text:
\begin{document}

\maketitle
\begin{abstract}
We introduce a higher dimensional Atkin-Lehner theory for
Siegel-parahoric and Borel congruence subgroups of $GSp(2g)$. Old
Siegel forms are induced by geometric correspondences on Siegel
moduli spaces which commute with almost all local Hecke algebras.
Any satisfactory local definition of $p$-old Siegel forms which
guarantees that, any new eigenform of almost all local Hecke
algebras is an eigenform of all local Hecke algebras prime to the
level, will imply that our correspondences introduce a geometric
formulation for the notion of $p$-old Siegel forms inside all
Siegel forms of Siegel parahoric level $pn$. We also prove an
injection result for Siegel modular forms in finite
characteristics.
\end{abstract}

\subsection*{Introduction}

Classicial Atkin-Lehner theory for $GL(2)$ has two conceptual
ingredients: Casselman's theory of local new-forms for $GL(2)$
[Cas], together with strong multiplicity one for cuspidal
automorphic representations of $GL(2)$. The first ingredient is
generalized to $GSp(4)$-case by R. Schmidt for square-free level
[Sch]. But, at the moment there is no general local theory of
new-forms is available for $GSp(2g)$. Multiplicity one fails to
hold in higher $GSp(2g)$'s. Therefore, we regard all Siegel forms
with the same Hecke eigenvalues as old Siegel-forms to be old
forms again. This is what we have to do anyway, if we are
interested in getting congruences between old and new forms.
Despite this loss, Atkin-Lehner geometric theory can be
generalized to Siegel-parahoric congruence subgroups of $GSp(2g)$.

In the first part of this paper, we use elements of the Weyl group
of $GSp(2g)$ to construct new congruence subgroups sandwiched
between the Siegel-parahoric congruence group
$$
\Gamma^P(n)=\{\gamma\in Sp(2g,\Z)|\gamma\equiv
\left(\begin{array}{cc} *&*\\0&* \end{array}\right), (\textmd{mod}
\hspace{.05in}n)\}
$$
and the following congruence group
$$
\Gamma^T(n)=\{\gamma\in Sp(2g,\Z)|\gamma\equiv
diag(*,...,*),(\textmd{mod} \hspace{.05in}n)\}.
$$
These groups are all defined in terms of the mod-$n$ reduction of
elements in $Sp(2g,\Z)$.

In the second part of the paper, we use these congruence subgroups
to introduce geometric correspondences on Siegel moduli spaces
with explicit moduli interpretations. This allows one to extend
them to a model over $\Z$ with some primes inverted. These
correspondences induce an injection of a number of copies of
Siegel modular forms with $n$-level structure inside the space of
Siegel modular forms of Siegel-parahoric level $pn$. Since our
correspondences commute with all $q$-Hecke correspondences for
$(q,pn)=1$, any satisfactory local definition of $p$-old Siegel
forms will imply that our correspondences introduce a higher
dimensional Atkin-Lehner theory for $p$-old Siegel forms of
Siegel parahoric level $pn$. In a satisfactory local theory of
new-forms any new eigenform of almost all local Hecke algebras
should be an eigenform of all local Hecke algebras prime to the
level. Also, eigenforms with eigenvalues repeated in lover levels
should be considered as old-forms.

In the final part, we prove an Ihara-type result by getting an
injection result in positive characteristic in the case of Siegel
spaces of genus $\geq 2$. Then using Ihara-type results, we get
congruences between old-forms and new-forms.

\section*{Notations}

The Chevalley group scheme $GSp(2g)$ is defined as the set of
matrices $g\in GL(2g)$ with $^tgJg=\lambda(g)J$ where
$\lambda(g)\in GL(1)$ and $J=antidiag(I_g,-I_g)$ where $I_g$
denotes the $g\times g$ identity matrix. The representation
$\lambda$ is called the multiplier representation. Similar to
above, we also use two by two matrices whose entries are $g\times
g$ matrices to represent elements of $GSp(2g)$. The symplectic
group scheme $Sp(2g)$ is defined to be the kernel of the
multiplier representation, which is the space of transformations
on the symplectic space $\Z^{2g}$ with its standard alternating
form:
$$
<,>:\Z^{2g} \times \Z^{2g} \to \Z
\\ \hspace{.5in}<(u,v) , (z,w)>
\mapsto u.^tw-v.^tz
$$
Let $T\cong G_m^{g+1}$ denote the maximal torus in $GSp(2g)$. An
element of $T$ is a diagonal matrix
$diag(a_1,...,a_g,d_1,...,d_g)$ with $a_id_i$ equal to the
multiplier. Let $M$ denote the subgroup of $GSp(2g)$ which
respects the standard decomposition $\Z^{2g}\cong\Z^g\oplus\Z^g$.
The subgroup $M$ consists of diagonal elements in $GSp(2g)$ in
their two by two representation. These elements are of the form
$diag(A,D)$ with $A.^tD=\lambda I_g$. The subgroup of $Sp(2g,\Z)$
which fixes only the first direct summand $\Z^g\subset\Z^{2g}$ is
denoted by $U$ which is the space of all $\Z$-valued bilinear
symmetric forms. Elements of $U$ are upper-triangular with $I_g$
on diagonal entries and a symmetric matrix $B$ on the upper right
corner. The subgroup $P=M\ltimes U$ is a maximal parabolic
subgroup whose elements are zero in the lower left $g\times g$
corner. This parabolic subgroup is also called the
Siegel-parahoric subgroup. Fix the Borel subgroup contained in $P$
consisting of the matrices with $A,B,0,D$ entries with $A$
upper-triangular and $D$ lower-triangular. Weyl groups of
$GSp(2g)$ and $P$ with respect to the maximal torus $T$ are
denoted by $W_G$ and $W_P$ respectively. $W_G\cong S_g\ltimes (\pm
1)^g$ and $W_P\cong S_g$ act on diagonal matrices
$diag(a_1,...,a_g,d_1,...,d_g)$ by permutation or exchange of the
$a_i$'s and the $d_i$'s.

\section{Congruence subgroups of $Sp(2g,\Z)$}

A discrete subgroup $\Gamma\subset Sp(2g,\Q)$ is called a
congruence group, if it contains $\Gamma^{id}(n)$ for some
positive integer $n$, where $\Gamma^{id}(n)$ is the kernel of
reduction map modulo $n$ on $Sp(2g,\Z)$:
$$
\Gamma^{id}(n)=\{\gamma \in Sp(2g,\Z)| \gamma \equiv id\in
Sp(2g,\Z/n\Z) \hspace{.05in}(\textmd{mod} \hspace{.05in} n) \}.
$$
$\Gamma^P(n)$ and $\Gamma^T(n)$ are examples of congruence groups.
The significance of congruence groups is that they carry
arithmetic information. In this paper, we are only interested in
congruence subgroups of $Sp(2g,\Z)$.

\subsection{Parabolic congruence subgroups}

By the theory of Tits systems [Hum], every parabolic subgroup of
$GSp(2g)$ is conjugate to a "standard" parabolic subgroup which
contains $B$. Each standard parabolic subgroup $P_I$ corresponds
to one of the $2^{g+1}$ subsets $I\subset S$ where $S$ is a
minimal generating set for the Weyl group $W_G$ consisting of
involutions $\rho_i$ which are elements of order 2. $S$ can be
taken as the set of simple reflections corresponding to the base
of the root system determined by B. In fact, $P_I=BW_I B$ where
$W_I\subseteq W_G$ is the subgroup generated by elements in $I$.
In particular, $P_{\emptyset}=B$ and $P_S=GSp(2g)$.

By Bruhat decomposition theorem, $GSp(2g)=\cup B\sigma B$ is a
decomposition to disjoint subsets where $\sigma$ runs in the Weyl
group $W_G$. Two such double cosets coincide if and only if the
middle Weyl elements coincide. One can assume that axioms of Tits
systems are satisfied. Namely,

(T1) For $\rho\in S$ and $\sigma\in W_G$ we have $\rho
B\sigma\subset B\sigma B \cup B\rho\sigma B$.

(T2) For $\rho\in S$ we have $\rho B\rho \neq B$.
\\An expression $\sigma=\rho_1...\rho_k$ with $\rho_i\in S$ is
called reduced if $k$ is as small as possible. the minimal length
$k$ of a reduced expression is denoted by $\ell(\sigma)$. By
convention, $\ell(\sigma)=0$ if and only if $\sigma=id$ and
$\ell(\sigma)=1$ if and only if $\sigma\in S$. Tits axioms imply
that for $\rho\in S$ we have $\ell(\rho\sigma)=\ell(\sigma)\pm 1$.
This implies that for the reduced form $\sigma=\rho_1...\rho_k$
and $I=\{\rho_1,...,\rho_k\}$, the parabolic subgroup $P_I$ is
generated $B$ and $\sigma B\sigma^{-1}$ or by $B$ and $\sigma$.
Given a choice of a Borel subgroup $S$ is precisely the set of
elements in $W_G$ such that $B\cup B\sigma B$ is a group. $P_I$ is
conjugate to $P_J$ implies that $P_I=P_J$ and $P_I\subset P_J$
implies that $I\subset J$.

We define a parabolic congruence subgroup $\Gamma^{P_I}(n)\subset
Sp(2g,\Z)$ to be the set of elements which reduce to $P_I$ modulo
$n$:
$$
\Gamma^{P_I}(n)=\{\gamma \in Sp(2g,\Z)| \gamma \in P_I\subset
GSp(2g,\Z/n\Z) \hspace{.05in}(\textmd{mod} \hspace{.05in} n) \}.
$$

\subsection{A nested family of congruence subgroups}

Fix a generating set $S$ for $W_G$ consisting of $g-1$ pairs in
$S_g\subset W_g$ and a nonzero element in $(\pm 1)^g$. Now, there
exists a generating set $w_1,...,w_g$ for the weyl group such that
$w_i$ have increasing length if represented in reduced form in
term of involutions in $S$.

Assume that $w_{k+1}=\rho_k w_k$for all $k$ where $\rho_k$ is an
involution. Also assume $w_1$,...,$w_{g-1}$ generate $S_g$. We
have $\Gamma^{P_{I_{g-1}}}(n)=\Gamma^P(n)$ where $P$ is the
maximal parabolic we fixed in notations. The fact that $B$
together with $I_k=\{w_1,...,w_k\}$ can not generate any of
$w_{k+1}$,...,$w_g$ implies that each $P_{I_k}$ has exactly
$\frac{g!}{(k+1)!}2^g$ conjugates of the form $\sigma
P_{I_k}\sigma^{-1}$ for $\sigma$ in $W_G$. To these we associate
$\frac{g!}{(k+1)!}2^g$ parabolic congruence subgroups of the form
$\sigma \Gamma^{P_{I_k}}(n)\sigma^{-1}$ for $\sigma$ in $W_G$. In
particular, $\Gamma^B(n)$ has $g! 2^g$ conjugates in $Sp(2g,\Z)$
and each $\Gamma^{P_{I_k}}(n)$ has $k+2$ conjugates in
$\Gamma^{P_{I_{k+1}}}(n)$.

We have made a nested family of parabolic congruence groups
contained in the maximal parabolic congruence group $\Gamma^P(n)$
consisting of levels $0$ to $g-1$. The $k$-th level is formed by
$\frac{g!}{(k+1)!}2^g$ congruence groups. Each group in level $k$
contains $k+1$ groups in level $k-1$ for $k=1$ to $g-1$. There
are $2^g$ congruence groups in level $g-1$ and the congruence
groups in level $0$ are the $g!2^g$ conjugates by elements in
$W_g$ of $\Gamma^B(p)$ which are lying inside $Sp(2g,\Z)$.

\section{Higher Atkin-Lehner theory}

The Atkin-Lehner involution can be easily generalized from
$GL(2)$ to the higher dimensional case $GSp(2g)$. This
generalization involves Siegel-parahoric and Borel congruence
groups. On the other hand, local considerations show that $p$-old
Siegel modular forms with respect to the Siegel-parahoric or Borel
congruence groups of level $pn$ contain several copies of Siegel
forms of level $n$. This implies that a single Atkin-Lehner
involution would not do the job of geometrically generating the
$p$-old part. In this section we intend to introduce geometric
correspondences which complement the role of Atkin-Lehner
involution.

\subsection{Siegel moduli spaces}

A general reference for the arithmetic of Siegel moduli spaces is
[Fal-Cha]. Let $A_g$ denote the moduli stack of principally
polarized abelian schemes of relative dimension $g$. By a
symplectic principal level-$n$ structure, we mean a symplectic
isomorphism $\a:A[n]\to(\Z/n\Z)^{2g}$, where $(\Z/n\Z)^{2g}$ is
equipped with the standard non-degenerate skew-symmetric pairing.

Let $\z_n$ denote an $n$-th root of unity for $n \geq 3$. The
moduli scheme classifying the principally polarized abelian
schemes over $\Sn$ together with a symplectic principal level-$n$
structure is a scheme over $\Sn$ and will be denoted by $\Agn$.
The symplectic group $Sp(2g,\Z/n\Z)$ acts on $\Agn$ as a group of
symmetries by acting on level structures. We will recognize these
moduli spaces and their etale quotients under the action of
subgroups of $Sp(2g,\Z/n\Z)$ as Siegel spaces.

A $\Gamma^P(n)$-level structure of type I on $(A,\l)$ is choice
of a subgroup $H\subset A[n]$ of order $n^g$ which is totally
isotropic with respect to the Weil pairing induced by $\l$. A
$\Gamma^P(n)$-level structure of type II on $(A,\l)$ is choice of
a principally polarized isogeny $(A_1,\l_1)\to (A_2,\l_2)$ of
degree $n^g$. By a principally polarized isogeny, we mean an
isogeny $\s :A_1 \to A_2$ such that $\s\circ \l_2\circ \s^t\circ
\l_1^{-1}$ is multiplication by an integer. For $n\geq 3$ type I
and type II $\Gamma^P(n)$-level structures induce isomorphic
moduli schemes over $Spec(\Z [1/n])$ [DJ]. We denote this moduli
scheme by $\Aogn$. There exists a natural involution
$$
w^P_n:\Aogn \to \Aogn
$$
taking $(\s :(A_1,\l_1)\to (A_2,\l_2))$ to $(\s^t
:(A_2,(\l_2^t)^{-1}) \to (A_1,(\l_1^t)^{-1}))$ which we call the
Atkin-Lehner involution.

A $\Gamma^B(n)$-level structure of type I on $(A,\l)$ is choice
of $g$ subgroups $H_i\subset A[n]$ of order $n^i$ with $H_1
\subset ... \subset H_g$ where $H_g$ is totally isotropic. A
$\Gamma^B(n)$-level structure of type II on $(A,\l)$ is choice of
a chain of $g$ isogenies $(A_0,\l_0) \buildrel \a \over \to ...
\buildrel \a \over\to (A_g,\l_g)$ each of degree $n$ which
satisfy $n.id_{A_i}=\a^i\circ \l_0^{-1}\circ (\a^t)^g\circ \l_g
\circ \a^{g-i}$ for all $i=1,...,g$. In case $n\geq 3$ type I and
type II $\Gamma^B(n)$-level structures induce isomorphic moduli
schemes over $Spec(\Z [1/n])$ [DJ]. We denote this moduli scheme
by ${\cal{A}}^B_{g}(n)$. There also exists a natural involution
$$
w^B_n:\Abgn\to \Abgn
$$
taking $ ((A_0,\l_0) \buildrel \a \over \to ... \buildrel \a \over
\to (A_g,\l_g))$ to $((A_g,(\l_g^t)^{-1}) \buildrel \a^t \over \to
... \buildrel \a^t \over \to (A_0,(\l_0^t)^{-1}))$ which commutes
with the Atkin-Lehner involution under the natural projection
between the Siegel spaces.
$$
\begin{array}{ccc}
\Abgn & \buildrel {w^B_n} \over \longrightarrow & \Abgn\\
\downarrow &                     & \downarrow \\
\Aogn & \buildrel {w^P_n} \over  \longrightarrow & \Aogn
\end{array}
$$

A $\Gamma^T(n)$-level structure on $(A,\l)$ is choice of $2g$
subgroups $H_i\subset A[n]$ for $i=1$ to $2g$, each isomorphic to
$(\Z/n\Z)$ such that $H_1\oplus ...\oplus H_g$ and $H_{g+1}\oplus
...\oplus H_{2g}$ are totally isotropic subgroups of order $n^g$
which do not intersect with $H_i\oplus H_{g+i}$ hyperbolic for
$i=1$ to $g$. For $A$ and $A'$ abelian schemes over the schemes
$S$ and $S'$ respectively, we define a morphism from $(S,A,\l
,H_1,...,H_{2g})$ to $(S',A',\l' ,H'_1,...,H'_{2g})$ to be a pair
of morphisms $(f,g)$ where $f:S\to S'$ and $g:A\to A'$ satisfy
$g^*(\l')=\l$ and $g(H_i)=H'_i$ for all $1\leq i \leq 2g$. Also
we want the pair $(f,g)$ to induce an isomorphism $A \simeq
S\times_{S'}A'$. Having these morphisms defined, we have formed a
category $\Ab ^T_g(n)$. The functor $\pi :\Ab ^T_g(n)\to Sch$
defined by $\pi (S,A,\l ,H_1,...,H_{2g})=S$ makes $\mathbf A ^T
_g(n)$ into a stack in groupoids over $S$. The $1$-morphism of
stacks $\pi':\mathbf A ^T_g(n)\to \mathbf A_g$ defined by
$\pi'(S,A,\l ,H_1,...,H_{2g})=(S,A,\l )$ is representable and is
a proper surjective morphism. For $n\geq 3$ we get a separated
scheme of finite type $A^T_g(n)$ which is smooth over
$Spec(\Z[1/n])$.

Let $\H_g$ denote the Siegel upper half-space, which consists of
the set of complex symmetric $g \times g$ matrices $\O$ with
$\Im(\O)$ positive definite. As a complex analytic stack ${\cal
{A}}_{g/\C}$ is the quotient of Siegel upper half-space $\H_g$ by
the action of $Sp(2g,\Z)$ via M\"obius transformations. The
family of principally polarized abelian varieties over $\H_g$ is
given by $A(\O)=\C^g/(\Z^g \oplus \O.\Z^g)$. To any congruence
subgroup $\Gamma\subset Sp(2g,\Z)$ one can associate the quotient
$\Gamma\backslash \H_g$ which is a Siegel moduli-space with some
extra level structure. The corresponding level structure can be
made explicit. Indeed, $\Gamma^{id}(n)\backslash\H_g$ corresponds
to $\Agn_{/\C}$ whose quotient under the action of the symplectic
group $Sp(2g,\Z/n\Z)$ is ${\cal {A}}_{g/\C}$. Any congruence
subgroup $\Gamma^{id}(n)\subset \Gamma \subset Sp(2g,\Z)$
corresponds to the quotient of $\Agn_{/\C}$ under the action of
the finite group $\Gamma/\Gamma^{id}(n)$. This helps to associate
explicit level structures to the space $\Gamma\backslash\H_g$
which makes it a moduli space.

\subsection{The $\Gamma^{P_I}(p)$-level structure}

Let $p$ be a prime not dividing the positive integer $n$ and let
${\cal {A}}^T_g(p)_{/\C}$ and ${\cal {A}}^{T,P}_g(p,n)_{/\C}$
denote the Siegel spaces associated to congruence groups
$\Gamma^T(p)$ and $\Gamma^{T,P}(p,n)=\Gamma^T(p)\cap
\Gamma^{P}(n)$, respectively. The group $\Gamma^T(p)$ remains
invariant under conjugation by elements in $(\pm 1)^g\subset
W_g$. So $(\pm 1)^g$ acts on ${\cal {A}}^T_g(p)_{/\C}$ and ${\cal
{A}}^{T,P}_g(p,n)_{/\C}$ by $2^g$ involutions. Let ${\cal
{A}}^{P_{I_k}}_g(p)_{/\C}$ and ${\cal
{A}}^{P_{I_k},P}_g(p,n)_{/\C}$ denote the Siegel spaces
associated to the congruence groups $\Gamma^{P_{I_k}}(p)$ and
$\Gamma^{P_{I_k},P}(p,n)=\Gamma^{P_{I_k}}(p)\cap \Gamma^{P}(n)$
respectively. We have a chain of etale maps
$$
{\cal {A}}_g^{B,P}(p,n) \rightarrow ... \rightarrow {\cal
{A}}^{P_{I_k},P}_g(p,n) \rightarrow {\cal
{A}}^{P_{I_{k+1}},P}_g(p,n) \rightarrow ... \rightarrow {\cal
{A}}_g^{P}(pn).
$$
Since each congruence group $\Gamma^{P_{I_{k+1}}}(p)$ on the
$(k+1)$-th level contains $k+2$ conjugates (by Weyl elements) of
$\Gamma^{P_{I_k}}(p)$
 on the $k$-th level, we expect that for each $k$
we get $k+2$ copies of forms on ${\cal
{A}}^{P_{I_{k+1}},P}_g(p,n)$ injecting in forms on ${\cal
{A}}^{P_{I_k},P}_g(p,n)$. We will use the geometry of ${\cal
{A}}_g^{B,P}(p,n)$ to give a geometric construction of these $k+2$
copies.

In order to simplify the notations, let us forget the
$\Gamma^{P}(n)$-level structure which is auxiliary. We get a
chain of etale maps
$$
{\cal {A}}_g^{B}(p) \rightarrow ... \rightarrow {\cal
{A}}^{P_{I_k}}_g(p) \rightarrow {\cal {A}}^{P_{I_{k+1}}}_g(p)
\rightarrow ... \rightarrow {\cal {A}}_g^{P}(p)
$$
which corresponds to a chain of congruence groups
$$
\Gamma^{B}(p) \hookrightarrow ... \hookrightarrow
\Gamma^{P_{I_k}}(p) \hookrightarrow \Gamma^{P_{I_{k+1}}}(p)
\hookrightarrow ... \hookrightarrow \Gamma^{P}(p).
$$
Each $\Gamma^{P_{I_k}}(p)$ maps to $\Gamma^{P_{I_{k+1}}}(p)$ by
$k+1$ maps: natural inclusion and conjugation by representatives
$\sigma_{k+1}\in W_k\backslash W_{k+1}$ followed by inclusion,
where $W_k$ is the subgroup of the Weyl group generated by
$w_1,...,w_k$. Inclusion induces the natural projection map
$\pi_{P_{I_k},P_{I_{k+1}}}:{\cal {A}}^{P_{I_k}}_g(p) \rightarrow
{\cal {A}}^{P_{I_{k+1}}}_g(p)$. Conjugation by $\sigma_{k+1}$
induces an inclusion
$$
\sigma_{k+1}\Gamma^{P_{I_k}}(p)\sigma_{k+1}^{-1}=
\Gamma^{\sigma_{k+1}P_{I_k}\sigma_{k+1}^{-1}}(p)\hookrightarrow
\Gamma^{P_{I_{k+1}}}(p).
$$
This inclusion corresponds to another projection from a different
moduli-space
$$
{\cal {A}}^{\sigma_{k+1}P_{I_k}\sigma_{k+1}^{-1}}_g(p)
\rightarrow {\cal {A}}^{P_{I_{k+1}}}_g(p).
$$
Conjugation by $\sigma_{k+1}$ identifies ${\cal
{A}}^{P_{I_k}}_g(p)$ with ${\cal
{A}}^{\sigma_{k+1}P_{I_k}\sigma_{k+1}^{-1}}_g(p)$. The
moduli-space ${\cal {A}}^T_g(p)$ is the appropriate moduli space
to geometrically realize all the endomorphisms
$$
v^{\sigma}_p:{\cal {A}}^T_g(p) \to {\cal {A}}^T_g(p)
$$
induced by conjugation via elements $\sigma$ in the Weyl group
$W_G$. In fact, the following diagrams are commutative

$$
\begin{array}{ccc}
{\cal {A}}_g^T(p) & \buildrel {v^{\sigma}_p} \over \longrightarrow
& {\cal {A}}_g^T(p)\\
\downarrow &                     & \downarrow \\
{\cal {A}}_g^{P_I}(p) & \buildrel {\sigma .} \over \longrightarrow
& {\cal {A}}_g^{\sigma P_I \sigma^{-1}}(p) \\
\downarrow &                     & \downarrow \\
{\cal {A}}_g^{P_J}(p) & \buildrel {\sigma .} \over \longrightarrow
& {\cal {A}}_g^{\sigma P_J \sigma^{-1}}(p)
\end{array}
$$

\begin{defn} We say that two points $x$ and $y$ on ${\cal
{A}}_g^{P_I,P}(p,n)$ are $\sigma$-connected, for an element
$\sigma$ in the Weyl group, if there exists a chain of points
$x=x_1,...,x_t=y$ on ${\cal {A}}_g^{P_I,P}(p,n)$ such that for
each $i$ there are points $x'_i$ and $x'_{i+1}$ on ${\cal
{A}}^{T,P}_g(p,n)$ mapping to $x_i$ and $x_{i+1}$ respectively,
with $x_{i+1}=v^{\sigma}_p(x_i)$ where
$$
v^{\sigma}_p:{\cal {A}}^{T,P}_g(p,n) \to {\cal {A}}^{T,P}_g(p,n)
$$
is the endomorphism induced by the action of $\sigma$ on
$p$-level structure.
\end{defn}

\begin{prop} Every fiber of the map $\pi_k:{\cal {A}}^{P_{I_k},P}_g(p,n)
\rightarrow {\cal{A}}^{P_{I_{k+1}},P}_g(p,n)$ is an equivalence
class of $\rho_{k+1}$-connected points.
\end{prop}
$\textbf {Proof}$. The $n$-level structure is auxiliary. Let $x$
be a point on the Siegel upper half-plane $\H_g$ and let $[x]^T$
denote the equivalence class containing $x$ defined by left
quotient of the Siegel upper space by $\Gamma^T(p)$. We have
$\sigma .[y]^T=[\sigma.y]^T$. The group $\Gamma^{P_{I_{k+1}}}(p)$
is generated $\Gamma^{P_{I_k}}(p)$ and $\rho_{k+1}$. So every
element in $\Gamma^{P_{I_{k+1}}}(p)$ can be written as a product
of elements of the form $\gamma_i \rho_{k+1}$ with $\gamma_i \in
\Gamma^{P_{I_k}}(p)$. Define the equivalence class
$[x]^{P_{I_k}}$ similarly. The classes $[x]^{P_{I_k}}$ and
$[\gamma\rho_{k+1}.x]^{P_{I_k}}$ are $\rho_{k+1}$-connected. So
the equivalence class $[x]^{P_{I_{k+1}}}$ is obtained by joining
the $\rho_{k+1}$-connected points.$\square$

\begin{defn} For a subset $W\subset W_g$, we say that
two points $x$ and $y$ on ${\cal{A}}_g^{P_I,P}(p,n)$ are
$W$-connected, if there exists a chain of points
$x=x_1,...,x_t=y$ on ${\cal{A}}_g^{P_I,P}(p,n)$, such that for
each $i$, $x_i$ and $x_{i+1}$ are $\sigma$-connected for some
$\sigma\in W$.
\end{defn}

\begin{prop} Every fiber of the map ${\cal{A}}_g^{B,P}(p,n)\to
{\cal{A}}_g^{P_{I_k},P}(p,n)$ is an equivalence class of
$W_k$-connected points.
\end{prop}

\begin{prop} Every fiber of the map ${\cal{A}}_g^P(pn)\to
{\cal{A}}_g^P(n)$ is an equivalence class of $\sigma$-connected
points for some nonzero representative $\sigma$ of $W_P \backslash
W_G$.
\end{prop}
$\textbf {Proof}$. The Bruhat-Tits decomposition modulo $p$
implies that the congruence groups $\Gamma^P(p)$ and
$\sigma\Gamma^P(p)\sigma^{-1}$ generate $Sp(2g,\Z)$. This is a
consequence of the fact that any two conjugates of a maximal
parabolic subgroup over $\Bbb {F}_p$ generate the whole algebraic
group $GSp(2g,\Bbb {F}_p)$. $\square$

\subsection{$p$-old Siegel modular forms on ${\cal {A}}_g^{P}(p)$}

Let $\Bogn$ denote the universal abelian variety over $\Aogn$.
The Hodge bundle $\o$ is defined to be the pull back via the zero
section $i_0:\Aogn \to \Bogn$ of the line bundle
$\wedge^{top}\Omega^1_{\Bgn/\Agn}$. The Hodge bundle is an ample
invertible sheaf on $\Aogn$. Let $R$ be a $\Z [1/n]$-module. By a
$\Gamma^P(n)$-Siegel modular form of weight $k$ with coefficients
in $R$, we mean an element in $H^0(\Aogn /R,\o^{\otimes k}/R)$.
The same notation is used for other congruence subgroups, but in
this paper we focus on Siegel modular forms with respect to
Siegel parahoric congruence subgroup $P$ and Borel congruence
subgroup $B$.

If we pull back the Hodge bundle $\o$ to the Siegel upper
half-space, the pull back is canonically isomorphic to
${\cal{O}}_{\H_g} \otimes_{\C} \wedge^g(\C^g)$. A complex modular
form of weight $k$ becomes an expression of the form
$f(\O).(dz_1\wedge...\wedge dz_g)^{\otimes k}$ where $f$ is an
$\Gamma^P(n)$-invariant complex holomorphic function on $\H_g$
which is holomorphic at $\infty$. For genus $\geq 2$ the
condition, holomorphic at infinity, is automatically satisfied by
Koecher principle. Trivializing $\o$ on $\H_g$, complex modular
forms of weight $k$ are identified with holomorphic functions
$f(\O)$ on $\H_g$, satisfying the transformation rule
$f|[\gamma]_k=f$ for all $\gamma \in \Gamma^P(n)$ where
$$
f|[\gamma]_k(\O)=\et(\gamma)^{gk/2}det(C.\O+D)^{-k}f(\gamma.\O).
$$

Let $l$ be a prime not dividing $pn$. To any Siegel modular form
$f$ of weight $k$ and level $n$, one associates an irreducible
admissible representation $\pi=\bigotimes \pi_{\ell}$ of
$GSp(2g,\Ad_f)$ over $\Q_l$ [Ca]. This association is not unique,
but we use it as a motivation to understand the notion of $p$-old
form. Let $U$ denote the open subgroup of $GSp(2g,\Ad_f)$
associated to the congruence group $\Gamma^P(n)$. If $(p,n)=1$ and
$\pi^U\neq 0$, then $\pi_p$ is spherical, and it is the unique
irreducible subquotient of some unramified principal series
representation $\pi_{\k}$ with respect to Borel subgroup
$B(\Q_p)$. One can show that $\pi_{\k}^{GSp(2g,\Z_p)}$ is
one-dimensional. So the number of copies of modular forms with
respect to $\Gamma^P(n)$ inside modular forms with respect to
$\Gamma^P(pn)$ is equal to the dimension of
$\pi_{\k}^{\Gamma^P(p)}$. The mod-$p$ Bruhat-Tits decomposition
implies that, we have the following decomposition
$$
GSp(2g,\Q_p)=\coprod_{\sigma\in (\Z/2\Z)^g \subset W_g}
B(\Q_p)\sigma\Gamma^P(p).
$$
So to specify $f\in \pi_{\k}^{\Gamma^P(p)}$ it is enough to
specify it on elements $\sigma\in (\Z/2\Z)^g$. Because
$W_{P}\simeq S_g$ and the subgroup $(\Z/2\Z)^g \subset W_G$ is a
complete set of representatives for $W_{P}\backslash W_G$.
Therefore, the space of $p$-old forms on ${\cal {A}}_g^P(pn)$
consists of $2^g$ copies of forms on ${\cal {A}}_g^P(n)$. The
vector space $\pi_{\k}^{\Gamma^P(p)}$ has a basis consisting of
functions $f_1,...,f_{2^g}$ where $f_i$ is supported on
$B(\Q_p)\sigma_i\Gamma^P(p)$ and $f_i(\sigma_i)=1$. The group of
involutions $(\Z/2\Z)^g\subset W_G$ acts on the space of $p$-old
forms by $f(z)\mapsto f^{\sigma}(z):=f(\sigma .z)$ for $\sigma\in
(\Z/2\Z)^g$.

Similar considerations show that we expect $g!2^g$ copies of forms
on ${\cal {A}}_g^P(n)$ inside the space of $p$-old forms on ${\cal
{A}}_g^{B,P}(p,n)$. Following Atkin-Lehner theory, we need a
geometric characterization of the space of $p$-old forms.

\subsection{Atkin-Lehner correspondences}

Let $\pi^{}_{T,P_I}:{\cal {A}}_g^T(p)\to {\cal {A}}_g^{P_I}(p)$
and $\pi^{}_{P_I,P_J}:{\cal {A}}_g^{P_I}(p)\to {\cal
{A}}_g^{P_J}(p)$ denote the natural projection maps induced by
inclusions of the corresponding congruence groups. Pulling back
forms from level $n$ to level $np$ using the natural projection
map
$$
\pi_n:{\cal {A}}_g^{B,P}(p,n)\rightarrow {\cal {A}}_g^P(n)
$$
induces the first copy of $p$-old forms in $H^0({\cal
{A}}_g^{B,P}(p,n) ,\o^{\otimes k})$. For simplicity, let us forget
the auxiliary level structure and consider the projection
$$
\pi_1:{\cal {A}}_g^{B}(p)\rightarrow {\cal {A}}_g
$$
and $p$-old forms in $H^0({\cal {A}}_g^B(p) ,\o^{\otimes k})$. At
first glance, it seems that geometric correspondences of the form
$D^{\sigma}_B(p)=\pi^{}_{T,B*} v^{\sigma *}_p\pi^{*}_{T,B}$ should
induce more copies of $p$-old Siegel forms on ${\cal {A}}_g^B(p)$
out of the pull-back copy.
$$
\begin{array}{ccc}
{\cal {A}}_g^T(p) & \buildrel {v^{\sigma}_p} \over \longrightarrow
& {\cal {A}}_g^T(p)\\
\downarrow &                     & \downarrow \\
{\cal {A}}_g^B(p) &                         & {\cal {A}}_g^B(p)
\end{array}
$$
But $\pi_{T,B}\circ \pi_1$ commutes with $v^{\sigma}_p$ for all
$\sigma\in W_G$. Therefore, correspondences of above type
generate the same copy as the pull back copy. To disturb the
symmetry of the picture, we use Atkin-Lehner involution.
$\pi_{T,B}\circ w^B_p \circ \pi_1$ no longer commutes with
$v^{\sigma}_p$ and we can hope that correspondences of the form
$C^{\sigma}_P(p)=\pi^{}_{T,B*} v^{\sigma
*}_p\pi^{*}_{T,B}w^{B*}_p$ could generate more copies of $p$-old
forms.
$$
\begin{array}{ccccccc}
& & {\cal {A}}_g^T(p) & \buildrel {v^{\sigma}_p} \over
\longrightarrow
& {\cal {A}}_g^T(p)& & \\
& & \downarrow &                     & \downarrow & & \\
& &{\cal {A}}_g^B(p) & & {\cal {A}}_g^B(p) & \buildrel {w^B_p}
\over \longrightarrow & {\cal {A}}_g^B(p)
\end{array}
$$
To generate $p$-old forms in $H^0({\cal {A}}_g^B(p) ,\o^{\otimes
k})$ we should use $g!2^g$ correspondences $C^{\sigma}_B(p)$ for
$\sigma\in W_G$.

\begin{mthm}
The linear subspaces of $H^0({\cal {A}}_g^{B,P}(p,n) ,\o^{\otimes
k})$ generated by Atkin-Lehner correspondences
$C^{\sigma}_B(p)\pi^*_n$ where $C^{\sigma}_B(p)$ is defined by
$\pi^{}_{T,B*} v^{\sigma *}_p\pi^{*}_{T,B}w^{P*}_p$
$$
\begin{array}{ccccccc}
& & {\cal {A}}_g^{T,P}(p,n) & \buildrel {v^{\sigma}_p} \over
\longrightarrow
& {\cal {A}}_g^{T,P}(p,n)& & \\
& & \downarrow &                     & \downarrow & & \\
& &{\cal {A}}_g^{B,P}(p,n) & & {\cal {A}}_g^{B,P}(p,n) & \buildrel
{w^B_p} \over \longrightarrow & {\cal {A}}_g^{B,P}(p,n)
\end{array}
$$
for $\sigma$ varying in $W_G$ give $g!2^g$ linearly independent
copies of $H^0({\cal {A}}_g^P(n) ,\o^{\otimes k})$ inside $p$-old
forms of level $pn$ living on ${\cal {A}}_g^{B,P}(p,n)$.
\end{mthm}

The corresponding theorem for $H^0({\cal {A}}_g^P(p) ,\o^{\otimes
k})$ can also be proved. However, in order to generate $p$-old
forms in $H^0({\cal {A}}_g^P(p) ,\o^{\otimes k})$ we should
divide the space of $p$-old forms on ${\cal {A}}_g^B(p)$ by the
action of $(\Z/2\Z)^{2g}\subset W_G$.

Main theorem is proved in a few stages. In the first stage, we
prove that Atkin-Lehner involution induces a second copy of
level-$n$ forms inside $p$-old part of level-$np$ forms which has
trivial intersection with the pull-back copy.
\begin{prop}
$\pi_{B,P_I}^*H^0({\cal {A}}_g^{P_I,P}(p,n) ,\o^{\otimes k})$ and
$w^{B*}_p\pi_{B,P_I}^*H^0({\cal {A}}_g^{P_I,P}(p,n) ,\o^{\otimes
k})$ as subspaces of $H^0({\cal {A}}_g^{B,P}(p,n) ,\o^{\otimes
k})$ have trivial intersection.
\end{prop}
\textbf{Proof.} Let $f$ and $g$ be nonzero Siegel modular forms in
$\pi_{B,P_I}^*H^0({\cal {A}}_g^{P_I,P}(p,n) ,\o^{\otimes k})$ with
$w^{B*}_pf=g$. Since $w^{B*}_p$ is an involution, $f\pm g$ are
eigenforms of $w^{B*}_p$ and the proposition follows from the
following
\begin{lem}
Any Siegel form which is eigenform of $w^{B*}_p$ on ${\cal
{A}}_g^{B,P}(p,n)$ and also pull back of a Siegel form on ${\cal
{A}}_g^{P_I,P}(p,n)$ vanishes if $P_I\subsetneqq B$.
\end{lem}
\textbf{Proof.} The zero locus of such an eigenform is pull back
of the zero locus of a form living on ${\cal {A}}_g^{P_I,P}(p,n)$
and also $w^{B*}_p$-invariant. This contradicts density of Hecke
orbit. $\square$

In the second stage, we show that Atkin-Lehner correspondences
induce $g!2^g$ non-intersecting copies of level-$n$ forms inside
$p$-old part of level-$np$ forms.

\begin{lem} Let $D^{\sigma}_B(p)=\pi^{}_{T,B_*} v^{\sigma
*}_p\pi^{*}_{T,B}$ where $\pi^{}_{T,B}:{\cal {A}}_g^{T,P}(p,n) \to
{\cal {A}}_g^{B,P}(p,n)$ is the natural projection. Then, for
$\sigma,\sigma'\in W_G$ the correspondences
$D^{\sigma}_B(p)D^{\sigma'}_B(p)$ and $D^{\sigma\sigma'}_B(p)$
acting on any linear subspace of $H^0({\cal {A}}_g^{B,P}(p,n)
,\o^{\otimes k})$ generate the same image subspaces.
\end{lem}
\textbf{Proof.} For simplicity, let us forget the auxiliary
$n$-level structure. Then, lemma follows from commutativity of
the following diagram
$$
\begin{array}{ccccccc}
{\cal {A}}_g^{T}(p)& \buildrel {v^{\sigma}_p} \over
\longrightarrow & {\cal {A}}_g^{T}(p) & \buildrel {v^{\sigma'}_p}
\over \longrightarrow
& {\cal {A}}_g^{T}(p)& & \\
\downarrow & & \downarrow &  & \downarrow & & \\
{\cal {A}}_g^{\sigma\sigma' B\sigma'^{-1}\sigma^{-1}}(p) &
\buildrel {\sigma} \over \longrightarrow & {\cal {A}}_g^{\sigma'
B\sigma'^{-1}}(p) & \buildrel {\sigma'} \over \longrightarrow &
{\cal {A}}_g^B(p) & &
\end{array}
$$
and, $v^{\sigma}_p\circ v^{\sigma'}_p=v^{\sigma\sigma'}_p $ and
$\sigma\circ \sigma'=\sigma\sigma'$ hold for $\sigma,\sigma'\in
W_G$.$\square$

\begin{prop}
Every two linear subspaces of $H^0({\cal {A}}_g^{B,P}(p,n)
,\o^{\otimes k})$ generated by correspondences
$C^{\sigma}_B(p)\pi^*_n$ acting on $H^0({\cal {A}}_g^{P}(n)
,\o^{\otimes k})$ for $\sigma\in W_G$ have trivial intersection.
\end{prop}
\textbf {Proof.} By the previous lemma, it is enough to show that
the linear subspaces generated by correspondences $\pi^{}_{T,B_*}
v^{\sigma *}_p\pi^{*}_{T,B}w^{B*}_p\pi_1^*$ and $w^{B*}_p\pi_1^*$
have such intersection. Suppose $\pi^{}_{T,B_*} v^{\sigma
*}_p\pi^{*}_{T,B}f=g$ for nonzero Siegel modular forms on ${\cal
{A}}_g^B(p)$ which are elements of $w^{B*}_p\pi_n^*H^0({\cal
{A}}_g^P(n) ,\o^{\otimes k})$. Let $d=deg(\pi^{}_{T,B})$. Then
$df\pm g$ are eigenforms of $\pi^{}_{T,B_*} v^{\sigma
*}_p\pi^{*}_{T,B}$ with eigenvalue $\pm d$. Now, the truth of
proposition is a consequence of the first stage and the following
\begin{lem}
Any nonzero Siegel modular form on ${\cal {A}}_g^{B,P}(p,n)$ which
is an eigenform of $\pi^{}_{T,B_*} v^{\sigma *}_p\pi^{*}_{T,B}$
for a non-zero $\sigma$ in $W_G$ is pull back of a Siegel form on
${\cal {A}}_g^{P',P}(p,n)$ where $P'$ is the parabolic subgroup
generated by $B$ and $\sigma B\sigma^{-1}$.
\end{lem}
\textbf {Proof.} This is a consequence of propositions 2.2 and
2.3. $\square$

In the final stage we show that the above-mentioned $g!2^g$
copies of level-$n$ Siegel forms inside the space of Siegel forms
of level $pn$ are indeed $p$-old forms of level-$pn$ and they are
linearly independent.
\begin{prop}
The Atkin-Lehner correspondences $C^{\sigma}_B(p)\pi^*_n$ commute
with all Hecke correspondences which generate the local Hecke
algebras $H_q$ for $q$ relatively prime to $pn$.
\end{prop}
\textbf {Proof.} The action of Atkin-Lehner correspondences can
be interpreted in terms of the $p$-torsion of abelian varieties
representing points of the moduli-space. By geometric base-change
one can see that such an action commutes with those interpreted
in terms of the $q$-torsion points.$\square$ \\\textbf{Proof of
the main theorem.} The $g!2^g$ copies of level-$n$ Siegel forms
induced by Atkin-Lehner correspondences are contained in the
space of $p$-old forms by previous proposition. Recall that in
this paper Siegel modular forms with eigenvalues repeated from
lover levels are considered to be old. The space of Siegel forms
on ${\cal {A}}_g^{B,P}(p,n)$ is finite-dimensional and has a basis
of all prime to $pn$ Hecke eigenforms. So is the case for any of
the $g!2^g$ Atkin-Lehner copies, by previous proposition. Fix a
basis for the space of Siegel forms of level $pn$ whose elements
are eigenforms of all prime to $pn$ local Hecke algebras. Suppose
the $g!2^g$ Atkin-Lehner copies are linearly dependent. It means
that a non-zero eigenform $f$ of all prime to $pn$ local Hecke
algebras is generated by some basis elements of the Atkin-Lehner
copies which have the same eigenvalues. Consider the vector space
$V$ of all Siegel forms with the same Hecke eigenvalues $f$ and
consider the $g!2^g$ Atkin-Lehner copies in it. $V$ is invariant
under the action of all the $g!2^g$ correspondences
$D^{\sigma}_B(p)=\pi^{}_{T,B_*} v^{\sigma *}_p\pi^{*}_{T,B}$ for
$\sigma\in W_G$, because these correspondences commute with the
Atkin-Lehner copies. By this symmetry, $df+\pi^{}_{T,P_*}
v^{\sigma *}_p\pi^{*}_{T,P} f$ is also a Hecke eigenform in $V$
with the same Hecke eigenvalues as $f$. There exists a $\sigma\in
W_G$ which gives a nonzero Siegel form $df+\pi^{}_{T,B_*}
v^{\sigma *}_p\pi^{*}_{T,B} f$ generated by the Atkin-Lehner
copies. But such a vector is a Siegel form which can be pulled
back from a lower Siegel space by proposition 2.3. This
contradicts linear dependency.$\square$

Having constructed the Atkin-Lehner copies of $p$-old forms on
${\cal {A}}_g^{B,P}(p,n)$ one can get the $2^g$ copies of $p$-old
forms on ${\cal {A}}_g^P(pn)$ by pushing forward all the $g!2^g$
$p$-old copies down to ${\cal {A}}_g^P(pn)$.

\begin{thm}
The linear subspaces of $H^0({\cal {A}}_g^P(pn) ,\o^{\otimes k})$
generated by Atkin-Lehner correspondences $C^{\sigma}_P(p)\pi^*_n$
for $\sigma$ varying in $W_G$ where $C^{\sigma}_P(p)$ is defined
by
$$
\sum \pi^{}_{T,P*} v^{\eta\sigma *}_p\pi^{*}_{T,B}w^{B*}_p
$$
where the sum ranges over $\eta\in S_g\subset W_G$ give $2^g$
linearly independent copies of $H^0({\cal {A}}_g^P(n)
,\o^{\otimes k})$ inside $p$-old forms of level $pn$ living on
${\cal {A}}_g^P(pn)$.
\end{thm}
\textbf {Proof.} This is a direct consequence of the main theorem
and proposition 2.2. $\square$

\section{Congruences between old and new forms}

We expect that the $2^g$ geometric correspondences we introduced,
generate the whole space of $p$-old forms. But we can't prove it,
since, neither we have a multiplicity one result available for
Siegel modular forms, nor we have any general local theory of old
forms developed for higher $GSp(2g)$. What we can do is to find
congruences between the vector space generated by the
Atkin-Lehner copies (which we artificially call "the space of
$p$-old forms") and its orthogonal complement with respect to
Weil-Peterson pairing ("new forms"). For this purpose we need an
Ihara type result.

\subsection{Ihara type result}

Let us discuss models of the Siegel moduli spaces and
Atkin-Lehner correspondences over $\Z[1/pn]$. ${\cal
{A}}^T_g(pn)$ and ${\cal {A}}^P_g(pn)$ have natural models over
$\Z[1/pn]$. Let $T(\Z/n\Z)$ denote the diagonal subgroup of
$Sp(2g,\Z/n\Z)$. The quotient $Sp(2g,\Z/n\Z)/T(\Z/n\Z)$ acts on
${\cal {A}}^T_g(pn)$ by acting on the $n$-level structures. The
quotient $\Gamma^P(n)/\Gamma^T(n)$ is a subgroup of
$Sp(2g,\Z/n\Z)/T(\Z/n\Z)$. We can think of ${\cal
{A}}^{T,P}_g(p,n)$ as equivariant etale quotient of ${\cal
{A}}^T_g(pn)$ under the action of $\Gamma^P(n)/\Gamma^T(n)$. This
action is defined over $\Z[1/pn]$. Therefore ${\cal
{A}}^{T,P}_g(p,n)$ also has a natural model over $\Z[1/pn]$.
Involutions $v^{\sigma}_p$ which are defined in terms of the
$p$-torsions of the universal variety can be extended to
involutions of ${\cal {A}}^{T,P}_g(p,n)$ over $\Z[1/pn]$. This
way we get a model for all Atkin-Lehner correspondences over
$\Z[1/pn]$.

Let $S$ be a scheme over $\mathbb {F}_q$, and $A\to S$ be a
semi-abelian scheme over $S$. Frobenius morphism induces a
homomorphism $Fr^*:e^*\Omega^1_{A/S}\to
(e^*\Omega^1_{A/S})^{(q)}$, where $e$ is the zero section and
$(e^*\Omega^1_{A/S})^{(q)}$ is the same as $e^*\Omega^1_{A/S}$,
except that the underlying $\cal O_S$-module structure is
combined with Frobenius. Therefore we get a homomorphism
$det(Fr^*):\o_{A/S}\to \o^{\otimes q}_{A/S}$. Since
$det({e^*\Omega^1_{A/S}}^{(q)} )=det(e^*\Omega^1_{A/S})^{\otimes
q}$, over $\mathbb {F}_q$ this morphism corresponds to a section
$h$ of $\o^{\otimes (q-1)}$. The section $h$ is called the Hasse
invariant.

\begin{thm}
Let $q\leq k+1$ be a prime number not dividing
$pn[\Gamma^P(p):\Gamma^T(p)]$. The linear subspaces of $H^0({\cal
{A}}_g^P(pn)_{/\F_q} ,\o^{\otimes k})$ generated by Atkin-Lehner
correspondences $C^{\sigma}_P(p)\pi^*_n$ for $\sigma$ varying in
$(\Z/2\Z)^{2g}\subset W_G$ give $2^g$ linearly independent copies
of $H^0({\cal {A}}_g^P(n)_{/\F_q} ,\o^{\otimes k})$.
\end{thm}
\textbf{Proof.} The three stages in the proof of main theorem
work line by line because everything can be interpreted in terms
of the torsions of the representing abelian varieties. One only
misses the local Hecke algebra $H_q$. The following lemma needs
more consideration.
\begin{lem}
Let $q$ be a prime number not dividing
$pn[\Gamma^B(p):\Gamma^T(p)]$ where $q-1\nmid k$. Any section of
the Hodge bundle $\o^{\otimes k}$ which is an eigenform of
$w^{B*}_p$ on ${\cal {A}}_g^{B,P}(p,n)_{/\F_q}$ and also pull back
of a form on ${\cal {A}}_g^{P_I,P}(p,n)_{/\F_q}$ vanishes if
$P_I\subsetneqq B$.
\end{lem}
\textbf{Proof.} We follow the idea of G. Pappas [Pa] to reduce
this step to proving density of a Hecke orbit of the type
discussed in [Ch]. The idea of using density results to get
cokernel torsion-freeness results first appeared in [Di-Ta]. Let
$g$ be as such. The zero locus of $g$ is $w^{B*}_p$-invariant.
This contradicts density of the Hecke orbit of an ordinary point
in the moduli space if the zero locus contain an ordinary point.

Now we show that if $g$ vanishes only on non-ordinary points, then
$k=m(q-1)$. There exists an embedding of $\mathbb {P}^1(\F_q)$ in
${\cal {A}}_g^{B,P}(p,n)_{/\F_q}$ such that the restriction of
$\o$ to the embedded $\mathbb {P}^1(\F_q)$ is ${\cal O}(q-1)$
(Look at [MB]). Therefore the tensor powers of the Hodge bundle
are non-trivial. Let $S\subset {\cal {A}}_g^{B,P}(p,n)_{/\F_q}$
and $\bar S\subset \bar {\cal {A}}_g^{B,P*}(p,n)_{/\F_q}$ denote
the subschemes defined by $h=0$ on the Siegel moduli space and its
minimal compactification. The subscheme $\bar S$ is codimension
one, because it does not contain any component of the boundary.
Also $\bar S$ is the reduced Zariski closure of $S$ [Ko]. $S$ is
geometrically irreducible [Oo] and so is $\bar S$. If $g$
vanishes only on points of $\bar S$ then for some positive integer
$m$ we have $g/h^m$ is a global nonzero section of $\o^{\otimes
k-m(q-1)}$. Existence of a nowhere vanishing modular form of
nonzero weight $k-m(q-1)$ implies triviality of $\o^{\otimes
k-m(q-1)}$ and hence triviality of ${\cal O}(k-m(q-1))$ on the
embedded $\mathbb {P}^1(\F_q)$. So we get $k=m(q-1)$ and $g=\l
h^m$ for some $\l\in \F_q$.$\square$

The above Ihara result can be used to obtain congruences between
Siegel modular forms. The idea of using Ihara lemma to get
congruences is due to Ribet [Ri]. We leave calculation of the
congruence kernel to elsewhere.

\section*{Acknowledgements}

For this research we have benefited from conversations with J. de
Jong, A. Genestier, G. Pappas, A. Rajaei, P. Sarnak, C. Skinner,
R. Takloo-bighash, J. Tilouine and A. Wiles. This is a polished
version of an old article [Ra]. We shall thank A. Genestier and J.
Tilouine heartily for many useful comments and questions on an
early version which lead us to correction of many mistakes.

\thispagestyle{plain}
\addcontentsline{toc}{chapter}{Bibliography}
\bibliographystyle{amsplain}
\bibliography{Bibliography}


TITLE: Distributions and Random Variables
QUESTION [2 upvotes]: I am confused about the relation between random variables and probability distributions in applications.
Let $\Omega$ be a probability space. Often, $\Omega$ is some measure space that is turned into a  probability space by a probability distribution. Assume that $\Omega$ is $\mathbb R$ with a probability distribution $f$. Let $X$ be a random variable on $\Omega$. Then, the image of $X$ can be thought of as a probability space by defining the probability of a measurable subset $Y$ in its image to be
$$
\int_{X^{-1}(Y)} X \cdot f.
$$
Furthermore, now that the image of $X$ is considered a probability space, there exists a probability distribution $g$ that defines it, i.e., for each measurable subset $Y$ in the image of $X$,
$$
\int_{X^{-1}(Y)} X \cdot f = \int_{Y} g.
$$
What is an example (a real-world application) of when one cares about each of these objects, i.e., when would one want to defined a probability distribution on $\mathbb R$, then a random variable and then the probability distribution on the image of that random variable?

REPLY [1 votes]: I think the essence of your question is the following:

What is the point of using random variables if everything will boil
  down to a distribution in the end?

My personal feeling is that in the limited situation you are describing (when the measure space is $\Omega=\mathbb R$ and only allowed to consider one random variable at a time taking values in $\mathbb R$) there is indeed not that much to be gained by using random variables, other than convenience of notation.
However, working with random variables becomes essential on $\mathbb R^n$ for $n>1$, and I bet you are already encountering this case in practical applications without realizing it. For instance, any time you want to consider/compare two random variables on the same probability space, you are actually working with a random element of $\mathbb R^2$. This is the idea behind the concept of coupling. Indeed, just to describe the marginal law of a random variable $X$ given the joint distribution $(X,Y)$ requires random variables.
This is the point in the post where I will start using notations from measure theory, the rigorous foundation for modern probability - see here if you are not familiar with this notation.
Going back to the previous example, to describe the marginal law of a random variable $X$ given its joint distribution $(X,Y)$ with some other random variable $Y$, one must make use of a (measurable) mapping between measure spaces - in this case, $\mathbb R^2$ equipped with the Borel $\sigma$-algebra and a measure (perhaps given by a density similar to how you described, or perhaps not) and a mapping to $\mathbb R$ given by projecting onto the first coordinate.
Typically in a first-year graduate course in probability, one starts by learning measure theory then transitioning into probability theory - with the transition point usually being the moment that independence is introduced. Of course, independence can be studied purely in the context of measure theory under the name of "product measures", but much of the intuition is lost without thinking in terms of random variables. And this point is also the moment when the power/usefulness of working with random variables, and not just with distributions, comes into its own.
Now to wildly extrapolate the essence of your question to a much broader context. Many classes of mathematical objects can be categorized in a systematic way - not surprisingly, called category theory - with probability being no exception. Categories have objects, and maps between objects (one famous example being the category whose objects are sets, and whose mappings are functions). One can consider a category whose objects are probability spaces $(\Omega,\mathcal F,\mathbb P)$, and whose mappings are random variables (more precisely, random elements - since people typically reserve "random variable" for the case when the codomain is $\mathbb R$). From this perspective, the reason we care about random variables is the same reason we care about mappings in any category: even if the most "tangible" part of a category is its objects, all the most interesting stuff happens in the mappings between them. This is why (in my opinion) probabilists seem to talk a lot more about random variables than about probability distributions.


TITLE: Means of Promoting Mathematics in Young Countries!
QUESTION [28 upvotes]: We all know mathematics is life, this question is for Mankind. It's mathoverflow here when some parts of the world we have mathunderflow! I think we can do something through ideas.  A  similar question "Good ways to engage in mathematics outreach" has been featured but this is a different question all together.
In this question, am interested in understanding what the so called super countries in mathematics  done right utilizing the little resources they have in the promotion(development) of mathematics. How can very young  country (mathematics development is wanting, the research output is low, some fundamental courses are not even taught at the first place due to lack of resource persons) move on? What are some of the ideas which have helped countries with a small economy grow? Are there mathematicians who have been involved in development of mathematics in developing/ less developed countries which are experiencing an upward trend? How did you do it?

REPLY [19 votes]: Dear Ongaro Nyang'
Although Israel is no longer so young, it was young, (even
younger than most other countries,) not so long ago, and
it is a small and rather isolated place, with some difficulties.
So some lessons from Israeli mathematics, especially in its early days
may be relevant.

A) Immigration

1. Immigration
Israeli mathematics was initially based on and had benefited all the
times from immigration of mathematicians to Israel. The ability to
absorb immigration, in general, and to attract and absorb immigrating scientists, in particular,
is crucial.(Keeping the relations with mathematicians who immigrated from
Israel is also an important issue.)

B) Financial matters.

Investing resources is, of course, crucial. There was a large public investment
in universities in Israel's first years even when the country itself
was in rather bad economic shape. Overall, theoretical academic
subjects like mathematics are "cheaper." Keeping the right balances
regarding policies for spending the money is very important.
Let me mention two items.
2. Sabbatical/Travel money
1.1 Good Sabbatical opportunities: Israeli mathematicians (and scientists
in general) had relatively good sabbatical terms which make it possible
for them to get (from the Israeli institution) a reasonable European/US
salary while in sabbatical abroad. This was especially effective when the
Israeli salaries were very low compared to salaries abroad. The academic
system is built on 15% or so of the faculty being on sabbatical at
any time. (Often people spend additional time abroad on leave.)
2.2 In addition, Israeli scientist (with academic university positions) and graduate students (who works as T.A.'s)  have funds for short-term
travels: (Fixed amounts depending on the academic rank per year) This
enable participation in conferences and joint research.
Both the sabbatical and travel money apply to all people with academic
positions (Sabbatical only to lecturer/asst professor position and above)
and are essentially automatic. (Minimal amount of bureaucracy, no
committees to judge qualification and to decide on amounts, no requirement
to be an invited speaker in a conference, etc. etc..)
3. Salaries
Keeping the right balances when it comes to salaries is also important.
Low salary gives incentives to leave but very high salaries (compared to
average salaries in the country) are morally problematic and may
give incentive for corrupting the hiring and promoting system.
The salary system in Israel is based on essentially equal salary for equal
academic rank and (overall) there are no substantial salary awards for
academic excellence beside the academic rank. (There is some (modest)
awards for people getting external grants and larger but still rather small
for people serving in administrative positions.)
I think that not having overly differential salaries and not
being "in the game" of offers and counteroffers is actually beneficial.

C) Activities

4. Activities for national math society
There are, since early times,
regular annual meeting of the Israeli Mathematics Union and some
other local activities.
5. National mathematical journal
There was a substantial effort, again from early times,
to create and maintain an Israeli journal of
mathematics. (In the 40th there was even a professional research level
journal in Hebrew for a few years.)
6. Conferences and visitors.
There was, again from early times, some resources devoted to conferences
and visitors. Carefully administered and with attention to
the added value for the local people this can be very fruitful.
Arranging visits of top people in mathematics for lecture series and visits
can also be useful. It seems that it is a good policy (when the country is not
rich)  not to over-pay for such visitors (among other reasons, because this set standards which push the cost of visitors in general too high.)
Of course, warm hospitality is priceless.

D) Content

7. Maintaining a sense of tradition.
Basing activities on areas with long tradition of success which are
identified with the country's mathematical strength can be a successful
and well excepted by the whole mathematical community.
8. Self-breeding can work
The success of Israeli mathematical departments was largely based on
successful self-breeding, namely absorbing as faculty member people who
graduated at the department.
9. Self-confidence, Tolerance for "sporadic"(or non-main stream) areas
(and tolerance in general)
This seems to me a strength of Israeli mathematics and looks (to me) a
good attitude especially for a peripheral and somewhat isolated place.
Tolerance is important especially since mathematical quality is rather
high dimensional (some dimensions being
importance/depth/visibility/applicability/usefulness.)
(There is also
complete tolerance and essentially indifference in the context of
mathematical life towards matters of politics, attitude towards religion,
etc, issues that Israel is very torn apart about.)
10. Patience, realistic goals, unrealistic dreams
Building a good mathematical activity takes time, and there are
ups and downs as well.

E) Outreach

11. Issues concerning high school mathematics
There is some efforts to promote interest in mathematics among gifted high
school students: special mathematical journal (in Hebrew), some "clubs"
and "summer camps" math Olympiads etc. I think this had some factor in
promoting math among young people.  Usually, what it takes is some
mathematician in academics which is devoted to this issue and some (not
large) budget. Giving an incentive for such an activity and such a
mathematician may be a good idea. Popular Math books and text books in
Hebrew had substantial influence. A. Frankel (the set theorist)
wrote  wonderful five-volume Hebrew books introducing
mathematics (It was called "An Introduction to Mathematics") in the 40s/50s.
(More recently, the Hebrew edition of Singh's
book on Wiles proof of FLT increased popularity of mathematics.)

F) Relations with other areas

12. Relation with CS, physics and other disciplines
In Israel CS department were largely built out of math departments (not
electrical engineering) and there are still strong academic ties between
these communities. Connection with CS seems valuable. Of course, relations with physics are very important (not so strong in Israel).  Relations between math and economics seems strong in Israel.
(In Jerusalem there is an interdisciplinary "center for
rationality" which involves people from math/economics and some from
statistics/psychology/philosophy/law/biology.)
In summary, when it comes to mathematical life in a small somewhat isolated and
at times a bit troubled place, it seems that it is valuable to make the
right balances in the local mathematical community between competition and
solidarity, to practice a lot of patience and tolerance, to be open to new people and new directions, and to be careful about incentives.
Like in Brazil the efforts of few pivotals mathematicians was very crucial.
As usual, luck is useful too. Good luck.

REPLY [6 votes]: As it was said in an other answer it seems like a good idea to send students abroad to study at strong places and have them come back, or if they don't, keep contact, have them give courses in their home country etc.
A scientifically excellent place helping with precisely such a policy is the ICTP in Trieste, see the "Education and Training" and the "Research "sections of their website (they are called institute of theoretical physics, but they have a strong mathematics section). Concretely they provide training programmes for undergrads to prepare them for a PhD - see here - and they also offer short and longer term stays for working scientists from developing countries, as well as institution partnerships, see here.


TITLE: Is any $C_c^\infty(\mathbb R)$-function the product of two $C_c^\infty(\mathbb R)$-functions?
QUESTION [2 upvotes]: Let $f\in C_c^\infty(\mathbb R)$. Are we able to write $$f=gh$$ for some $g,h\in C_c^\infty(\mathbb R)$? Unfortunately, I've no idea how to I could prove or disprove this.

REPLY [5 votes]: There exists a smooth compactly supported function $g$ that is $1$ on an open ball containing the support of $f$. Then $f=fg$.
For the definition of $g$: let $h(r)=0$ if $r \leq 0$ and $h(r)=e^{-1/r}$ if $r > 0$. It is well-known that $h$ is smooth. Let $h_1(r)=\frac{h(r)}{h(r)+h(1-r)}$: $h_1$ is smooth, zero on $(-\infty,0]$ and $1$ on $[1,\infty)$.
Take now $g(x)=h_1\left(2-\frac{|x|^2}{R^2+1}\right)$ where $\{f \neq 0\} \subset \mathcal{B}(0,R)$.