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\par\noindent
{Preprint (\today), Version 0.5}
\vspace{15ex}

\par\noindent
{\bf \LARGE Component groups of optimal quotients\\
of Jacobians}\\
\vspace{3ex}
\par\noindent
{\large W.A. Stein}\\
{\small Department of Mathematics, University of California, Berkeley,
CA 94720, USA}

\section*{Introduction}
Let $A$ be an abelian variety over a finite extension $K$ of the
$p$-adic numbers $\Qp$.  Let $\O$ be the ring of integers of $K$,
$\m$ its maximal ideal and $k=\O/\m$ the residue class field.
The N\'{e}ron model of $A$ is a smooth commutative group scheme $\A$
over $\O$ such that $A$ is its generic fiber and satisfying the
property:\\
the restriction map
$$\Hom_\O(S,\A)\lra \Hom_K(S/K,A)$$
is bijective for all schemes $S$ over $\O$.
The special fiber $\A_k$ is a group scheme over $k$,
which need not be connected.  Denote by
$\A_k^0$ the connected component containing the identity.
There is an exact sequence
$$0\lra\A_k^0\lra\A_k\lra\Phi_A\lra0$$
with $\Phi_A$ a finite \'{e}tale group scheme over $k$,
i.e., a finite abelian group equipped with an action of
$\Gal(\kbar/k)$.

In this paper we study the group $\Phi_A$
with particular emphasis on quotients $A$ of Jacobians
of modular curves $X_0(N)$.  When $A$ has semistable reduction
Grothendieck described the component group in terms
of a monodromy pairing on certain free abelian groups.
When $A=J$ is the Jacobian of $X_0(N)$, this pairing
can be explicitly computed, hence so can $\Phi_J$,
as has been done in many cases in
\cite{mazur:eisenstein} and \cite{edixhoven:eisen}.
Suppose now that $A$ is a simple quotient of $J$ and
that the kernel of the map $J\ra A$ is connected.
There is a natural map $\Phi_J\ra \Phi_A$.  In this
paper we give a formula which can be used to compute
the image and the order of the cokernel.

We now state our main result in more precise language.
Suppose $\pi:J\ra A$ is an optimal quotient,
with $J$ a semistable Jacobian and $A$ purely toric.
We express the component group of $A$ in terms of
the monodromy pairing associated to $J$.
Let $m_A=\sqrt{\deg(\theta_A)}$ where
$\theta_A:A'\ra A$ is induced by
the canonical principal polarization of $J$.
Let $X_J$ be the character group of the toric part
of the special fiber of $J$.  Let $\L$ be the saturation
of the image of $X_A$ in $X_J$.  The monodromy pairing
defines a map $\alp:X_J\ra \Hom(\L,\Z)$.
Let $\Phi_X$ be the cokernel of $\alp$ and
$m_X=[\alp(X_J):\alp(\L)]$ be the order of the finite
group $\alp(X_J)/\alp(\L)$.   We prove that
$$\frac{|\Phi_A|}{m_A} = \frac{|\Phi_X|}{m_X}.$$
More precisely, $\Phi_X$ is the image of $\Phi_J \ra \Phi_A$
and the cokernel of $\Phi_J \ra \Phi_A$ has order $m_A/m_X$.
If the optimal quotient $J\ra A$ arises from a modular
form on $\Gamma_0(N)$,
then the quantities $m_A$, $m_X$ and $\Phi_X$ can
be explicitly computed, hence so can $|\Phi_A|$.
Having done this, we present some tables and conjectures
which they suggest.

{\bf Acknowledgement: }
I am deeply grateful for conversations with
A. Agashe, R. Coleman, B. Edixhoven, D. Lorenzini,
B. Mazur, L. Merel, K. Ribet, and S. Takahashi.

\section{Optimal quotients of jacobians}
Let $J$ be a Jacobian equipped with its canonical principal
polarization $\theta_J$.
An {\bf optimal quotient} of $J$ is an
abelian variety $A$ and a surjective
map $\pi: J \ra A$ whose
kernel is an abelian subvariety $B$ of $J$.
Denote by $J'$ and $A'$ the abelian varieties dual to $J$ and $A$,
respectively.
Dualizing $\pi$ and composing with $\theta_J'=\theta_J$
we obtain a map
$A'\xrightarrow{\pi'} J'\xrightarrow{\theta_J} J$.
\begin{proposition}
$A'\ra J$ is injective.
\end{proposition}
\begin{proof}
Since $\theta_J$ is an isomorphism it suffices to prove
that $\pi'$ is injective.
Since the dual of $\pi'$ is $(\pi')'=\pi$ and $\pi$ is surjective,
$\pi'$ must have finite kernel.
Thus $A' \ra C=\im(\pi')$ is
an isogeny.  Let $G$ denote the kernel,
and dualize. By \cite[\S11]{milne:abvars} we have have
$$\xymatrix{ G\ar[r] & A'\[email protected]{->>}[r] \ar[dr]_{\pi'} & C\ar[d]\\ && J' }\qquad \quad \xrightarrow{\textstyle \text{dualize}} \quad \qquad \xymatrix{ A & C\ar[l] & G'\ar[l] \\ & J\ar[u]_{\vphi}\ar[ul]^{\pi} }$$
with $G'$ the Cartier dual of $G$.
Since $G'$ is finite, $\ker(\vphi)\subset\ker(\pi)$
is of finite index.
Since $\ker(\pi)$ is an abelian  variety it is divisible.
Thus $\ker(\vphi)=\ker(\pi)$ so $G'=0$ and hence $G=0$.
\end{proof}
We denote the map $A'\ra J$ by $\pi'$.
The kernel of $\theta_A$ measures the intersection of
$A'$ and $B=\ker(\theta_A)$ inside of $J$
as shown in the following diagram.
$$\xymatrix{ A'\intersect B\ar[r]\ar[d] & B\ar[d] \\ A'\[email protected]{^(->}[r]^{\pi'}\ar[dr]_{\theta_A} & J\ar[d]^\pi\\ & A }$$

Because $\theta_A$ is a polarization, $|\ker(\theta_A)|$ is
a square \cite[Theorem 13.3]{milne:abvars}.   The
{\bf congruence modulus} is the integer
$$m_A=\sqrt{|\ker(\theta_A)|}.$$

\section{The special fiber of the N\'{e}ron model}
Let $K$ be a finite extension of $\Qp$ with ring of integers $\O$
and residue class field $k$.
Let $A/K$ be an abelian variety and denote its N\'{e}ron model
by $\cA$.
Let $\Phi_A$ be the group of connected components of
the special fiber $\cA_k$. This group
is a finite \'{e}tale group scheme over $k$, i.e.,
a finite abelian group equipped with an action of
$\Gal(\kbar/k)$. There is an exact sequence of group schemes
$$0 \ra \cA_k^0 \ra \cA_k \ra \Phi_A \ra 0.$$
The group $\cA_k^0$ is an extension of an abelian variety
$\cB$ of dimension $a$ by the product of a torus $\cT$ of
dimension $t$ and of a unipotent group $\cU$ of dimension $u$:
$$0\ra \cU\cross \cT \ra \cA_k^0 \ra \cB \ra 0.$$
The abelian variety $A$ is said to have {\bf purely toric} reduction
if $t=\dim A$, and is {\bf semistable} if $u=0$.
The character group $X_A = \Hom(\cT,\Gm)$ is a free abelian
group of rank $t$ contravariantly associated to $A$.
If $A$ is semistable there is a monodromy pairing
$X_A\cross X_{A'}\ra \Z$ and an exact sequence
$$0 \ra X_{A'} \ra \Hom(X_{A},\Z) \ra \Phi_A \ra 0.$$
\comment{
Suppose again that $J\ra A$ is a symmetric optimal quotient, that $J$
has semistable reduction and $A$ has purely toric reduction.
Since $J$ is canonically self-dual the monodromy pairing
defines a map $X=X_J \ra \Hom(X,\Z)$.  By functoriality there
is a map $X\ra X_{A'}$.
Let $\alpha:X\ra \Hom(X_{A'},\Z)$ be the resulting map.
Let $L \subset X$ be the saturation of the image of $X_{A}$
in $X$, i.e.,the image of $X_{A}$ in $L$ has finite index
and $X/L$ is torsion free.  The {\bf character group congruence modulus}
of the optimal quotient $J\ra A$ is the integer
$$m_x=[\alpha(X):\alpha(L)].$$
We now state our main result.
$$|\Phi_A| = |\image(\Phi_J\ra\Phi_A)|\cdot \frac{m_\theta}{m_x}$$
This is an expression for the cokernel of the map $\Phi_J\ra \Phi_A$
as the quotient of two congruence moduli.}

\section{Rigid uniformization}
In this section we review the rigid analytic uniformization of
a semistable abelian variety over a finite extension $K$ of the
maximal unramified extension $\Qp^{\ur}$ of $\Qp$.   We use this
to prove that if $A$ is purely toric, and $\phi:A'\ra A$ is an
isogeny, then
$$\deg(\phi) = |\coker(X_A\ra X_{A'})|^2.$$
We also prove a few lemmas about the character groups $X_A$.

\subsection{Raynaud-van der Put uniformization}\label{subsec:raynaud}

\begin{theorem}[Raynaud, van der Put]\label{raynaud}
If $A$ is a semistable Abelian variety, its universal
covering is isomorphic to an extension $G$ of an abelian
variety $B$ with good reduction by a torus $T$, the
covering map from $G$ to $A$ is a homomorphism and
its kernel is a twisted free Abelian group $\Gamma$ of finite rank.
\end{theorem}
This may be summarized by the following diagram,
$$\xymatrix{ &\Gamma\ar[d] \\ T\ar[r] & G\ar[r]\ar[d] & B\\ & A }$$
which we call the {\bf uniformization cross} of $A$.
The group $\Gamma$ can be identified with the character
group $X_A$ of the previous section.
The uniformization cross of the dual abelian variety
$A'$ is
$$\xymatrix{ &\Gamma'\ar[d] \\ T'\ar[r] & G'\ar[r]\ar[d] & B'\\ & A' }$$
where $\Gamma'=\Hom(T,\Gm)$ and $T'=\Hom(\Gamma,\Gm)$
and the morphisms $\Gamma'\ra G'$ and $T'\ra G'$ are the
one-motive duals of the morphisms $T\ra G$ and $\Gamma\ra G$,
respectively.
For more details see \cite{coleman:monodromy}.

\begin{example}[Tate curve]
If $E/\Qp$ is an elliptic curve with multiplicative reduction
then the uniformization is $E=\Gm/q^\Z$ where $q=q(j)$ is
obtained by inverting the expression for $j$ as a function of
$q(z)=e^{2\pi iz}$.
\end{example}

\subsection{Some lemmas}
Let $\pi:J\ra A$ be an optimal quotient,
with $J$ semistable and $A$ purely toric.
\begin{lemma}\label{lem:surj}
The map $\Gamma_J\ra \Gamma_A$ induced by $\pi$ is surjective.
\end{lemma}
\begin{proof}
Because $G_J$ is simply connected,
$\pi$ induces a map
$G_J\ra T_A$ and a map $\Gamma_J\ra \Gamma_A$.
Because $\pi$ is surjective and $T(A)$ is a
torus, the map $G_J\ra T_A$ is surjective.
The snake lemma applied to the following diagram gives
a surjective map from $B=\ker(\pi)$ to
$M=\coker(\Gamma_J\ra\Gamma_A)$.
$$\xymatrix{ & \Gamma_J\ar[r]\ar[d] & \Gamma_A\ar[r]\ar[d] & M\ar[r]\ar[d]& 0 \\ & G_J\ar[r]\ar[d] & T_A\ar[d]\ar[r] & 0 \\ B\ar[r] & J\ar[r]^{\pi}& A }$$
Because $\pi$ is optimal, $B$ is connected so $M$ must also be connected.
Since $M$ is discrete it follows that $M=0$.
\end{proof}

\subsection{Purely toric abelian varieties}
Assume that $A$ is purely toric. Then
$B=0$, and  the uniformization cross becomes
$$\xymatrix{\Gamma\ar[d]\\ T\ar[d] \\ A}$$
Let $\vphi:A'\ra A$ be a {\bf symmetric isogeny}, i.e.,
$\vphi':A'\ra (A')'=A$ is equal to $\vphi$.
Denote by $\vphi_t:T'\ra T$ and $\vphi_a:\Gamma'\ra\Gamma$ the
induced maps.
\begin{proposition}\label{prop:kerphi}
There is an exact sequence
$$0\ra\ker(\vphi_t) \ra \ker(\vphi) \ra \coker(\vphi_a)\ra 0,$$
and $\ker(\vphi_t)$ is dual to $\coker(\vphi_a)$.
\end{proposition}
\begin{proof}
Since $\vphi$ is an isogeny we have the following diagram:
$$\xymatrix{ 0\ar[r]\ar[d] & \Gamma'\ar[r]\ar[d] & \Gamma\ar[r]\ar[d] & \coker(\vphi_a)\ar[d]\\ \ker(\vphi_t)\ar[r]\ar[d]& T'\ar[d]\ar[r] & T\ar[r]\ar[d] & 0\\ \ker(\vphi)\ar[r] & A'\ar[r]^{\vphi} & A}$$
The snake lemma then gives the claimed exact sequence.
For the second assertion observe that the one-motive dual of the diagram
$$\xymatrix{ & \Gamma'\ar[r]\ar[d] & \Gamma\ar[d]\ar[r] & \coker(\vphi_a) \\ \ker(\vphi_t)\ar[r] & T'\ar[r]^{\vphi} & T}$$
is the diagram
$$\xymatrix{ & T & T'\ar[l]_{\vphi'} & \coker(\vphi_a)'\ar[l]\\ \ker(\vphi_t)'& \Gamma\ar[l]\ar[u] &\Gamma'\ar[l]\ar[u] }$$
Since $\vphi$ is symmetric, $\vphi'=\vphi$ and so
$$\ker(\vphi_t) = \coker(\vphi_a)'.$$
\end{proof}

\begin{lemma}\label{lem:isogcoker}
$|\ker(\vphi)|=|\coker(\vphi_a)|^2$
\end{lemma}
\begin{proof}
The order of a finite group scheme equals the order of its
dual.
\end{proof}

\section{The main theorem}
Let $\pi:J\ra A$ be an optimal quotient,
with $J$ a semistable Jacobian and $A$ purely toric.
Let $X_A$, $X_{A'}$, and $X_J$ denote the
character groups of the toric parts of the
special fibers.

\subsection{Monodromy description of the component group}
There is a pairing
$X_A\cross X_{A'}\ra \Z$ called
the monodromy pairing.  We have an exact sequence
$$0 \ra X_{A'} \ra \Hom(X_{A},\Z) \ra \Phi_A \ra 0.$$
If $J$ is a Jacobian then $J$ is canonically self-dual so
the monodromy pairing on $J$
can be viewed as a pairing $X_J\cross X_J \ra \Z$ and
there is an exact sequence
$$0 \ra X_{J} \ra \Hom(X_J,\Z) \ra \Phi_J \ra 0.$$

\begin{example}[Tate curve]
Suppose $E=\Gm/q^{\Z}$ is a Tate curve.
The monodromy pairing on $X_E=q^{\Z}$ is
$$\langle q, q\rangle = \ord_p(q)=-\ord_p(j).$$
Thus $\Phi_E$ is cyclic of order $-\ord_p(j)$.
\end{example}

\subsection{Proof of the main theorem}
We now prove the main theorem.
The key diagrams are
$$\[email protected]=3pc{A' \[email protected]{^(->}[r]^{\pi'}\ar[dr]_{\theta} & J \[email protected]{->>}[d]^{\pi}\\ &A} \qquad\qquad\qquad \[email protected]=3pc{X_A \[email protected]{^(->}[r]^{\pi^*} \ar[dr]^{\theta^*} & X_J \[email protected]{->>}[d]^{\pi_*} \\ & X_{A'}\[email protected]/^1.5pc/[ul]^{\theta_*}}$$
The surjectivity of $\pi_*$ was proved in Lemma~\ref{lem:surj}.
The injectivity of $\pi^*$ follows because
$$\theta_*\pi_*\pi^*=\theta_*\theta^*=\deg(\theta)\neq 0,$$
and multiplication by $\deg(\theta)$ on a free abelian
group is injective.

Let
$$\alp:X_J\ra \Hom(\pi^* X_A,\Z)$$
be the map defined by the monodromy pairing restricted
to $X_J\cross \pi^* X_A$.
\begin{lemma}\label{lem:twokers}
$\ker(\pi_*) = \ker(\alp)$
\end{lemma}
\begin{proof}
Suppose $x\in \ker(\pi_*)$ and let $y=\pi^* z$ with
$z\in X_A$.  Then
$$\langle x, y \rangle = \langle x, \pi^* z \rangle = \langle \pi_* x, z \rangle = 0$$
so $x\in\ker(\alp)$.
Next let $x\in\ker(\alp)$.
Then for all $z\in X_A$,
$$0 = \langle x, \pi^* z\rangle = \langle \pi_* x, z\rangle$$
so $\pi_* x$ is in the kernel of the
monodromy map
$$X_{A'} \ra \Hom(X_A,\Z).$$
Since $X_{A'}$ and $\Hom(X_A,\Z)$ are free of the same rank
and the cokernel is torsion, the monodromy map is injective.
Thus $\pi_* x=0$ and $x\in\ker(\pi_*)$.
\end{proof}

\begin{lemma}\label{lem:compphi}
There is an exact sequence
$$X_J \ra \Hom(\pi^* X_A,\Z) \ra \Phi_A \ra 0.$$
\end{lemma}
\begin{proof}
Lemma~\ref{lem:twokers} gives the following
commutative diagram with exact rows
$$\xymatrix{0\ar[r] & X_J/\ker(\alp)\ar[d]^{\isom} \ar[r] & {\Hom(\pi^* X_A,\Z)}\ar[r] \ar[d]^{\isom} & \coker(\alp)\ar[r]\ar[d] & 0\\ 0\ar[r] & X_{A'}\ar[r] & {\Hom(X_A,\Z)}\ar[r] & {\Phi_A}\ar[r] & 0}$$
By Lemma~\ref{lem:twokers}, the first vertical map is an isomorphism.
The second is an isomorphism because it is induced by the
isomorphism $\pi^*:X_A\ra \pi^* X_A$.  It follows that
$\coker(\alp)\isom \Phi_A$, as claimed.
\end{proof}

Let $\L$ be the {\bf saturation} of $\pi^* X_A$ in $X_J$, i.e.,
$[\L:\pi^*X_A]$ is finite and $X_J/\L$ is torsion free.
Suppose $L$ is of finite index in $\L$.
Define the {\bf congruence modulus} of $L$
$$m_L = [\alp(X_J):\alp(L)]$$
and the {\bf component group} by
$$\Phi_L = \coker( X_J \ra \Hom(L,\Z)).$$
When $L=\L$ we often set $m_X=m_\L$ and $\Phi_X=\Phi_\L$
and think of $m_X$ and $\Phi_X$ as the character group
congruence modulus and component group.''

\begin{lemma}\label{lem:homog}
The rational number $\ds \frac{|\Phi_L|}{m_L}$ does not
depend on the choice of $L$.
\end{lemma}
\begin{proof}
If $L'$ is another choice let $n=[L:L']\in\Q$.
Then since $\alp$ is injective when restricted to $\L$,
$$m_{L'} = [\alp(X_J):\alp(L')] = [\alp(X_J):\alp(L)]\cdot[\alp(L):\alp(L')] = m_L\cdot n$$
and similarly $|\Phi_{L'}| = |\Phi_L|\cdot n$.
\end{proof}

Recall that we defined
\begin{eqnarray*}
m_A &=& \sqrt{\deg(\theta)}\\
\Phi_A &=& \coker(X_{A'}\ra \Hom(X_A,\Z))
\end{eqnarray*}

\begin{theorem}\label{formula}
For any $L$ of finite index in $\L$
the following relation holds:
$$\frac{|\Phi_A|}{m_A} = \frac{|\Phi_L|}{m_L}.$$
\end{theorem}
\begin{proof}
By Lemma~\ref{lem:homog} we may assume that $L=\pi^*X_A$.
With this choice of $L$, Lemma~\ref{lem:compphi} says that
$\Phi_L \isom \Phi_A$.
By Lemma~\ref{lem:twokers}, properties of the index,
and Lemma~\ref{lem:isogcoker} we have
\begin{eqnarray*}
m_L&=&[\alp(X_J):\alp(L)] \\
&=& [\pi_*(X_J):\pi_*(L)]\\
&=& [X_{A'}:\pi_*(\pi^*X_A)]\\
&=& [X_{A'}:\theta^* X_A]\\
&=& \coker(\theta^*) \\
&=& \sqrt{\deg(\theta)} = m_A.
\end{eqnarray*}
\end{proof}

\begin{proposition}\label{prop:compim}
$$\image(\Phi_J\ra\Phi_A) \isom \Phi_\L.$$
\end{proposition}
\begin{proof}
Because $\pi^*X_A\subset \L \subset X_J$, by
Lemma~\ref{lem:compphi} we obtain a commutative diagram
with exact rows
$$\xymatrix{ X_J\ar[r]\[email protected]{=}[d]& \Hom(X_J,\Z)\ar[r]\ar[d]& \Phi_J \ar[r]\ar[d]& 0\\ X_J\ar[r]\[email protected]{=}[d]& \Hom(\L,\Z)\ar[r]\ar[d]& \Phi_\L \ar[r]\ar[d] & 0\\ X_J\ar[r]& \Hom(\pi^*X_A,\Z)\ar[r]& \Phi_A \ar[r]& 0 }$$
The map $\Hom(\L,\Z)\ra\Hom(\pi^*X_A,\Z)$ is an isomorphism
so $\Phi_\L\ra\Phi_A$ is injective, hence
$$\image(\Phi_J\ra\Phi_A) \isom \image(\Phi_J\ra \Phi_\L).$$
The cokernel of $\Hom(X_J,\Z)\ra\Hom(\L,\Z)$
surjects onto the cokernel of $\Phi_J\ra \Phi_\L$.
Using the exact sequence
$$0\ra \L \ra X_J \ra X_J/\L \ra 0,$$
we find that
$$\coker(\Hom(X_J,\Z)\ra\Hom(\L,\Z)) \subset \Ext^1(X_J/\L,\Z)=0,$$
where $\Ext^1$ vanishes because $\L$ is saturated
so that $X_J/\L$ is torsion free.  Thus the cokernel of
$\Phi_J\ra\Phi_\L$ is $0$, from which the proposition follows.
\end{proof}

The following corollary
follows from Theorem~\ref{formula} and Proposition~\ref{prop:compim}.
\begin{corollary}\label{cor:div}
$$|\coker(X_J\ra X_A)| = \frac{m_A}{m_\L}.$$
As a consequence, $m_\L | m_A.$
\end{corollary}

\section{Optimal quotients of $J_0(N)$}
Let $X_0(N)/\Q$ be the modular curve associated to the congruence
subgroup $\Gamma_0(N)\subset\sltwoz$ of matrices which are upper
triangular modulo $N$.  Let $p$ be a prime divisor of $N$ which is
coprime to $M=N/p$.  The Jacobian $J=J_0(N)$ of $X_0(N)$ has semistable
reduction at $p$.   The Hecke algebra
$$\T=\Z[\ldots T_n\ldots]\subset\End(J)$$
is a commutative ring of endomorphisms of $J$ of $\Z$-rank $=\dim J$.
The character group $X_J$ is equipped with a
functorial action of $\T$.
The Hecke algebra $\T$ also act on the cusp
forms $$S = S_2(\Gamma_0(N),\C).$$
A newform $f$ is an eigenform normalized so that the coefficient
of $q$ in the expansion of $f$ at the cusp $\infty$ is $1$, and
such that $f$ does not occur at any level $N'\mid N$ with $N'\neq N$.
If $f$ is a newform, let $I_f$ be the ideal in $\T$ of
elements which annihilate $f$.  Then $\O_f=\T/I_f$ is an
order in the ring of integers of the totally real number field
$K_f$ obtained by adjoining the Fourier coefficients of $f$ to $\Q$.
The quotient
$$A_f = J_0(N)/ I_f J_0(N)$$
is a purely toric optimal quotient of dimension $[K_f:\Q]$.

Let $H=H_1(X_0(N),\Z)$ be the integral homology of the
complex algebraic curve $X_0(N)$.  Integration defines a
$\T$-equivariant nondegenerate
pairing  $S \cross H \ra \C$ which we view as a map
$\alp: H \ra \Hom_\C(S,\C)$.

\begin{theorem}\label{Af}
We have the following commutative diagram of $\T$-modules:
$$\xymatrix{ H[I_f] \[email protected]{^(->}[r]\[email protected]{^(->}[d] & H \[email protected]{->>}[r]\[email protected]{^(->}[d] & \alp(H)\ar[d]\[email protected]{^(->}[d]\\ \Hom_\C(S,\C)[I_f]\[email protected]{^(->}[r]\[email protected]{->>}[d] & \Hom_\C(S,\C)\[email protected]{->>}[r]\[email protected]{->>}[d] &\Hom_\C(S[I_f],\C)\[email protected]{->>}[d]\\ A_f'(\C)\[email protected]{^(->}[r]\[email protected]/_2pc/_{\theta_A}[rr] & J(\C) \[email protected]{->>}[r] & A_f(\C) \\ }$$
\end{theorem}
\begin{proof}
This can be deduced from \cite{shimura:factors}.
\end{proof}

\begin{corollary}\label{moduluscomp}
$m_A^2 = [\alp(H) : \alp(H[I_f])]$.
\end{corollary}
\begin{proof}
Recall that $m_A$ is defined to be $\sqrt{\deg(\theta_A)}$.
The kernel of an isogeny of complex tori is
isomorphic to the cokernel of the induced map
on lattices.  The corollary now follows from
the diagram of Theorem~\ref{Af}
which indicates that the index $[\alp(H):\alp(H[I_f])]$
is the cokernel of the map $H[I_f]\ra \alp(H).$
\end{proof}

Let $\Frob_p:X_J\ra X_J$ denote the map induced by Frobenius.
One has $\Frob_p=-W_p$, where $W_p$ is the map induced
by the Atkin-Lehner involution on $J_0(p)$.
Let $f$ be a newform, $A=A_f$ the corresponding optimal
quotient, and $w_p$ the sign of the eigenvalue of
$W_p$ on $f$.
\begin{proposition}
$$\Phi_A(\Fp) = \begin{cases} \Phi_A(\Fpbar) & \text{if w_p=-1},\\ \Phi_A(\Fpbar)[2] & \text{if w_p=1.} \end{cases}$$
\end{proposition}
\begin{proof}
If $w_p=-1$, then $\Frob_p=1$ and the $\Gal(\Fpbar/\Fp)$-action
of $\Phi_A(\Fpbar)$ is trivial.  Thus in this case,
$\Phi(\Fp)=\Phi(\Fpbar)$.
Next suppose $w_p=1$.  We have an exact sequence
$$0\ra X_{A'} \ra \Hom(X_A,\Z) \ra \Phi_A \ra 0.$$
Since $W_p$ acts as $+1$ on $f$ it acts as $+1$ on
$A$, on $X_A$, on $\Hom(X_A,\Z)$, and on $\Phi_A$.
Thus $\Frob_p=-W_p$ acts as $-1$ on $\Phi_A$.  The $2$-torsion
in a finite abelian group equals the fixed points under $-1$.
\end{proof}

\subsection{Computation}
Suitable generalizations of the algorithms described in
\cite{cremona:algs} can be used to enumerate the optimal
quotients $A_f$ and to compute $m_A$.  These will be
described in the author's Berkeley
Ph.D. thesis \cite{stein:phd}.
The method of graphs \cite{mestre:graphs} and \cite{kohel:hecke}
can be used to compute $X=X_{J_0(N)}$ with its $\T$-action
and the monodromy pairing.  We can then compute
$$\L=\bigcap_{t\in I_f} \ker(t|_X),$$
$m_X:=m_\L$, and $\Phi_X:=\Phi_\L$.
By Theorem~\ref{formula} we can now compute
$$|\Phi_A| = |\Phi_X| \cdot \frac{m_A}{m_X}.$$
We have computed $\Phi_A$ in a number of cases.  In the
next subsection we discuss two conjectures suggested by
our numerical computations.

\subsection{Conjectures}
Our numerical computations suggest the following conjectures.
Suppose that $N=pM$ with $(p,M)=1$.
Let
$$H_{\new} = \ker\,\Bigl( H_1(X_0(N),\Z)\lra (H_1(X_0(M),\Z)\oplus H_1(X_0(M),\Z)\Bigr),$$
where the map is induced by the two natural
degeneracy maps $X_0(N)\ra X_0(M)$.
The Hecke algebra $\T$ acts on $H_{\new}$,
and on the submodule $H_{\new}[I_f]$ of elements annihilated
by $I_f$. Integration defines a map
$\alp: H_{\new}\ra \Hom(S[I_f],\C)$.
Define the homology congruence modulus $m_H$ by
$$m_H^2 = [\alp(H_{\new}) : \alp(H_{\new}[I_f])].$$
We expect that there is a very close relationship
between $m_X$ and $m_H$.
\begin{conjecture}\label{conj:deg}
Up to powers of $2$,
$$m_X = m_H.$$
\end{conjecture}

When $N=p$ is prime we make the following conjecture.
\begin{conjecture}\label{conj:iso}
Let $p$ be a prime and let $f_1,\ldots,f_n$ be a set of
representatives of the Galois conjugacy classes for newforms
in $S_2(\Gamma_0(p))$. Let $A_1,\ldots, A_n$ be the corresponding
optimal quotients.  Then the natural maps
\begin{eqnarray*}
\Phi_{J_0(p)} &\lra& \prod_{i=1}^n \Phi_{A_i}\\
J_0(p)(\Q) &\lra& \prod_{i=1}^n A_i(\Q)
\end{eqnarray*}
are isomorphisms.
\end{conjecture}
We offer the tables in the next section as evidence that
an assertion such as the above two conjectures may be true.

\section{Tables}
We computed several component groups of optimal quotients
$A_f$ of $J_0(N)$ associated to newforms $f$.
We denote such an optimal quotient by
\begin{center}
{\bf  N\, isogeny-class\, dimension}
\end{center}
The dimension frequently determines the factor, so it
is included in the notation.

\subsection{Table 1: Some large component groups predicted by
the Birch and Swinnerton-Dyer conjecture}
Using the algorithm described in \cite{stein:vissha} we computed
the special value $L(A,1)/\Omega$ (up to a Manin constant)
for every optimal quotient $A=A_f$ of level $\leq 1500$.
We found exactly five for which the numerator of
$L(A,1)/\Omega$ is nonzero and divisible by a
prime number $>10^9$.
These are given below.
$$\begin{array}{|lcc|}\hline A & N & \text{\qquad L(A,1)/\Omega\cdot \text{Manin constant}\qquad }\\\hline \text{\bf 1154E20}&2\cdot 577 & 2^?\cdot 85495047371/17^2\\ \text{\bf 1238G19}& 2\cdot 619 & 2^?\cdot 7553329019/5\cdot 31\\ \text{\bf 1322E21}& 2\cdot 661 & 2^?\cdot 57851840099/331\\ \text{\bf 1382D20}& 2\cdot 691 & 2^?\cdot 37 \cdot 1864449649 /173\\ \text{\bf 1478J20}& 2\cdot 739 & 2^?\cdot 7\cdot 29\cdot 1183045463 / 5\cdot 37\\\hline \end{array}$$
The Birch and Swinnerton-Dyer conjecture predicts that these large
prime divisors must divide either $|\Phi_A|$ or
the Shafarevich-Tate group of $A$.  We computed $\Phi_A$ and
found that this was the case.

$$\begin{array}{|lccccc|}\hline A & p & w & |\Phi_X| & m_X & |\Phi_A| \\\hline \text{\bf 1154E20}&2 & - & 17^2 & 2^{24} & 2^?\cdot 17^2 \cdot 85495047371 \\ &577& + & 1 & 2^{26}\cdot85495047371 & 2^? \\ \vspace{-1ex}&&&&&\\ \text{\bf 1238G19}& 2 &- & 5\cdot31 & 2^{26} &2^?\cdot 5\cdot31\cdot7553329019 \\ & 619 &+ & 1 & 2^{28}\cdot 7553329019 & 2^? \\ \vspace{-1ex}&&&&&\\ \text{\bf 1322E21}& 2 & - & 331 & 2^{28} & 2^?\cdot 331 \cdot 57851840099\\ & 661& + & 1 & 2^{32}\cdot 57851840099 & 2^?\\ \vspace{-1ex}&&&&&\\ \text{\bf 1382D20}& 2 & - & 173 & 2^{29} & 2^?\cdot 37\cdot173\cdot 1864449649 \\ & 691 & + & 1 & 2^{31}\cdot 37\cdot 1864449649 & 2^?\\ \vspace{-1ex}&&&&&\\ \text{\bf 1478J20}& 2 & - & 5\cdot37 &2^{31} & 2^?\cdot5\cdot 7\cdot29\cdot37\cdot1183045463 \\ & 739 & + & 1 &2^{33}\cdot 7\cdot 29\cdot 1183045463 & 2^? \\ \hline \end{array}$$

\subsection{Table 3: Some quotients of $J_0(N)$}
In this table we give the invariants defined above for
the optimal quotients of levels $65$, $66$, $68$, and $69$.
$$\begin{array}{|lccccccc|}\hline A & p & w & |\Phi_X| & m_X & m_H & m_A & |\Phi_A| \\\hline \text{\bf 65A1} & 5 & +& 1 &2 & ? & 2 & 1\\ & 13 &+& 1 & 2& ? & & 1\\ \text{\bf 65B2} & 5 &+& 3 & 2^2&? & 2^2 & 3\\ & 13 &- & 3 & 2^2&? & & 3\\ \text{\bf 65C2} & 5 &-& 7 & 2^2&? & 2^2 &7 \\ & 13 &+ & 1 & 2^2&? & & 1\\ \vspace{-1ex} & & & & & & & \\ \text{\bf 66A1}& 2 &+ & 1 &2 & ?& 2^2&2 \\ & 3 &- & 3 &2^2 & ?& & 3\\ & 11 &+ & 1 &2^2 & ?& &1 \\ \text{\bf 66B1}& 2 &- & 2 &2 & ?& 2^2& 2^2\\ & 3 &+ & 1 &2^2& ?& & 1\\ & 11 &+ & 1 &2^2 &? & & 1\\ \text{\bf 66C1}& 2 & -& 1 & 2&? & 2^2\cdot 5& 2\cdot5\\ & 3 &- & 1 & 2^2&? & & 5\\ & 11 &- & 1 & 2^2\cdot5&? & &1 \\ \vspace{-1ex} & & & & & & & \\ \text{\bf 68A2} &17&+&2 &2\cdot3 &? &2\cdot3 & 2 \\ \vspace{-1ex} & & & & & & & \\ \text{\bf 69A1} &3 &-&2 &2 & ?&2& 2\\ &23 &+& 1&2 &? & & 1\\ \text{\bf 69B2} &3 &+&2 &2 &? &2\cdot11& 2\cdot11 \\ &23 &-&2 &2\cdot11 &? && 2 \\ \hline \end{array}$$

\subsection{Table 3: Some quotients of $J_0(p)$}
Using the method of graphs and modular symbols we computed
the quantities $m_A$, $m_L$ and $\Phi_L$ for each abelian
variety $A=A_f$ associated to a newform of prime level
$p\leq 757$.  The results were as follows:
\begin{enumerate}
\item In all cases $m_A=m_L$, so the map $\Phi_J\ra \Phi_A$
is surjective.
\item $\Phi_A=1$ whenever the sign of the Atkin-Lehner involution
$w_p$ on $A$ is $1$.
\item $\prod |\Phi_A| = |\Phi_J|$
\end{enumerate}
Table 1 lists those $A$ of level $\leq 631$ for which $w_p=-1$, along with
the order of the component group.

\newpage
Table 1: Some quotients of $J_0(p)$~%
$$\begin{array}{|lc}\hline \vspace{-2ex}\\ A & |\Phi_A| \\ \vspace{-2ex}\\\hline 11\text{A}1&5\\ 17\text{A}1&2^2\\ 19\text{A}1&3\\ 23\text{A}2&11\\ \vspace{-2ex} &\\ 29\text{A}2&7\\ 31\text{A}2&5\\ 37\text{B}1&3\\ 41\text{A}3&2\cdot5\\ \vspace{-2ex} &\\ 43\text{B}2&7\\ 47\text{A}4&23\\ 53\text{B}3&13\\ 59\text{A}5&29\\ \vspace{-2ex} &\\ 61\text{B}3&5\\ 67\text{A}1&1\\ 67\text{C}2&11\\ 71\text{A}3&5\\ \vspace{-2ex} &\\ 71\text{B}3&7\\ 73\text{A}1&2\\ 73\text{C}2&3\\ 79\text{B}5&13\\ \vspace{-2ex} &\\ 83\text{B}6&41\\ 89\text{B}1&2\\ 89\text{C}5&11\\ 97\text{B}4&2^3\\ \vspace{-2ex} &\\ 101\text{B}7&5^2\\ 103\text{B}6&17\\ 107\text{B}7&53\\ 109\text{A}1&1\\ \vspace{-2ex} &\\ 109\text{C}4&3^2\\ 113\text{A}1&2\\ 113\text{B}2&2\\ 113\text{D}3&7\\ \vspace{-2ex} &\\ 127\text{B}7&3\cdot7\\ 131\text{B}10&5\cdot13\\ 137\text{B}7&2\cdot17\\ 139\text{A}1&1\\ \vspace{-2ex} &\\ 139\text{C}7&23\\ 149\text{B}9&37\\ 151\text{B}3&1\\ 151\text{C}6&5^2\\ \hline\end{array} \begin{array}{lc}\hline \vspace{-2ex}\\ A & |\Phi_A| \\ \vspace{-2ex}\\\hline 157\text{B}7&13\\ 163\text{C}7&3^3\\ 167\text{B}12&83\\ 173\text{B}10&43\\ \vspace{-2ex} &\\ 179\text{A}1&1\\ 179\text{C}11&89\\ 181\text{B}9&3\cdot5\\ 191\text{B}14&5\cdot19\\ \vspace{-2ex} &\\ 193\text{C}8&2^4\\ 197\text{C}10&7^2\\ 199\text{A}2&1\\ 199\text{C}10&3\cdot11\\ \vspace{-2ex} &\\ 211\text{A}2&5\\ 211\text{D}9&7\\ 223\text{C}12&37\\ 227\text{B}2&1\\ \vspace{-2ex} &\\ 227\text{C}2&1\\ 227\text{E}10&113\\ 229\text{C}11&19\\ 233\text{A}1&2\\ \vspace{-2ex} &\\ 233\text{C}11&29\\ 239\text{B}17&7\cdot17\\ 241\text{B}12&2^2\cdot5\\ 251\text{B}17&5^3\\ \vspace{-2ex} &\\ 257\text{B}14&2^6\\ 263\text{B}17&131\\ 269\text{C}16&67\\ 271\text{B}16&3^2\cdot5\\ \vspace{-2ex} &\\ 277\text{B}3&1\\ 277\text{D}9&23\\ 281\text{B}16&2\cdot5\cdot7\\ 283\text{B}14&47\\ \vspace{-2ex} &\\ 293\text{B}16&73\\ 307\text{A}1&1\\ 307\text{B}1&1\\ 307\text{C}1&1\\ \vspace{-2ex} &\\ 307\text{D}1&1\\ 307\text{E}2&3\\ 307\text{F}9&17\\ 311\text{B}22&5\cdot31\\ \hline\end{array} \begin{array}{lc}\hline \vspace{-2ex}\\ A & |\Phi_A| \\ \vspace{-2ex}\\\hline 313\text{A}2&1\\ 313\text{C}12&2\cdot13\\ 317\text{B}15&79\\ 331\text{D}16&5\cdot11\\ \vspace{-2ex} &\\ 337\text{B}15&2^2\cdot7\\ 347\text{D}19&173\\ 349\text{B}17&29\\ 353\text{A}1&2\\ \vspace{-2ex} &\\ 353\text{B}3&2\\ 353\text{D}14&2\cdot11\\ 359\text{D}24&179\\ 367\text{B}19&61\\ \vspace{-2ex} &\\ 373\text{C}17&31\\ 379\text{B}18&3^2\cdot7\\ 383\text{C}24&191\\ 389\text{A}1&1\\ \vspace{-2ex} &\\ 389\text{E}20&97\\ 397\text{B}2&1\\ 397\text{C}5&11\\ 397\text{D}10&3\\ \vspace{-2ex} &\\ 401\text{B}21&2^2\cdot5^2\\ 409\text{B}20&2\cdot17\\ 419\text{B}26&11\cdot19\\ 421\text{B}19&5\cdot7\\ \vspace{-2ex} &\\ 431\text{B}1&1\\ 431\text{D}3&1\\ 431\text{F}24&5\cdot43\\ 433\text{A}1&1\\ \vspace{-2ex} &\\ 433\text{B}3&1\\ 433\text{D}16&2^2\cdot3^2\\ 439\text{C}25&73\\ 443\text{C}1&1\\ \vspace{-2ex} &\\ 443\text{E}22&13\cdot17\\ 449\text{B}23&2^4\cdot7\\ 457\text{C}20&2\cdot19\\ 461\text{D}26&5\cdot23\\ \vspace{-2ex} &\\ 463\text{B}22&7\cdot11\\ 467\text{C}26&233\\ 479\text{B}32&239\\ 487\text{A}2&1\\ \hline\end{array} \begin{array}{lc|}\hline \vspace{-2ex}&\\ A & |\Phi_A| \\ \vspace{-2ex}&\\\hline 487\text{B}2&3\\ 487\text{C}3&1\\ 487\text{D}16&3^3\\ 491\text{C}29&5\cdot7^2\\ \vspace{-2ex} &\\ 499\text{C}23&83\\ 503\text{B}1&1\\ 503\text{C}1&1\\ 503\text{D}3&1\\ \vspace{-2ex} &\\ 503\text{F}26&251\\ 509\text{B}28&127\\ 521\text{B}29&2\cdot5\cdot13\\ 523\text{C}26&3\cdot29\\ \vspace{-2ex} &\\ 541\text{B}24&3^2\cdot5\\ 547\text{C}25&7\cdot13\\ 557\text{B}1&1\\ 557\text{D}26&139\\ \vspace{-2ex} &\\ 563\text{A}1&1\\ 563\text{E}31&281\\ 569\text{B}31&2\cdot71\\ 571\text{A}1&1\\ \vspace{-2ex} &\\ 571\text{B}1&1\\ 571\text{C}2&1\\ 571\text{D}2&1\\ 571\text{F}4&1\\ \vspace{-2ex} &\\ 571\text{I}18&5\cdot19\\ 577\text{A}2&3\\ 577\text{B}2&1\\ 577\text{C}3&1\\ \vspace{-2ex} &\\ 577\text{D}18&2^4\\ 587\text{C}31&293\\ 593\text{B}1&2\\ 593\text{C}2&1\\ \vspace{-2ex} &\\ 593\text{E}27&2\cdot37\\ 599\text{C}37&13\cdot23\\ 601\text{B}29&2\cdot5^2\\ 607\text{D}31&101\\ \vspace{-2ex} &\\ 613\text{C}27&3\cdot17\\ 617\text{B}28&2\cdot7\cdot11\\ 619\text{B}30&103\\ 631\text{B}32&3\cdot5\cdot7\\ \hline\end{array}$$

\bibliography{biblio}

\end{document}

$$\begin{array}{|lcccccc|}\hline A & p & w & |\Phi_X| & m_X & m_A & |\Phi_A| \\\hline \text{\bf 1154E20}&2 & - & 17^2 & 2^{24}& 2^?\cdot 85495047371 & 2^?\cdot 17^2 \cdot 85495047371 \\ &577& + & 1 & 2^{26}\cdot85495047371 & & 2^? \\ \vspace{-1ex}&&&&&&\\ \text{\bf 1238G19}& 2 &- & 5\cdot31 & 2^{26}& 2^?\cdot7553329019 &2^?\cdot 5\cdot31\cdot7553329019 \\ & 619 &+ & 1 & 2^{28}\cdot 7553329019 & & 2^? \\ \vspace{-1ex}&&&&&&\\ \text{\bf 1322E21}& 2 & - & 331 & 2^{28} & 2^?\cdot57851840099 & 2^?\cdot 331 \cdot 57851840099\\ & 661& + & 1 & 2^{32}\cdot 57851840099 & & 2^?\\ \vspace{-1ex}&&&&&&\\ \text{\bf 1382D20}& 2 & - & 173 & 2^{29} & 2^?\cdot37\cdot1864449649 & 2^?\cdot 37\cdot173\cdot 1864449649 \\ & 691 & + & 1 & 2^{31}\cdot 37\cdot 1864449649 & & 2^?\\ \vspace{-1ex}&&&&&&\\ \text{\bf 1478J20}& 2 & - & 5\cdot37 &2^{31} & 2^?\cdot7\cdot29\cdot1183045463 & 2^?\cdot5\cdot 7\cdot29\cdot37\cdot1183045463 \\ & 739 & + & 1 &2^{33}\cdot 7\cdot 29\cdot 1183045463 & & 2^? \\ \hline \end{array}$$