_{A^0_m}/k(\sigma)$  is identified with
the subgroup of $\Gamma(1)/\Gamma( p^m)$ formed by the classes of
$\{R_i\}_{1\leq i< p^m } \in \Gamma(1)$, where
$$R_i\equiv   \left(\begin{matrix} i   & 0
  \\0 & i^{-1}
  \end{matrix}\right)\mathrm{mod}\,\Gamma(  p^m) \text{, } R_i\equiv  \mathrm{Id}\,\mathrm{mod} \,\Gamma (  4),$$
  with $1\leq i < p^m$ and $(i,p)=1$.

\begin{rem}\label{tri}
Let $\pi: Y_{q(\sigma)}\to ({\mathcal C}_{A^0_m})_k$ be the morphism induced by the inclusion $\Sigma_{A^0_m}\subset K_{q(\sigma)}$. Let   us denote  $\rho:Gal(K_{q(\sigma)}/k(\sigma))\to ({\Bbb Z}/p^{m})^\times,$ the surjective morphism
between the Galois groups of the extensions $K_{q(\sigma)}/k(\sigma) $ and $\Sigma_{A^0_m}/k(\sigma)$.
The pushforward to $({\mathcal C}_{A^0_m})_k\times ({\mathcal C}_{A^0_m})_k$ of the correspondence $D_k$,
   $$(\pi\times \pi)_*(D_k)=\mathrm{deg}(\pi)\cdot\sum_{i=0}^{d-1}
  (\underset { \mathrm{deg}(r(\sigma))=d-1- i}  {\sum_{r(\sigma)\text{(monic)}}}
\Gamma(R_{r(\sigma)}) *{\Gamma}(\mathrm{Fr}^i))$$
   is again a trivial correspondence,    up to vertical and horizontal
correspondences, where  $R_{ {r(\sigma)}}=\rho(\beta_{r(\sigma)})$. Thus, as the ring of   classes of  correspondences is ${\Bbb Z}$-free-torsion, $\mathrm{deg}(\pi)^{-1}\cdot (\pi\times \pi)_*(D_k)$ is also trivial.

Note that  given a proper morphism  $f:Z\to \bar Z$,  between   varieties,  if
$U$ is a $k$-cycle of $Z$, with  $dim \,U=dim\, f(U)$, then the
pushforward  $f_*(U)$  is defined by
$$f_*(U)=[\Sigma_U :\Sigma_{f(U)}]\cdot f(U). $$ Where, $\Sigma_U $ and $\Sigma_{f(U)}$  are the
function fields associated with $U$ and $f(U)$, respectively.

\end{rem}

\begin{rem} Let $Y$ be a   proper curve  over a field $k$  and $ \widetilde {Y}_{\mathrm{red}}$ the normalization of the largest reduced subscheme $Y_{\mathrm{red}}$ of $Y$.  Then, by \cite{BLR}, 9.3 Corollary 11,
   $(\mathrm{Pic}^0_{Y/k})^{ab} \simeq \mathrm{Pic}^0_{\widetilde {Y}_{\mathrm{red}}/k}$. We set $N=\mathrm{dim}  (\mathrm{Pic}^0_{\widetilde {M_1(4p^m)}_k/k})^{ab}$.
\end{rem}

\begin{thm}\label{oper} The operator $$D_K=\sum _{i=0}^{d-1}
 (\underset { \mathrm{deg}(r(\sigma))=d-1- i}  {\sum_{r(\sigma)\text { (monic)}} }
 R_{ r(\sigma) } \cdot  T_{p^{i\delta}})$$ annihilates  a space of dimension $N$ of cusp forms of level $\Gamma(4)\cap \Gamma_1(p^m)$ and weight $2$.
 \end{thm}
 \begin{proof} We have  $(M^0_1(4p^m )_k)_{\mathrm{red}}=\cup^{m}_{i=0} \Psi_i(({\mathcal
C}_{A^0_i})_k)$ (see Remark \ref{psi}) and by the last Remark we have a surjective morphism
$$\mathrm{Pic}^0_{  { M_1(4p^m )_k }  /k }\to \prod^{m}_{i=1}  \mathrm{Pic}^0_{({\mathcal C}_{A^0_i} )_k/k } .$$
Since   $A^0_i\subset A_m^0$, we have  morphisms of curves $\pi_i:({\mathcal C}_{A^0_m})_k\to ({\mathcal C}_{A^0_i})_k$ defined by   $\pi_i(E,\iota_4 , \phi_{p^m} )= (E,\iota_4 , \overline \phi_{p^i} )$, with
$\overline \phi_{p^i}=  {\phi_{p^m}}_{\vert p^{m-i}({\Bbb Z}/p^m)^{\oplus 2}}$. Moreover, because $\pi_i\cdot \mathrm{Fr}=\mathrm{Fr}\cdot \pi_i$  we have that
$(\pi_i
\times \pi_i)_* \Gamma(\mathrm{Fr}) =deg(\pi_i)\cdot  \Gamma(\mathrm{Fr}).$  Here, we have
denoted   the   Frobenius morphism   over ${\mathcal C}_{A^0_m}$ and $ {\mathcal C}_{A^0_i}$ in the
same way. This latter statement also is true for the correspondences associated with the graphics of the  automorphisms $R_j$.

By Remark \ref{tri}, since  $D_k$  annihilates $\mathrm{Pic}^0_{  ({{\mathcal C}_{A^0_m}})_k/k }$, the correspondence $deg(\pi_i)^{-1}\cdot(\pi_i\times \pi_i)_*(D_k)$ also  annihilates $\mathrm{Pic}^0_{  ({{\mathcal C}_{A^0_i}})_k/k }$, with $0\leq i\leq m$.

The Theorem is deduced from the Eichler-Shimura Theorem  (c.f: Lemma \ref{es}),   from Lemma \ref{ab}
 applied to $\mathrm{Pic}^0_{M_1(4p^m)_K/K}$ and to the endomorphism  $D_K$,  knowing that $D_k$  annihilates
 $\prod_{i=1}^m\mathrm{Pic}^0_{  ({{\mathcal C}_{A^0_i}})_k/k }$.

 We bear in mind that the homomorphism $$(\widetilde f_k)^*:(\mathrm{Pic}^0_{\widetilde {M_1(4p^m)}_k/k})^{ab}\to (\mathrm{Pic}^0_{\overline {M  }_k/k})^{ab}$$ has a finite kernel  because
   $\widetilde f_k:    (\widetilde {\overline M_k})_{\mathrm{red}}\to {( \widetilde  {M_1(4p^m)}_k)}_{\mathrm{red}}$ is surjective.
 \end{proof}

\subsection{An explicit calculation}\label{cal}

The morphisms $ ({{\mathcal C}_{A^0_1}})_k\to ({{\mathcal C}_{A^0_0}})_k$ and $ ({{\mathcal C}_{A^0_m}})_k\to ({{\mathcal C}_{A^0_1}})_k$ give  us extensions  $k(\sigma)\subset\Sigma_{A^0_1} $ and
$\Sigma_{A^0_1}\subset\Sigma_{A^0_m} $, respectively, which  are
  abelian extensions of groups $({\Bbb Z}/p )^\times$ and $ {\Bbb Z}/p^{m-1}  $, respectively . One can
obtain the generator of the extension
$k(\sigma)\subset\Sigma_{A^0_1} $,  because it is a totally ramified
extension and  its ramification is given by the polynomial
$p(\sigma)$, (c.f.\cite{I}). Thus, $\Sigma_{A^0_1}=k(\sigma,
p(\sigma)^{1/p-1})$.

According to  \cite{K}, Lemma 2.5   and (5.5),  together with  the following two Lemmas,  allow us to make
explicit calculations for the  Artin-Schreier generators for  the
   extensions
 $\Sigma_{A^0_1} \subset\Sigma_{A^0_m}.$
We denote ${\mathcal P}(a)=a^{p }-a$.
\begin{lem}   The extension $k((t))({\mathcal P}^{-1}( 1/t^n))/k((t))$ is totally ramified over $t=0$.
\end{lem}
\begin{proof} An element $1/x\in {\mathcal P}^{-1}( 1/t)$ satisfies     $x^p+x^{p-1}\cdot t-t=0$. This equation   is a non-singular curve   at the point $(0,0)$.
The Lemma is proved for $n=1$   by considering the support of the differential $k[[t]][x]$-module
$$\Omega_{(k[[t]][x]/x^p+x^{p-1}\cdot t-t)/k[[t]]}.$$
Bearing in mind  the totally ramified extension $k((t^n))\to k((t))$ and $k((t))( {\mathcal P}^{-1}(1/t^n))=k((t^n))( {\mathcal P}^{-1}(1/t^n))\otimes_{k((t^n))}k((t))$, one proves the Lemma  for any $n$.
\end{proof}

\begin{lem} If $k(t, {\mathcal P}^{-1}( r(t)/s(t)))/k(t)$ is an extension such that its ideal of ramification divides the polynomial  $p(t)$, then there exists $l\in {\Bbb N}$ with $r(t)/s(t)=h(t)/p(t)^l+{\mathcal P}(u(t)/v(t))$ and $\mathrm{deg}(h(t))\leq  \mathrm{deg}(p(t)^l)$.
\end{lem}
\begin{proof}  This is deduced from the above Lemma   bearing in mind  the decomposition in simple fractions   of $r(t)/s(t)$ and the fact that ${\mathcal P}$ is additive.
\end{proof}
By the previous Lemma to make explicit calculations   of the Artin-Schreier generators of $\Sigma_{A^0_1}\subset\Sigma_{A^0_m} $ it suffices to calculate $l$ and $h(t)$ for these extensions; $p(t)$ is the polynomial $p(\sigma)$ defined in section \ref{an}.

As example we shall make this calculation precise for $m=2$.  Following the notation of \cite{K} {\S}2 and \cite{Se}  2.2,  the Artin-Shreier generator for $\Sigma_{A_1}\subset
\Sigma_{A_2}$ is the reduction   $\mathrm{mod}\, p$  of
$\psi:=\beta(1)(\frac{a_p-1}{p })=\frac{a_p-1}{p\cdot E_{p-1} }\in   {\Bbb
F}_p((q))$, where $a_p$  is a modular form of weight $p-1$. By
\cite{Se}, 2.2, we have that
$$(\psi)^p-\psi= -\frac{1}{24}\cdot
\beta(1)(\theta^{p-2}(E_{p+1}))=\frac{\theta^{p-2}(E_{p+1})}{E_{p-1}^{3}}\quad (\mathrm{mod}\, p),$$
 with $\theta=q \frac{d}{dq}$ and $\theta^{p-2}(E_{p+1})$    a modular form of weight
 $ 3(p-1)$.

Here, $E_{p+1}$ and $E_{p-1}$ denote  the normalization,  (constant
term = $1$), of the Eisenstein series of weights $p+1$ and $p-1$,
respectively.

The algebra of modular functions is given by the graded ring of
polynomials  ${\Bbb Z}[g_2, g_3]$, where $g_2$ and $g_3$ have grades
 $4$ and $6$, respectively. If
$A(g_2,g_3)(\frac{dx}{2y})^{\otimes p-1}$, ($A(g_2,g_3)\in  {\Bbb
F}_p[g_2, g_3]$), is  the Hasse invariant
 of the universal
elliptic curve in the Weierstrass form  $y^2=4x^3-g_2x-g_3$  then
the class of $E_{p-1}$ in ${\Bbb F}_p[g_2, g_3]$ is $A(g_2,g_3)$.

Since the Deuring polynomial gives the Hasse invariant for the
universal elliptic curve in the Legendre form
$y^2=x(x-1)(x-\lambda)$, we have that
 $\Sigma_{A_2}=\Sigma_{A_1}[\psi]$
with
 $\psi^p-\psi-\frac{h(\sigma)}{p(\sigma)^3}.  $ Here,
$p(\sigma)=\sigma^{p-1}\cdot H(\lambda)$ with $\lambda=(\sigma+1/2\sigma)^2$.

Moreover, because
$\frac{\theta^{p-2}(E_{p+1})}{E_{p-1}^{3}}$ only has poles over
the zeroes of $E_{p-1}(z)dz$ we have that $\mathrm{deg}(h(\sigma))\leq \mathrm{deg}(p(\sigma)^3)(= 3(p-1))$. To calculate  $h(\sigma)$   it suffices to
bear  in mind the expansions    on the parameter  $q^{1/4}$  of
$\sigma$, $\frac{\theta^{p-2}(E_{p+1})}{E_{p-1}^{3}}$  and the
congruence
$$  \frac{\theta^{p-2}(E_{p+1})}{E_{p-1}^{3}} \equiv
 \frac{h(\sigma)}{p(\sigma)^3} \quad (\mathrm{mod}\, \mathfrak p)  .$$

Bearing in mind this latter calculation  and the action of  $(k[\sigma]/p(\sigma)^3
)^\times $ on $\psi$, given by
the class field theory, (c.f. \cite{Ha} Chapter 9),    one can explicitly obtain the
group homomorphism
$$\rho:(k[\sigma]/p(\sigma)^3 )^\times  \to  ({\Bbb
Z}/p^2)^\times.$$

\subsection{A way to calculate the $p$-term   of the Hasse-Weil $L$-function of $M_1(4p^m)$}

To obtain the $p$-term in the Euler product of the Hasse-Weil $L$-function of $M_1(4p^m)$ it suffices to obtain the Euler zeta function of the curve $ ({{\mathcal C}_{A^0_m}})_k$.

 We, now, consider the incomplete $L$-function associated with the Galois extension $K_{q(\sigma)}/k(\sigma)$:
$$\prod_{  x\in
\vert {\Bbb P}^1(k)\vert\setminus T} (1-F_x\cdot t^{deg(x)})^{-1}=\frac{U(t)}{1-p^{\delta}\cdot t},$$
(C.f \cite{An} 1.3), where $U(t)$ is an explicit polynomial in $  {\Bbb Z}[G][t]$; $G= (k
[\sigma]/p(\sigma))^\times $; $F_x$ is the Frobenius element for $x\in
\vert {\Bbb P}^1(k)\vert\setminus T$,   $T$ being the places in $\vert {\Bbb P}^1(k)\vert$ given by the divisors of $q(\sigma){\mathfrak m}_\infty$.

We define $\text{Norm}_{{\Bbb Z}[G]/{\Bbb Z}[H]}( U(t) )$   as the determinant of the  morphism of multiplication by $U(t)$ in the ${\Bbb Z}[t][H] $-module ${\Bbb Z}[t][G] $. $H$ being the kernel of the group morphism
$$\rho:(k[\sigma]/q(\sigma))^\times  \to  ({\Bbb
Z}/p^m)^\times$$
given by the Galois extensions $k(\sigma)\subset \Sigma_{A^0_m}\subset K_{q(\sigma)}$.

Let
$\pi:({{\mathcal C}_{A^0_m}})_k\to {\Bbb P}^1(k)$
be the morphism of smooth curves given by   $k(\sigma)\subset \Sigma_{A^0_m}$.
Bearing in mind that
 $$\text{Norm}_{{\Bbb Z}[G]/{\Bbb Z}[H]} (1-F_x\cdot t^{deg(x)}) =\prod_{ z\in
  \pi^{-1}(x)} (1-F_z\cdot t^{deg(z)})^{-1},$$
we can calculate the  incomplete  $L$-function
$$\prod_{ z\in
\vert ({{\mathcal C}_{A^0_m}})_k\vert\setminus \pi^{-1}(T)} (1-F_z\cdot t^{deg(z)})^{-1}$$
as a faction of polynomials $\frac{V(t)}{W(t)},$
where
$$\frac{V(t)}{W(t)}=\frac{\text{Norm}_{{\Bbb Z}[G]/{\Bbb Z}[H]}( U(t) )}{\text{Norm}_{{\Bbb Z}[G]/{\Bbb Z}[H]}(1-p^{\delta}\cdot t)}.$$

By making $h=1$ for each $h\in H$ in the quotient $\frac{V(t)}{W(t)}$, we obtain  a fraction of polynomials   $\frac{v(t)}{w(t)}$. The Euler zeta function of $ ({{\mathcal C}_{A^0_m}})_k$ is
$$\frac{v(t)}{w(t)}\cdot \prod_{  z\in
 \pi^{-1}(T)} (1-  t^{deg(z)})^{-1} .$$
By taking account the above section we can obtain $H$ in an explicit way  and we can calculate $\frac{v(t)}{w(t)}$.
Moreover, we obtain the finite product $\prod_{  z\in
 \pi^{-1}(T) } (1-  t^{deg(z)})^{-1}  $ by considering   $\pi^{-1}(\infty)$ and the roots of
$q(\sigma)$, which are given by the Deuring polynomial $H(\lambda)$.

\section{The cuspidal principal part ($\mathrm{mod} \,p$) of certain meromorphic modular functions}

\subsection{Line bundles from an adelic point of view}
 Let  $Y$  be a smooth, projective  and geometrically irreducible curve over a field $k$ of $q=p^\delta$ elements; $\Sigma$ denotes its function field. Let $\mathfrak q$ be a place of
$\Sigma$,  $t_\mathfrak q$ a local parameter for $\mathfrak
q$  and $k( \mathfrak q)$ its residual field. For each $k$-algebra $R$, we consider the group
 $(R\otimes k(\mathfrak q))((t_\mathfrak q)) :=(R\otimes k(\mathfrak q))[[t_\mathfrak q]][t_\mathfrak q^{-1}]   $
and the adele group
$${\Bbb A}_\Sigma(R):=\prod'_{\mathfrak q\in
\vert Y\vert} (R\otimes
 k(\mathfrak q))((t_\mathfrak q)),$$
where $\prod'$ refers to  the adeles with a non-trivial principal part
  only on a finite number of places of $\Sigma$. The idele group  $I_\Sigma(R)$  is the group of units of
${\Bbb A}_\Sigma(R)$. Thus,
 $$I_\Sigma(R) =(\prod'_{\mathfrak q\in \vert Y\vert} (R\otimes
 k(\mathfrak q))((t_\mathfrak q)))^\times.$$
Let $U_\Sigma(R)$ be the subgroup
 $ \prod_{\mathfrak q\in \vert Y\vert}(R\otimes k(\mathfrak q))[[t_\mathfrak q]]^\times .$
Let $\Sigma\otimes_k R$ be denoted by $\Sigma_R$. One can
embed  the group  $(\Sigma_R)^\times$  diagonally into $I_\Sigma(R)$.

 The quotient group
$$\frac{I_\Sigma(R)}{\Sigma^\times_R\cdot U_\Sigma(R)} $$
is isomorphic to the class group of line bundles $L$ on $Y\otimes
R$, such that if
$j: \mathrm{Spec}(\Sigma_R) \hookrightarrow Y_R$
is the natural inclusion  then the pullback  $j^*L$  is isomorphic
to the trivial line bundle. See, for example,  \cite{BL}, Section 2, and Lemma 3.4.

If we denote by  $J_Y$  the Jacobian variety of $Y$ and $\mu$ is an idele of degree $1$, we have a
monomorphism of groups
$$\frac{I_\Sigma(R)}{ \mu^{\Bbb Z}\cdot \Sigma^\times_R\cdot U_\Sigma(R)}\hookrightarrow
J_Y(R).$$

This monomorphism is an isomorphism if and only if $R$ is a finite
${k}$-algebra. Recall that $J_Y(R)$  is the group of isomorphism classes of line
bundles over $Y_R$ of degree $0$.

 In the following Lemma  we recall that the tangent
space  of the Jacobian on the zero element  can be identified with
the cohomology group $H^1(Y,\o_Y)$ and by duality with
$H^0(Y,\omega)^\vee$  ($\omega$ is the dualizating sheaf of $Y$):
 \begin{lem}\label{ep} We have an exact sequence of groups
 $$0\to H^1(Y,\o_Y) \to J_Y(k[\epsilon ]) \overset{\epsilon =0}\to J_Y(k
 )\to 0,\quad (\epsilon^2=0).$$
 \end{lem}
\begin{proof} It suffices to take into account that $k[\epsilon]$  is a finite $k$-algebra   and that
$$J_Y(k[\epsilon])\simeq \frac{I_\Sigma(k[\epsilon])}{\mu^{\Bbb Z} \cdot\Sigma^\times_{k[\epsilon]}\cdot U_\Sigma({\Bbb
F}_q[\epsilon])}=\frac{I_\Sigma  }{\mu^{\Bbb Z} \cdot\Sigma^\times \cdot
U_\Sigma}+\epsilon \cdot \frac{\Bbb A}{\Sigma +O_\Sigma},$$
where $O_\Sigma$ denotes  the integer adeles.
\end{proof}

\begin{rem}\label{mu}
Let $D$ be a Cartier divisor given by the class of an
idele
$$  \mu_0+\mu_1\epsilon\in \frac{I_\Sigma(k[\epsilon])}{  U_\Sigma(k[\epsilon])}=\frac{I_\Sigma
}{  U_\Sigma}+\epsilon \cdot \frac{\Bbb A}{ O_\Sigma}.$$
  According to  Lemma \ref{Fr}, $    \Gamma(\mathrm{Fr})(D) $ and $^t\Gamma(\mathrm{Fr})(D) $ are
given by
  $\mu_0+\mu_1\epsilon^q$ and $\mu^q_0+\mu_1^q\epsilon $, respectively. Note
 that
$ \epsilon^{q }=0$.
\end{rem}

\subsection{Correspondences in an additive setting}

We now prove a Lemma that will be used to make explicit calculations in the next section.
 Let $\mathrm{Spec}(B)\subset Y$  be an  open subscheme     and let  $C$ be a correspondence on $Y$ that is
   trivial  over $\mathrm{Spec}(B)\otimes \mathrm{Spec}(
  B)$. We set that  $C$  over $\mathrm{Spec}(B\otimes   B)$ will be  given by
the zero locus of a regular function $H(b,\bar b) \in B\otimes   B$. Let
$\mathfrak q\in \mathrm{Spec}(B) $ be a geometric point
and $t_\mathfrak q$   a local parameter for $\mathfrak q$. For
easy notation, we can assume $\mathfrak q$ to be rational. We
denote by $D^{-r\cdot \mathfrak q}$, $D_{ r\cdot \mathfrak q}$ the
Cartier divisors on $Y\otimes k[\epsilon]$ given by the line bundles associated with the
ideles of $I_\Sigma(k[\epsilon])$, whose entries are  $1-\epsilon
t_\mathfrak q^{-r}$  and $ t_\mathfrak q^{ r}-\epsilon $ at $\mathfrak q$ and $1$ elsewhere,
respectively.

\begin{lem}\label{re} With the   notations and conventions of section \ref{conv}, $C(D^{-r\cdot \mathfrak
q})$  is a Cartier divisor such that over the open subset  $
\mathrm{Spec}(B[\epsilon])\subset Y[\epsilon]$ is given by
 $1+\epsilon \mathrm{Res}_\mathfrak q(t_\mathfrak q^{-r} {\mathrm{d}}\,
\mathrm{log} \,H(t_\mathfrak q,\bar b))\in 1+\epsilon\cdot \Sigma.$

 To calculate this residue,
one considers   $H(b,\bar b)= H(t_{\mathfrak q},\bar b) \in B ((t_\mathfrak q))$ and $ \mathrm{d}
\,
\mathrm{log}\,H(t_\mathfrak q,\bar b)= \frac{\mathrm{d} \,
\mathrm{log} \,H(t_\mathfrak q,\bar b)}{\mathrm{d}\, t_\mathfrak q} \mathrm{d}\, t_\mathfrak q$.
\end{lem}
\begin{proof} We have that  $C(D_{ r\cdot \mathfrak q})$  over
$\mathrm{Spec}(B)$ is given by the kernel ideal of the $B$-algebra
morphism:
$$\phi:B[\epsilon]\to B[\epsilon, t_\mathfrak q]/
 (t_\mathfrak q^{ r}-\epsilon,H(t_\mathfrak q,\bar b) ), \quad \phi(\epsilon)=\epsilon. $$
Since $t_\mathfrak q^{ 2r}=0$ as an element of $$B[\epsilon,
t_\mathfrak q]/
 (t_\mathfrak q^{ r}-\epsilon,H(t_\mathfrak q,\bar b) ),$$ one can assume  that $H(t_\mathfrak q,\bar b)
 =u(t_\mathfrak q)=b_1t_\mathfrak q^{2r-1}+\cdots+b_{2r-1}t_\mathfrak q +b_{2r}$, with
 $b_i\in B$.

   The ideal  $\mathrm{Ker } ( \phi)$ is generated by  $\prod_i
 (\alpha_i^r-\epsilon)$; this product is taken  over the roots of
 $u(t_\mathfrak q)$.

 Because   $C(D^{ -r\cdot \mathfrak q})=C(-r\cdot \mathfrak q)C(D_{ r\cdot \mathfrak
 q})
  $   and because the Cartier divisor  $   C(-r\cdot \mathfrak q) $  is given over $\mathrm{Spec}(B)$ by the element $  \prod_i
  \alpha_i^{-r } $, we have that the Cartier divisor defined over $\mathrm{Spec}(B[\epsilon])$,    $C(D^{ -r\cdot \mathfrak q}) =   C(t_\mathfrak q^{ -r}) C(t_\mathfrak q^{ r}-\epsilon
  ) $  is given by the  element
$   \prod_i
 (1-\epsilon\alpha_i^{-r})=
  1-\epsilon(\sum_i\alpha_i^{-r}) .$

 By using the residues theorem for $1$-differential forms of $\Sigma(t_\mathfrak
 q)$, where the field of constants is $\Sigma$,
 we conclude  because
 $$ \mathrm{Res}_{ \mathfrak q }(t_\mathfrak q^{-r}\mathrm{d} \,
\mathrm{log}\,H(t_\mathfrak q,\bar b))=-\sum_y  \mathrm{Res}_y(t_{\mathfrak q}^{-r}\mathrm{d} \,
\mathrm{log}\,H(t_\mathfrak q,\bar b))=\sum_i
  \alpha_i^{-r}.$$
 The second sum is taken over the maximal ideals  $y$  of $\Sigma[t_\mathfrak q^{-1}]$.
  \end{proof}

\begin{rem}\label{to} It is not hard to prove that if $a\in B((t_\mathfrak q))$ with either $a\in
B$ or $a\in k((t_\mathfrak q))$ then
 $ \mathrm{Res}_{ \mathfrak q }(t_\mathfrak q^{-r}\mathrm{d} \,
\mathrm{log}\,(aH(t_\mathfrak q,\bar b)))= \mathrm{Res}_{ \mathfrak
q }(t_\mathfrak q^{-r}\mathrm{d} \,
\mathrm{log}\,H(t_\mathfrak q,\bar b))+\alpha,$  for  some $\alpha \in
k$. Here, $aH(t_\mathfrak q,\bar b)\in B((t_\mathfrak q))$.

From   Lemma \ref{re}   and the above Remark, to  calculate
 $C(D^{ -r\cdot \mathfrak q})$  as a Cartier divisor on  $Y_R$,
 one can forget the vertical and horizontal components of $C$.
Thus,  this Cartier divisor is given,  up
to  units of $k[\epsilon] $, by
 $1+\epsilon \cdot  \mathrm{Res}_{ \mathfrak q }(t_\mathfrak q^{-r} \mathrm{d}\,
\mathrm{log}\,H(t_\mathfrak q,\bar b)). $
\end{rem}

\subsection{The principal part  ($\mathrm{mod} \,p$) for   certain meromorphic modular functions of level $4p^m$ with poles over the cusps }

By \cite{DR} VI,  2.3.1,  we have a section
 $s_n:\mathrm{Spec}({\Bbb Z}[\zeta_n])\to {  M}(n) $. The completion of
 ${ M}(n) $ along $s_n$  is identified with
 $\mathrm{Spec}({\Bbb
Z}[\zeta_n][[q^{1/n}]])\to { M}(n).  $ This morphism can be
obtained from the Tate elliptic curve with an $n$-level structure.
The completion along the cusps of  ${  M}(n)$  is a finite
number   of copies of $\mathrm{Spec}({\Bbb Z}[\zeta_n][[q^{1/n} ]])$. From
this, one deduces that the abelian   ramified covering
 $( {\mathcal C}_{A^0_m})_k\to
M(4)_k$  splits completely over the cusps
$M(4)_k\setminus M^0(4)_k$.

By tensoring $s_{4p^m}$ by  $\otimes_{R_{\mathfrak p}} k$,
one
obtains a section
 $  (s_{4p^m})_k:\mathrm{Spec}(k)\to   M (4p^m)_k   $.
We denote by  $\infty$   the geometric point of $
{  M}(4p^m)_k$ given by $( s_{4p^m})_k$ and we denote by $
\overline{q}$ the local parameter $q^{1/4p^m}$ ($\mathrm{mod} \, \mathfrak
p$).

 Recall that    $
M(4p^m)_k=\cup^{l-1}_{i=0} ({\mathcal C}_{A^i_m})_k.$ We denote by  $B$
the ring such that $\mathrm{Spec}(B)={  M}^0(4p^m)$.

Bearing in mind \cite{DR}  V, 4.19 and 4.20, we have an
isomorphism
 $$   \phi:  B_k [p(\sigma)^{-1}] \simeq (H_{A^0_m})_{S_0}\times \cdots \times (H_{A^{l-1}_m})_{S_{l-1}},$$
where  we denote    $\mathrm{Spec}(H_{A^i_m})=({\mathcal C}^0_{A^i_m})_k$ and $  S_0, \cdots, S_{l-1}$  are multiplicative systems  given
by the powers of the image of the polynomial $p(\sigma) $ in
  $H_{A^0_m}, \cdots , H_{A^{l-1}_m}$, respectively.
\begin{thm}\label{prin} For each $m\in {\Bbb N}$, there exists an element of $ B_k [p(\sigma)^{-1}]$
  such that its principal part  is
$$\sum_{i=0}^{d-1}
(\underset { \mathrm{deg}(r(\sigma))=d-1- i}  {\sum_{r(\sigma)\text{(monic)} }}
  \frac{1}{ ({\overline
q^{m p^{i\delta}})^{R^{-1}_{r(\sigma)}}} }) .$$
\end{thm}
\begin{proof} We assume that $\infty \in ({\mathcal C}_{A^j_m})_k$ and we denote by $\Sigma_{A^j_m}$ its fraction field and
by $\mu_m $ the idele whose entries are $1+\epsilon \overline
q^{-m} $ at $ \infty$ and $1$ elsewhere.
 Following the notation of section 5.2,  let $D^{-m\infty}$ be the Cartier divisor associated with  $\mu_m $.
 Now, if we consider the transpose of the trivial correspondence $\mathrm{deg}(\pi)^{-1}(D_k)$ of the Remark \ref{tri} then
  $\mathrm{deg}(\pi)^{-1}(^tD_k)(D^{-m\infty}) $ is the idele on $k[\epsilon]$
  $$  \prod_{i=0}^{d-1}
(\underset { \mathrm{deg}(r(\sigma))=d-1- i}  {\prod_{r(\sigma)\text{(monic)}} }
  \frac{1}{ ({\mu^{m{p^{i\delta}}} )^{R^{-1}_{r(\sigma)}}} })\in 1+\epsilon \frac{\Bbb A}{ O_{\Sigma_{A^j_m}}}.$$

Bearing in mind   Lemmas \ref{ep} and \ref{re},  we obtain an element $b_m\in (H_{A^j_m})_{S_{j}}$ such that
$1+\epsilon b_m\in 1+\epsilon \frac{\Bbb A}{ O_{\Sigma_{A^j_m}}} $ is the above idele. We finish by considering the element  $\phi^{-1}(0,\cdots,b_m,\cdots,0)\in B_k [p(\sigma)^{-1}] $.
\end{proof}

{\bf{Acknowlegments}} The author wishes to thank a referee by the hard work offered in improving the original article.
I also would like to express my gratitude to  Ricardo Perez Marco. I  am very grateful to Ricardo Alonso and  Jesus Mu\~noz-Diaz for their encouragement and help.

\vskip2truecm


TITLE: Is it possible to solve a linear inequality system using SVD?
QUESTION [2 upvotes]: I have a large linear inequality system of the form $Ax ≤ 0$.
Is there a way to solve this system using linear algebra tools like SVD?

REPLY [2 votes]: No; loosely speaking this problem is more related to linear programming, which is not equal to linear algebra.
(Plugging in an SVD of $A$ will be too helpful as you probably have seen.)
The set $Ax\leq 0$ is a convex polytope and the representation $Ax\leq 0$ is called its H-representation because it is a description as intersection of half-spaces. Another representation is its V-representation which consists of a list of the vertices. The Wikipedia entry on convex polytopes has more pointers.


TITLE: Congruence Inconsistency
QUESTION [0 upvotes]: I have a question about congruency...
I understand that:
$$
12 \equiv 7 \bmod  5
$$
$$
\text {is equivalent to:}
$$
$$
5|12-7
$$
but this doesn't seem to hold for:
$$
2 \equiv 8 \bmod 6
$$
$$
\text {the conclusion i come to is:}
$$
$$
6|-6
$$
are these equivalent??

REPLY [1 votes]: Yes, $6\mid-6$. That simply means that there is some integer $k$ such that $-6=6k$, and there is: take $k=-1$. There is no requirement that the integer be positive.


TITLE: vector bundles on affine schemes
QUESTION [17 upvotes]: Serre's theorem (one of them) states that for a quasi-coherent sheaf $\mathscr F$ on an affine noetherian scheme $H^i(X,\mathscr{F})$ vanish for $i >0$. I used to think that this would imply that on an affine variety there cannot be non-trivial vector bundles, because such a bundle would define a non-trivial cocycle in $\check{H}^1(X, GL_n)$, and this is isomorphic to the derived functor cohomology $H^1(X, GL_n)$ for "good enough" schemes, and the latter vanishes by Serre's theorem.
But  here they give examples of non-trivial line bundles on affine varieties.
Where is my reasoning fallacious? Is it because the trivialising opens for the vector bundle can be non-affine?
Does it mean that in general there is no simple way to classify algebraic vector bundles even on an affine variety?

REPLY [10 votes]: $GL_n$ is not a coherent sheaf, because it is not a sheaf of $\mathcal{O}_X$-modules.


TITLE: A simple example of symmetry setting the properties of a Physical System
QUESTION [0 upvotes]: Does anybody know of an example were one could derive some important properties of a physical system from a symmetry of said system.
I´m specially looking for simple classical examples, which could serve to illustrate the importance of finding symmetries of a system to non-physicists (high school students or first year undergrads)

REPLY [1 votes]: My favourite was always angular momentum conservation of planetary orbits. The gravitational potential energy $$V(r) = - G\frac{M m}{r}$$ depends only on the radial distance $r$ and not the angle, i.e. it is rotationally symmetric. Therefore, the angular momentum of a planet orbiting the Sun is conserved. This all comes out rather beautifully from the Lagrangian formulation of classical mechanics, but for high school/first year students you might have to work a little harder to prove it. Of course, this is just a specific example of the Noether theorem mentioned by Qmechanic.


text:
\section{Theoretical results}

\subsection*{Outline of results}

\begin{enumerate}
  \item From previous papers: under broad conditions, we get pointwise convergence of our log-target or log-likelihood estimator, as long as we do not wander off to infinity.
  \begin{itemize}
    \item We do still need random refinement since \textcolor{red!90!black}{since we cannot rule out all of the points we happen to have sampled looking like near-perfect fits even when they are not.}
    \item[] \textcolor{red!90!black}{What would it mean for the points to look like ``near-perfect fits''?  The error indicator is not aware of any structure of the density itself.  It only ``knows'' the ball size.  Is it not enough to for the radius to go to zero everywhere?  I suppose to be rigorous, the ball size goes to zero in some region under the ``Lyapunov umbrella'' (not everywhere in state space).  Perhaps this where I'm confused.} AS: Oops - I was being dumb. Yes, this should be fine. See [ZZX].
  \end{itemize}
    \item If the original kernel $K$ satisfied a Lyapunov function, then \textit{all} of our approximate kernels $\hat{P}$ will satisfy the \textit{same} Lyapunov inequality.  If we only know good level sets for the Lyapunov function outside of some compact set $k$, then we will get the \textit{original} Lyapunov function outside of $K$.
    \item Consequence of 1 and 2: If the original kernel $K$ satisfied a \textbf{minorization condition} on a compact set, then the new kernel will eventually satisfy a minorization condition on the same compact set. The minorizing measure will be the same, but the minorizing consant will be slightly worse. Remark: We really do need (2) here, as the result is otherwise false in general. The issue is that the original minorization inequality talks about short walks started from anywhere within a small set $C$. These short walks might exit $C$, and so we need to be sure that (with very high probability) they stay within a slightly-larger compact set $C" \supset C$ over their short walk. The Lyapunov condition (2) will guarantee this. Of course, instead one could simply restrict to one-step minorization bounds, where this is free.
    \item Immediate consequence of 1,2,3: Under these conditions, the approximate kernel will eventually have a unique stationary measure. Furthermore, this stationary measure will give very little mass to the ``tails'' as measured by the Lyapunov function.
    \item Immediate consequence of 3: If the original kernel $K$ is ``nice'' in the above sense, then the new kernel will eventually have a CLT, concentration of measure, etc etc. Furthermore, the constants appearing in those bounds will not be ``too much worse'' than the original constants. Thus, the error of a ``large'' MCMC sample will be bounded by something like $\mbox{[(Relaxation time of original kernel]}*\mbox{[some small multiplicative factor, e.g. 10]}/\sqrt{\mbox{[running time]}} + \mbox{[error due to bias]}$. Furthermore, as a consequence of 4, this ``error due to bias'' will not have a large term coming from the tails. In particular, the heuristic bias calculations in Andy's Ph.D. should be rigorous up to (i) errors in the occupation measure of the tails (which are controllable by the Lyapunov function + Markov's inequality), and (ii) the usual fact that bounds you get from concentration-of-measure have an extra $\log{n}$-or-so factor relative to bounds you get from the CLT (details of this ``$\log{n}$'' will depend on e.g. the Lyapunov function).
    \item Consequence: since the error due to bias goes to $0$ more quickly than $1/\sqrt{\mbox{[number of model evaluations]}}$ in many situations, we should get the desired faster-than-sqrt(something) convergence.
\end{enumerate}

\textbf{Some small concern:} One of the original motivations was to concentrate samples in the ``real'' mode (which might look e.g. like a kidney bean). The above theory can do this if you have a good guess about the Lyapunov function, but otherwise it more-or-less requires a certain amount of infill for the entire small set of the Lyapunov function (which might look e.g. like a large ball containing the original kidney bean). On the one hand, it is nice that the theory doesn't require infill if you have a good guess; on the other hand, we probably don't have a good a-priori guess. On the gripping hand, in practice we expect that in fact we should get a good Lyapunov function for the kidney bean around the same time that we get one for the surrounding ball, so in fact the theory claims here should more-or-less line up with the practice, even if there are some calculations in the middle that are very pessimistic.

\subsection{Geometric ergodicity of approximate chain}

\begin{assumption}
  Given transition kernels $Q$ and $K$, the Lyapunov condition states that for $L \in \{Q, K\}$, $$(LV)(x) = \int V(y) L(x, dy) \leq (1-a)V(x)+b,$$ where $V \geq 0$ is a function and $a>0$, $b<\infty$ are constants.
  \label{assumption:lyapunov-condition}
\end{assumption}

 Aside: the sets $C$ and $C^{\prime}$ are compact sets that contain almost all of the mass of the target. Thus, informally we can think of $C$ as being the ``mode'' and $C^{c}$ as being the ``tail.''

\begin{assumption}
  Minorization condition: Let $C = \{x \,:\, V(x) \leq 4b/a\}$.  For all $x \in C$, $$K^{s}(x,\cdot) \geq \epsilon \pi(\cdot)$$ for some $\epsilon > 0$, some $s \in \mathbb{N}$, and some measure $\pi$.
  \label{assumption:minorization-condition}
\end{assumption}

\begin{assumption}
  Approximation Condition: Let $C^{\prime} = \{x \,:\, V(x) \leq 20 b s/(\epsilon a) \}$. For all $x \in C^{\prime}$, $$\|K(x,\cdot) - Q(x,\cdot)\|_{\mathrm{TV}} \leq \epsilon/2.$$
  \label{assumption:approximation-condition}
\end{assumption}

\begin{lemma}
Under assumptions \ref{assumption:lyapunov-condition}, \ref{assumption:minorization-condition}, and \ref{assumption:approximation-condition}, for all $x \in C$, $Q^{s}(x,\cdot) \geq \epsilon \pi(\cdot) / 4.$
\end{lemma}

\begin{proof}
For $x \in C$, let $\{X_{t}\} \sim Q$ with starting point $X_{0} = x$. Appling \ref{assumption:lyapunov-condition} repeatedly, $$\mathbb{E}[V(X_{t})] \leq (1-a)^{t} V(x) + b/a \leq 5b/a.$$  Thus, by Markov's inequality,

\begin{equation}
  Q^{t}(x,C') \geq 1 - \epsilon/4s
  \label{eq:EscapeProbabilityInequality}
\end{equation}

\noindent for all $t \in \mathbb{N}$. We then note, for $x \in C$,

\begin{eqnarray}
  \| K^{s}(x,\cdot) - Q^{s}(x,\cdot) \|_{\mathrm{TV}} &\leq& \sum_{t=0}^{s-1} (1 - Q^{t}(x,C')) + \sum_{t=0}^{s-1} \sup_{x \in C'} \| Q(x,\cdot) - K(x,\cdot) \|_{\mathrm{TV}} \\
  &\leq& \frac{\epsilon}{4} + \frac{\epsilon}{2},
\end{eqnarray}

\noindent where in the last line we used \eqref{eq:EscapeProbabilityInequality} to bound the first term and assumption \ref{assumption:approximation-condition} to bound the second.  Combining this with assumption \ref{assumption:minorization-condition} completes the proof.
\end{proof}

\subsection*{Technical Lemmas and Proof Sketch}

\subsubsection*{Comparing Densities}
First, we check that good pointwise estimates of the \textit{unnormalized} log-likelihood will lead to good pointwise estimates of the \textit{normalized} probability density:

\begin{lemma}
Let $p_1 = Z_1^{-1} f_1$, $p_2 = Z_2^{-1} f_2$ with $Z_{1}, Z_{2} \in (0, \infty)$ be two probability measures on the same state space. Then we have the pointwise inequality $$\vert p_1 - p_2 \vert \leq p_1 \frac{\vert Z_2 - Z_1 \vert }{Z_{2}} + \frac{|f_1 - f_2|}{Z_2}.$$
\label{lem:LemmaCloseNorm}
\end{lemma}
\begin{proof}
One has the pointwise identity\todo{was $Z_{2}$} $$p_1 - p_2 = \frac{1}{Z_{2}} \left( f_1 \frac{Z_{2} - Z_{1}}{\textcolor{red!90!black}{Z_{1}}} + f_{1} - f_{2} \right) = p_1 \frac{Z_{2} - Z_{1}}{Z_{2}} + \frac{f_{1} - f_{2}}{Z_{2}}.$$

The result follows from the triangle inequality.
\end{proof}

This gives the harder-to-state but easier-to-use corollary:

\begin{lemma}
Fix $f_{1} \,:\, \mathbb{R}^{d} \mapsto [0,\infty)$ with $Z_{1} \equiv \int f_{1}(x) dx < \infty$ and $\sup_{x} f_{1}(x) \equiv M < \infty$ and define $p_{1} = Z_{1}^{-1} f_{1}$. Assume there exists a compact set $C \subset \mathbb{R}^{d}$ for which \todo{was $C$}

\begin{equation}
\int_{\textcolor{red!90!black}{C^{c}}} p_{1}(x) dx < \epsilon,
\label{eq:CloseEst1}
\end{equation}

\noindent Then there exists $\delta = \delta(Z_{1}, M, \vol{(C)}) > 0$ \todo[color=red!25]{Why do we need to define $\delta$?} and $A = A(Z_{1},M, \vol{(C)}) < \infty$ so that for all $f_{2} \,:\, \mathbb{R}^{d} \mapsto [0,\infty)$ (and corresponding $Z_{2} = \int f_{2}(x) dx < \infty$, $p_{2} = Z_{2}^{-1} f_{2}$) satisfying:

\begin{enumerate}
\item for the same compact set $C$, \todo{was $C$} $$\int_{\textcolor{red!90!black}{C^{c}}} p_{2}(x) dx < \epsilon,$$
\item and
\begin{equation}
\sup_{x \in C} \vert f_{1}(x) - f_{2}(x) \vert < \epsilon,
\label{eq:CloseEst2}
\end{equation}
\end{enumerate}

\todo[inline,color=red!25]{Concern: in the local approximation algorithm we do satisfy (2) in $C$.  However, it is \emph{not} true that $\int f_{2}(x) dx < \infty$.  In fact, the opposite is almost definately true: $\int f_{2}(x) dx = \infty$.  We could tweak this by satisfying $Z_{2} = \int_{C} f_{2}(x) dx < \infty$ and $p_{2} = Z_{2}^{-1} f_{2}$ for $x \in C$ and $p_2 = 0$ otherwise.  This implies that $\int_{C^{c}} p_2(x) dx = 0$.}

\noindent we have $$\| p_{1} - p_{2} \|_{\mathrm{TV}} \leq A \epsilon.$$
\end{lemma}

\begin{proof}
By inequalities \eqref{eq:CloseEst1} and \eqref{eq:CloseEst2},

\begin{align*}
\vert Z_{1} - Z_{2} \vert &\leq \int_{C} \vert f_{1}(x) - f_{2}(x) \vert dx + \int_{C^{c}} \vert f_{1}(x) - f_{2}(x) \vert dx \\
&\leq \int_{C} \epsilon dx + \int_{C^{c}} f_{1}(x) dx + \int_{C^{c}} f_{2}(x) dx \\
&\leq \epsilon(\vol{(C)} + 2).
\end{align*}

\noindent Thus, for all $0 < \epsilon < \textcolor{red!90!black}{\frac{1}{2} Z_{1} / (\vol{(C)} + 2)}$\todo{was $\frac{1}{2} Z_{1} (\vol{(C)} + 2)$} sufficiently small, we have $\vert Z_{1} - Z_{2} \vert < \frac{1}{2} Z_{1}$ \textcolor{blue!90!black}{(note: this implies that $1/Z_{2} \leq 2/Z_{1}$)}. For such $\epsilon$, Lemma \ref{lem:LemmaCloseNorm} gives for all $x \in C$ the pointwise bound:

\begin{align*}
\vert p_{1}(x) - p_{2}(x) \vert &\leq  p_1 \frac{\vert Z_2 - Z_1 \vert}{Z_2}  + \frac{\vert f_1 - f_2\vert }{Z_2} \\
&\leq \frac{2M}{Z_{1}} |Z_{2} - Z_{1}| + \frac{2}{Z_{1}} \epsilon \\
&\leq \epsilon \left( \frac{2M}{Z_{1}}(\vol{(C)} + 2) + \frac{2}{Z_{1}} \right).
\end{align*}

\noindent Combining this with Inequality \eqref{eq:CloseEst2}, we have for all measurable subsets $E \subset \mathbb{R}^{d}$

\begin{align*}
|p_{1}(E) - p_{2}(E)| &\leq |p_{1}(E \cap C) - p_{2}(E \cap C)| + |p_{1}(E \cap C^{c}) - p_{2}(E \cap C^{c})| \\
&\leq \sup_{x \in C} |p_{1}(x) - p_{2}(x)| + p_{1}(C^{c}) + p_{2}(C^{c}) \\
&\leq \epsilon  \left( \frac{2M}{Z_{1}}(Vol(C) + 2) + \frac{2}{Z_{1}}  + 2 \right).
\end{align*}

\noindent This completes the proof.

\end{proof}

We quickly check that Lyapunov conditions imply small mass in the tails:

\begin{lemma}
Let $K$ be the transition kernel of an ergodic Markov chain that satisfies the  Lyapunov inequality

\begin{equation}
  (KV)(x) \leq (1-a) V(x) + b
  \label{eq:IneqLyapSimp}
\end{equation}

\noindent for some function $V \geq 0$ and constants $0 < a < 1$, $0 < b < \infty$. Let $\pi$ be the stationary measure of $K$. Then $$\pi(\{x \, : \, V(x) > C\}) \leq \frac{b}{aC}.$$
\end{lemma}

\begin{proof}
  Let $X_{0} \sim \pi$. Using stationarity in the first line, and Inequality \eqref{eq:IneqLyapSimp} in the second, $$\E[V(X_{0})] = \E[V(X_{1})] \leq (1 - a) \E[V(X_{0})] + b.$$  Solving, one has $$\E[V(X_{0})] \leq \frac{b}{a}.$$  The result then follows from Markov's inequality.
\end{proof}

Relatedly, we check that Lyapunov conditions imply all compact sets get hit often:

\begin{lemma}

Let $Q$ be a reversible transition kernel for which the measure $Q(x,\cdot)$ has a density $q_{x}(\cdot)$ that satisfies

\begin{equation}
\inf_{x \in A, y \in B} q_{x}(y) > 0
\label{eq:IneqMinComp}
\end{equation}

\noindent for all pairs of compact sets $A,B \subset \mathbb{R}^{d}$.

[AS: I will definitely have to clean up the following assumptions at some point.]
Let $\{p_{t}\}_{t \geq 0}$ be a sequence of probability densities. This sequence may be random, but we assume it is adapted to a filtration $\{ \mathcal{F}_{t} \}_{t \geq 0}$. Let $K_{t}$ be the Metropolis-Hastings kernel with proposal $Q$ and target $p_{t}$. Assume that, for all $t$, $K_{t}$ satisfies the Lyapunov inequality $$(K_{t}V)(x) \leq (1-a) V(x) + b$$
for the \textit{same} continuous function $V \geq 0$ and constants $a> 0$, $b <\infty$. Assume also $\lim_{r \rightarrow \infty} \inf_{x \, : \, |x| \geq r}V(x) \rightarrow \infty$.

Finally, We let $\{X_{t}\}_{t \geq 0}$ be the ``obvious" Markov chain associated with $\{K_{t}\}_{t \geq 0}$, and let $\{X_{t}^{\ast} \}_{t \geq 0}$ be the ``obvious" sequence of points that are proposed (but perhaps not accepted) during the Metropolis-Hastings algorithm. We assume that $$\mathbb{P}[X_{t+1} \in \cdot | \mathcal{F}_{t}, X_{t}] = K_{t}[X_{t},\cdot]$$
and $$\mathbb{P}[X_{t+1}^{\ast} \in \cdot | \mathcal{F}_{t}, X_{t}] = Q[X_{t},\cdot].$$

[AS: If we talk about $V$ growing with some rate, we can replace the following conclusion ``all compact sets are hit infinitely often" with actual bounds on how long it takes to hit a certain number of times. I think we probably want to do this, but I give the easy lemma first until we're a bit more sure of what we want.] Then for any measurable compact set $A \subset \mathbb{R}^{d}$ with strictly positive Lebesgue measure,
\[
\lim_{n \rightarrow \infty} |\{1 \leq m \leq n \, : \, X_{m}^{\ast} \in A\}| = \infty
\]
almost surely.

\end{lemma}

\begin{proof}
By the standard proof of Lyapunov stability,
\[
\lim_{n \rightarrow \infty} |\{1 \leq m \leq n \, : \, V(X_{n}) \leq \frac{4b}{a} \}| = \infty
\]
almost surely. Since $V$ is continuous and grows, the set $\{x \, :\, V(x) \leq \frac{4b}{a} \}$ is compact and so the result follows from Inequality \eqref{eq:IneqMinComp}.

\end{proof}

\section{Trying Again}

The theory in the previous section is for a fixed approximate chain. This isn't terrible, but it would probably be better to deal with the ``true" adaptive chain. We try to do that nw.

\subsection*{General Coupling and Variance Estimates for Good Approximation Processes}

We begin by setting up notation:

\begin{assumption}
Let $Q$ be a reversible transition kernel for which the measure $Q(x,\cdot)$ has a density $q_{x}(\cdot)$ that satisfies

\begin{equation}
\inf_{x \in A, y \in B} q_{x}(y) > 0
\label{eq:IneqMinComp}
\end{equation}

\noindent for all pairs of compact sets $A,B \subset \mathbb{R}^{d}$.

[AS: I will definitely have to clean up the following assumptions at some point.]
Let $\{p_{t}\}_{t \geq 0}$ be a sequence of probability densities. This sequence may be random, but we assume it is adapted to a filtration $\{ \mathcal{F}_{t} \}_{t \geq 0}$. Let $K_{t}$ be the Metropolis-Hastings kernel with proposal $Q$ and target $p_{t}$. Assume that, for all $t$, $K_{t}$ satisfies the Lyapunov inequality $$(K_{t}V)(x) \leq (1-a) V(x) + b$$
for the \textit{same} continuous function $V \geq 0$ and constants $a> 0$, $b <\infty$. Assume also $\lim_{r \rightarrow \infty} \inf_{x \, : \, |x| \geq r}V(x) \rightarrow \infty$.

Finally, We let $\{X_{t}\}_{t \geq 0}$ be the ``obvious" Markov chain associated with $\{K_{t}\}_{t \geq 0}$, and let $\{X_{t}^{\ast} \}_{t \geq 0}$ be the ``obvious" sequence of points that are proposed (but perhaps not accepted) during the Metropolis-Hastings algorithm. We assume that $$\mathbb{P}[X_{t+1} \in \cdot | \mathcal{F}_{t}, X_{t}] = K_{t}[X_{t},\cdot]$$
and $$\mathbb{P}[X_{t+1}^{\ast} \in \cdot | \mathcal{F}_{t}, X_{t}] = Q[X_{t},\cdot].$$

\end{assumption}

\subsection*{Guaranteeing Good Approximation Property}

\subsection*{Putting It Together}


TITLE: Why is the derivative of $\arctan(\frac{x}{\sqrt{1-x^2}})$ the same as the derivative of $\arcsin(x)$?
QUESTION [1 upvotes]: I'm self studying math through MIT opencourseware. Unfortunately the solutions of the problem sets are not very user-friendly and don't contain much explanation. There is a question which is bugging me and I couldn't come up with any explanation (there is no explanation provided in the solutions as well).
The question is:
Why is the derivative of $\arctan(\frac{x}{\sqrt{1-x^2}})$ the same as the derivative of $\arcsin(x)$?
I have solved the derivative of the arctan part and it's obvious to me how to get to the $\frac{1}{\sqrt{1-x^2}}$ answer. But I don't understand how this is related to the $\arcsin(x)$. Why are they the same?
Thank you in advance

REPLY [2 votes]: Conceptually, both $\theta_t(x)=\arctan\frac{x}{\sqrt{1-x^2}}$ and  $\theta_s(x)=\arcsin(x)$ represent angles as functions of $x$. Equivalently, the two expressions are $\tan( \theta_t (x))= \frac{x}{\sqrt{1-x^2}}$ and $\sin(\theta_s(x))=x$, respectively. Then, $\cos (\theta_s(x))= \sqrt{1-x^2}$ and
$$\tan(\theta_s(x) )= \frac {\sin(\theta_s(x))}{\cos(\theta_s(x))}= \frac{x}{\sqrt{1-x^2}}=\tan( \theta_t(x))
$$
Thus, $\theta_t(x)$ and $\theta_s(x)$ are the same. So are their derivatives.


TITLE: Finding a basis for subspaces of the complex vector space.
QUESTION [1 upvotes]: Each of the following sets spans a subspace of the complex vector space of all
functions from $\mathbb C$ to $\mathbb C$. In each case find a basis for the
subspace and prove it is a basis; state the dimension
{$\exp(iz); \cos z;\sin z$}
{$\exp(iz); \cosh z;\sinh z$}

REPLY [3 votes]: Hint for your first case $\;\{e^{iz}, \cos z, \sin z\}\;$:

Recall that $\;e^{iz} = \cos(z) + i\sin(z)$: this is a linear combination of $\cos z$ and $\sin z$.

What does this tell you about the linear-(in)dependence of the the vectors $e^{iz}, \cos z, \sin z\;$?
So the basis of the corresponding subspace must be: ... ?.
And the corresponding subspace then has dimension = ... ?


TITLE: Prove that this graph is k-connected
QUESTION [1 upvotes]: Let $G$ be a graph with $n$ vertices and minimum degree $\delta \ge (n+k-2)/2$, with $k\ge 2$. Prove that $G$ is $k$-connected.

REPLY [1 votes]: First remove any $k-1$ nodes. We get a new graph $G'$ with  $|V(G')|=n-k+1$ and a minimum of $\delta(G') = (n-k)/2$. Assume that $G'$ is not connected, then
there exist two distinct nodes with distinct neighbors. The neighbors are at
least $\delta(G')*2 \ge n -k$. Add the aforementioned nodes and we have a minimum
number of $n-k+2 > |V(G')|$ nodes. Therefore $G'$ is connected and thus $G$ is $k$-connected.


TITLE: Integral closure of a local ring is the intersection of valuation rings lying above it
QUESTION [8 upvotes]: Let $L/K$ be a finite field extension. Let $\mathcal{O}$ be a valuation ring of $K$. Let $R$ be the integral closure of $\mathcal{O}$ in $L$. Why is $R$ the intersection of all valuation rings of $L$ that lie above $\mathcal{O}$? (A valuation ring $B$ of $L$ lies above $\mathcal{O}$ if $B\cap K=\mathcal{O}$)

REPLY [7 votes]: Valuation rings are integrally closed, so the integral closure of $\mathcal{O}$ is contained in each valuation ring, hence in the intersection.
Conversely, let $x\in L$ be an element that is not integral over $\mathcal{O}$; then $x\notin \mathcal{O}[x^{-1}]$. Thus, $x^{-1}$ is not a unit in $\mathcal{O}[x^{-1}]$, so there is a maximal ideal $\mathfrak{M}$ of $\mathcal{O}[x^{-1}]$ that contains $x^{-1}$.
Consider the map $\mathcal{O}[x^{-1}]\to \mathcal{O}[x^{-1}]/\mathfrak{M}$, and then map the image of $\mathcal{O}$ (via the natural inclusion) into an algebraic closure; you can extend this map to some valuation ring $B$ of $L$ that lies over $\mathcal{O}$ in such a way that the kernel is precisely the maximal ideal of $B$; and since $x^{-1}$ maps to $0$, then $x\notin B$ (otherwise, you would have that the maximal ideal of $B$ contains a unit, a contradiction). Thus, $x$ is not in the intersection of all valuation rings of $L$ that lie above $\mathcal{O}$.


TITLE: There exist such f, f'
QUESTION [0 upvotes]: My question is:
There exist a function $f = f(x)$ that satisfy
$$\lim_{x \to +\infty} f(x) = +\infty$$
$$\lim_{x \to -\infty} f(x) = +\infty$$
$$\lim_{x \to +\infty} f'(x) = +\infty$$
$$\lim_{x \to -\infty} f'(x) = +\infty$$
Another (related) question:
There exist $g = g(x)$ that satisfy
$$\lim_{x \to +\infty} g(x) = +\infty$$
$$\lim_{x \to -\infty} g(x) = +\infty$$
And $g'(x) \ge 0 \ \forall x$?

REPLY [2 votes]: I'm assuming you're talking about real valued everywhere differentiable functions.
The second part is quick: No, no such $g$ can exist. $g^\prime \geq 0$ says that $g$ is non-decreasing. However, since $\lim_{x \rightarrow -\infty} g(x) = \infty$, it is the case that there is some $x <0$ such that $g(x) >g(0)$ (can you prove this?). So at some point $g$ is decreasing on the interval $(x,0)$.
The exact same trick works on the first part- try and write down a proof.


TITLE: What does it mean to prove that two systems have to same theorems?
QUESTION [1 upvotes]: Lets say there is two systems $L_1$ and $L_2$.If I get asked:

Show that $L_1$ and $L_2$ have the same theorems.

Is it sufficient to prove the axioms of $L_1$ using the axioms of $L_2$ or vice-versa ? Or do I have to show that a wf is a theorem of $L_1$ iff it is a theorem of $L_2$ ?

REPLY [3 votes]: It is indeed sufficient to show that each axiom of $L_1$ is derivable from the axioms of $L_2$ and vice versa. This implies that any wff is a theorem of $L_1$ iff it a theorem of $L_2$.
To see why assume $\phi$ is a theorem of $L_1$. The proof $\Pi$ of $\phi$ can only use finitely many axioms of $L_1$, say $\alpha_1, \ldots, \alpha_k$. If we have proofs $\Pi_1, \ldots, \Pi_k$  of $\alpha_1, \ldots, \alpha_k$ in $L_2$, we can combine $\Pi_1, \ldots, \Pi_k$ and $\Pi$ to give a proof of $\phi$ in $L_2$. So any theorem of $L_1$ is a theorem of $L_2$. Likewise, swapping $L_1$ and $L_2$ any theorem of $L_2$ is a theorem of $L_1$.
Aside: the above assumes that your systems have the same inference rules, e.g., if you are looking at a typical exposition of first-order logic, as a Hilbert style system with modus ponens as the only inference rule. If you are looking at systems with different inference rules (e.g, a natural deduction calculus on the one hand and a Hilbert-style system on the other), you will also have to show that each rule of $L_2$ is derivable (or more generally, what is called permissible) on the basis of the axioms and rules of $L_1$ and vice versa.


TITLE: Correct expansion for 2D anisotropic diffusion
QUESTION [2 upvotes]: I am trying to model a diffusion phenomenon with anisotropic and heterogeneous diffusion.
$$\nabla \cdot (D\nabla c)$$
where
$$D = \begin{bmatrix} D_{xx}(x,y)&D_{xy}(x,y)\\ D_{yx}(x,y)&D_{yy}(x,y) \end{bmatrix}$$
and $D_{xy} = D_{yx}$. From Wikipedia, I get
$$\nabla \cdot (D\nabla c) = \nabla \cdot[D(x,y)] \nabla c(x,y) + \mbox{tr} \left( D(x,y) \nabla \nabla ^T c(x,y) \right)$$
If I am correct, then
$$\mbox{tr} \left( D(x,y)\nabla \nabla ^Tc(x,y) \right) = D_{xx}\frac{\partial^2 c}{\partial x^2}+D_{yy}\frac{\partial^2 c}{\partial y^2}$$
but I am unsure how to expand the first term. Perhaps,
$$\left( \frac{\partial c}{\partial x} \hat i + \frac{\partial c}{\partial y} \hat j \right) (\nabla D_{xx}+2\nabla D_{xy}+\nabla D_{yy})$$
where $\nabla (\cdot) = \left(\frac{\partial (\cdot)}{\partial x} \hat i+\frac{\partial (\cdot)}{\partial y} \hat j \right)$
completely FOIL'd out?
Any help would be appreciated, as I am trying to use this for implementing a model. I also apologize for my notation.

REPLY [0 votes]: The expansion of the divergence is explained in this post, where intrinsic expressions are given. We might want to expand the divergence explicitly, noting that
\begin{aligned}
\nabla\cdot(D\nabla c) &= (D_{xx}c_{,x} + D_{xy}c_{,y})_{,x} + (D_{xy}c_{,x} + D_{yy}c_{,y})_{,y} \\
&= D_{xx,x}c_{,x} + D_{xy,x}c_{,y} + D_{xy,y}c_{,x} + D_{yy,y}c_{,y} \\
&\quad + D_{xx}c_{,xx} + 2\, D_{xy}c_{,xy} + D_{yy}c_{,yy} .
\end{aligned}
where we have used differential rules, the symmetry property $D_{yx}=D_{xy}$, and the definition $\nabla c = [c_{,x}, c_{,y}]^T$ of the gradient (indices after the comma denote partial differentiation). In other words, we get $$
\nabla\cdot(D\nabla c) = (\nabla\cdot D) \cdot \nabla c + D : \nabla \nabla c
\tag{*}
$$
as was found in the related post, where $$
\nabla\cdot D = \begin{bmatrix}
D_{xx,x}+D_{xy,y}\\
D_{xy,x}+D_{yy,y}
\end{bmatrix}, \qquad
\nabla \nabla c = \begin{bmatrix}
c_{,xx} & c_{,xy}\\
c_{,xy} & c_{,yy}
\end{bmatrix},
$$
and the colon denotes the Frobenius inner product (which is analogous to the vector dot product). In Eq. (*), the first term corresponds to the first line of the expanded expression above, and the second term corresponds to the second line.
Note: in this answer, $\nabla$ is viewed as the gradient operator, not as a vector of scalar differential operators.