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\begin{document}
\centerline{Bryden Cais  \\ Kris Klosin }
\centerline{\today}
\smallskip
\centerline{\sc Mazur Seminar}

\section{Introduction}

The goal of these notes is to show that the modular jacobian $J_0(N)_{/\Q}$ has a nontrivial quotient
of rank 0 for any prime $N$ such that the genus of $X_0(N)$ is positive.  

Recall the following theorem from James' talk:

\begin{theorem}\label{rk0}
	Let $A_{/\Q}$ be an abelian variety with good reduction outside of $N$ and purely toric reduction
	at $N$.  Suppose moreover that $\A[p]$ is admissible for
	some $p\neq N$, where $\A_{/\Z}$ is the N{\'e}ron model of
	$A$.  Then $A_{/\Q}$ has rank $0$.
\end{theorem}

Our aim is to construct a nonzero isogeny factor $J_{\p}$ of $J_0(N)$ satisfying the hypotheses of this theorem.
Before we construct $J_{\p}$ however, we will show that the above reduction properties of $A$
are inherited by any isogeny factor, and will discuss the relationship between isogenies $A\sim A'\times A''$
(with $A,A',A''$ abelian varieties over a field $K$) and idempotents in the ring of endomorphisms $\End^0(A)$ in the isogeny cateogory
of abelian varieties over $K$.

\section{Semi-abelian reduction}

In order to apply Theorem \ref{rk0} to the isogeny factor $J_{\p}$ of $J_0(N)$ that we will construct, we must show that $J_{\p}$ 
has good reduction outside $N$ and purely
toric reduction at $N$, and that $J_{\p}[p]$ is admissible.  
We first show that all isogeny factors of $J_{\p}$ inherit these reduction properties.
 
Let $K$ be the fraction field of a discrete valuation ring $\mathcal{O}$ with normalized valuation $v$ and residue
field $k$.
Fix a separable
closure $K_s$ of $K$ and a prime $v_s$
of $K_s$ lying over $v$, and denote by $I$ the inertia group of
$v_s$.
Let 
$A$ be an abelian variety over $K$, and 
denote 
by $\mathcal{A}$ its Neron model over $\mathcal{O}$, and by 
$\mathcal{A}^0$ 
the  connected component of the identity of $\mathcal{A}$. Recall that we say that 
$A$ has {\em semi-abelian reduction} with respect to $\O$ if $\mathcal{A}^0$ is an 
extension of an abelian variety by a torus.  

The aim of this section is to prove 
the following propositon:

\begin{proposition} \label{semiab}
Let the notation be as above. Suppose 
moreover that $A$ is isogenous to the product $A' \times A''$ of abelian 
varieties $A'$ and $A''$ over $K$. Then if $A$ has good $($resp. 
semi-abelian$)$ 
reduction, so do $A'$ and $A''$.
\end{proposition}

\begin{proof} 
We will only prove the case of 
semi-abelian reduction as the other cases are easier. Choose a prime $\ell \not= \Char k$.
 Let $V_{\ell}(A) := T_{\ell}(A) \otimes \Q$ denote the rational 
$\ell$-adic Tate module of $A$. First 
observe that 
$$V_{\ell}(A) = V_{\ell}(A' \times A'') = V_{\ell}(A') \times V_{\ell}(A'').$$
Grothendieck's criterion for semi-abelian reduction (cf. [1], theorem 6, 
page 184) 
asserts that $A$ has semi-abelian reduction if and only if there 
exists a subspace $V$ of $V_{\ell}(A)$ stable under the inertia group $I$ such that $I$ acts trivially on both $V$ and the quotient 
$V_{\ell}(A)/V$. (cf.  \cite[Theorem 6, p. 184]{ner})
Note that $V$ can (and will) be taken to be the maximal
subspace of
$V_{\ell}(A)$ on which the action of $I$ is trivial.


Let $V'$, $V''$ be the maximal subspaces of $V_{\ell}(A')$ and 
$V_{\ell}(A'')$ respectively, 
on 
which $I$ operates trivially. Obviously $V' \times V'' = V$.
Moreover, as $(V_{\ell}(A')/V') \times (V_{\ell}(A'')/V'') = V_{\ell}(A)/V$, we 
conclude that 
$I$ acts trivially on $V_{\ell}(A')/V'$ and on $V_{\ell}(A'')/V''$. This 
finishes 
the 
proof.  
\end{proof}

\begin{remark} \label{toric} Let us prove that if $A$ has toric
reduction, then so do $A'$ and $A''$. Assume that $A$ has toric 
reduction. First note that since $A$ and
$A' \times A''$ are isogenous, they are both semi-abelian as their 
rational Tate modules are isomorphic. Denote by $\phi_K$ an isogeny 
$A\rightarrow A' \times A''$. Then there exists an isogeny $\psi_K: A' 
\times A'' \rightarrow A$ and a nonzero integer $n$, such that $\phi_K 
\psi_K 
= n_{A' \times A''}$ and $\psi_K\phi_K = n_A$ \cite[p. 169]{mum}.
The Neron mapping property ensures that
 $\phi_K$ and $\psi_K$ extend to morphisms 
$\phi$ and $\psi$ between $\A$ and the Neron model $\A' \times \A''$ of 
$A' 
\times A''$. We have
$\text{Ner} (A' \times A'') = \A' \times \A''$ because of the Neron 
mapping property: if $Z$ is a smooth
$\mathcal{O}$-scheme, then 
\begin{align*}
\Hom_K (Z_K, A' \times A'') = \Hom_K (Z_K, A')\times \Hom_K (Z_K, A'') \\
&= \Hom_{\O} (Z, \A')  \times \Hom_{\O} (Z, \A'') =\Hom_{\O} (Z, \A' \times \A'').
\end{align*}
We will first show that $\bar{\phi}: \bar{\A}\rightarrow \bar{\A'} \times 
\bar{\A''}$ is an 
isogeny. First note that $\bar{\psi}\bar{\phi} = \bar{\psi \phi} = [n]$.
Since $A$ is semi-abelian, $\bar{\A}^0$ is an extension 
of 
an 
abelian variety by a torus. As multiplication by a nonzero integer is 
surjective 
on abelian varieties and on tori, we conclude 
that $[n]$ is surjective on $\bar{\A}^0$, hence 
an isogeny. Since $[n] = \bar{\psi} \bar{\phi}$,  and  
$\bar{\A}$ and $\bar{\A'} \times \bar{\A''}$ have the same 
dimension, we conclude that $\bar{\phi}$ is surjective on the identity 
components and has finite kernel, hence is an 
isogeny.

The isogeny between $\bar{\A}$ and $\bar{\A'} \times \bar{\A''}$ induces 
an isogeny between $\bar{\A}^0$ and $(\bar{\A'} \times \bar{\A''})^0 = 
\bar{\A'}^0 \times \bar{A''}^0$, so it remains to check that for connected 
commutative smooth algebraic groups over a field $k$, the property of 
being a torus is preserved under isogeny and under formation of direct 
factors. This is clear since by the structure theorem for commutative 
algebraic groups over $\bar{k}$ such a group $G$ is a torus if and only if 
$G$ is affine and $\text{dim}(V_{\ell}(G)) = \text{dim}(G)$, where $l\not= 
\text{char} \hspace{2pt} k$ is a prime.  

\end{remark}


\section{Idempotents inside the endomorphism ring of abelian varieties} 

In this section we remark upon some generalities concerning the relationship 
between idempotents in the ring $\End^0(A)$ for an abelian variety $A$ and
isogenies $A\sim A'\times A''$ with $A',A''$ abelian varieties.  

\begin{proposition} \label{idem}
 Let $A$ be an abelian variety over a field $K$. There is a one-to-one
correspondence between ordered pairs $(A', A'')$ of abelian subvarieties
of $A$ such that $A' \times_K A''\rightarrow A$ is an isogeny and
idempotents in the ring $\End^0(A)$; the operation $(A', A'') \mapsto 
(A'', A')$ corresponds to $e \mapsto 1-e$. \end{proposition}


\begin{proof} 
Suppose we are given a pair $(A', A'')$ of abelian subvarieties of $A$
such that the natural map $\phi: A' \times A'' \rightarrow A$ is an 
isogeny. Let
$\phi^{-1}$ denote the inverse of $\phi$ inside $\text{Hom}^0 (A, A'\times
A'')$. Consider the composition 

$$e: A \xrightarrow{\phi^{-1}} A'\times A'' \xrightarrow{\rm pr} A' \stackrel{\iota}{\hookrightarrow} A'\times A'' \xrightarrow{\phi} A,$$ 
where $\iota$ is inclusion and ${\rm pr}$ is projection. 
Note that $e$ is an idempotent in $\text{End}^0(A)$, and by definition of 
$\phi$, it lifts the identity on $A'$.

Conversely, given an idempotent $e \in \text{End}^0(A)$, choose a nonzero
integer $n$ such that $ne \in \text{End}(A)$, and set $A'$ to be the image
of $ne$. Note that $A'$ is independent of $n$, since multiplication by a 
nonzero
integer is surjective on an abelian variety. The Poincare Reducibilty 
Theorem ({\em c.f.} \cite{milne}) guarantees
the existence of a unique abelian subvariety $A''$ of $A$, such that $A'
\times A'' \rightarrow A$ is an isogeny (the map being addition). One
checks that the two processes are inverses of each other.


\end{proof}


\section{The optimal quotient $J_{\p}$ of $J_0(N)$}\label{optquo}

In this section we will construct a nonzero optimal quotient $J_{\p}$  of $J_0(N)$ such that the action
of $\T$ on $J_0(N)$ induces an action on $J_{\p}$.

Fix $N$ and let $\T$ denote the Hecke ring.  Let $\I\subseteq \T$
be the Eisenstein ideal let $\p\supseteq \I$ be a prime of residual characteristic $p\neq N$ (we have seen that 
such a $\p$ exists).
Observe that $\p_p:=\p (\T\otimes \Z_p)$ is a prime of $T\otimes \Z_p$.  

Consider the subring $\T\otimes_{\Z} \Q\subseteq \End^0(J_0(N))$ (we have seen that in fact equailty 
holds, but we do not use this).
By Proposition \ref{idem}, the decomposition of $\T\otimes \Q$
into a product of fields gives us the isogeny decomposition of $J_0(N)$ as a product of $\Q$-simple abelian subvarieties:
\begin{align}
	J_0(N)&\sim \prod_{\q\in{\rm MinSpec}(\T)} J_{\q}.
\end{align}
We put
$$\tilde{J}_{\p}=\prod_{\substack{\q\in {\rm MinSpec}(\T)\\ \q \subseteq \p}} J_{\q}.$$

\begin{definition}\label{idef}
	Let $A$, $A'$, and $A''$ be abelian varieties and suppose that $A'$ and $A''$ are quotients of $A$.
	We say that $A'$ and $A''$ are {\em isogenous as quotients} of $A$ if there exists an isomorphism $\phi:A'\rightarrow A''$
	in the isogeny category commuting (in that category) with the quotient maps $A\rightarrow A'$ and $A\rightarrow A''$.
\end{definition}

\begin{claim}\label{isog}
Let $A\rightarrow A''$ be a surjective map of abelian varieties over a field $K$.  
Then there is a unique quotient abelian variety
$A_{\rm opt}$, isogenous to $A''$ as a quotient of $A$, that is the quotient of $A$ by an abelian subvariety.
\end{claim}

\begin{proof}
We treat existence first.  By the Poincar{\'e} Reducibility Theorem over $K$, we see that $A$ is isogenous to a product
of simple abelian subvarieties, say $A\leftarrow \prod_{i=1}^n A_i$, and since $A''$ is a quotient of $A$ we 
have (renumbering if necessary) an isogeny $A''\sim \prod_{i=1}^k A_i$, as quotients of $A$, for some $k\le n$.
Define $A'=\prod_{k< i\le n} A_i$.  Then we have a morphism 
$$A'\stackrel{\iota}{\hookrightarrow} \prod_{i=1}^n A_i \xrightarrow{\varphi} A,$$
where $\varphi$ is an isogeny.  Observe that $\varphi\circ\iota (A')$ is an abelian subvariety
of $A$.  We set
$$A_{\rm opt}=A/\varphi\circ\iota (A').$$
It is clear that $A_{\rm opt}$ is the quotient of $A$ by an abelian subvariety and it follows (again from 
Poincar{\'e} Reducibility) that $A_{\rm opt}$ is isogenous to $A''$ as a quotient of $A$.

We claim that $A_{\rm opt}$ has the following universal property: if $q'':A\rightarrow \tilde{A}$ is any quotient isogenous
to the quotient $q:A\rightarrow A_{\rm opt}$ in the sense of Definition (\ref{idef}) then $q''$
factors uniquely through $q$ in the isogeny category.  Indeed, if $\tilde{A}$ and $A_{\rm opt}$
are isogenous as quotients, there exists an isomorphism (In the isogeny category) $\phi: A_{\rm opt}\rightarrow \tilde{A}$. 
By composing $\phi$ with multiplication by a large enough
integer, we obtain {\em honest} maps $n\circ q'':A\rightarrow \tilde{A}$ and $\psi=n\circ \phi: A_{\rm opt}\rightarrow A$
such that $\psi\circ q=n\circ q''$.  Hence $n\circ q'' (\ker q) = 0.$
But multiplication by $n$ is an isogeny, and hence has finite kernel,
while $q''(\ker q)$ is connected (as $\ker q$ is an abelian subvariety of $A$), so that
$q''(\ker q)=0$.

It follows that the map $A\rightarrow A''$ factors uniquely through $A_{\rm opt}$.
From this universal property of $A_{\rm opt}$, we see at once that $A_{\rm opt}$
is unique up to unique isomorphism.
\end{proof}

Applying the above claim to $A=J_0(N)$ and the isogeny factor $\tilde{J}_{\p}$, we obtain an optimal 
quotient $J_{\p}$ of $J_0(N)$.  We claim that $\T$ acts on $J_{\p}$.
This will follow from the following more general theorem:

\begin{theorem}\label{heckeact}
	Let $\pi:A\rightarrow A'$ be a surjective map of abelian varieties over a field $K$ of characteristic 0 having an abelian variety
	kernel $B$ and let $T\in \End(A)$.
	Assume there exists $T'\in\End(A')^0$ such that the following diagram commutes in the isogeny category:
	\begin{align}
	\begin{CD}
		A @>>\pi> A'\\
		@VTVV	@VVT'V\\
		A	@>\pi>> A'
	\end{CD}.
	\label{d1}
	\end{align}
	Then $T'\in \End(A')$.
	\end{theorem}
	
\begin{proof}	
	We have $T_0:=nT'\in\End(A')$ for some nonzero integer $n$, and by the universal property of the quotient map
	$A'\rightarrow A'$ having kernel $A'[n]$, we have $T'\in \End(A')$ if and only if $T_0$ kills $A'[n]$.
	Since $T_0 \pi=nT'\pi = n\pi T=\pi T n$ in the isogeny category, the genuine maps $T_0\pi$ and $\pi T n$
	agree so that $T_0\pi$ kills $A'[n]$ as $\pi T n$ obviously does.  To conclude that $T_0$ kills $A'[n]$,
	it therefore suffices to show that $\pi$ is faithfully flat on $n$-torsion, for which it 
	is enough to show surjectivity on $\overline{K}$-points (as finite $K$-groups are {\'e}tale so $A[n]\xrightarrow{\pi} A'[n]$
	is faithfully flat if and only if it is surjective on $\overline{K}$-points).  
	But we have the commutative diagram
	$$
	\begin{CD}
		0 @>>> B(\overline{K}) @>>> A(\overline{K}) @>>> A'(\overline{K}) @>>> 0\\
			&& @VnVV @VnVV @VnVV \\
					0 @>>> B(\overline{K}) @>>> A(\overline{K}) @>>> A'(\overline{K}) @>>> 0\\
	\end{CD}
	$$
	and since $A,A',B$ are abelian varieties and $K$ is of characteristic 0, the vertical maps
	are all surjective, so by the Snake lemma, 
	the map on $n$-torsion $A[n]\xrightarrow{\pi} A'[n]$ is surjective as desired.
	This completes the proof.	
\end{proof}	
	

Applying Theorem \ref{heckeact} to the optimal quotient $J_{\p}$ of $J_0(N)$ shows that we have an action of $\T$
on $J_{\p}$.    


\section{Admissibility}

In Brian's talk, it was explained why $J_{\p}[p]$ is an admissible group scheme over $\Z$.  In this section we will
recall the proof of this fact in more detail.

Since $\T$ is a finite $\Z$-module, the ring
$\T\otimes_{\Z} \Z_{p}$ is a finite $\Z_p$-module and hence is semi-local.
Moreover, we have seen in $\S\ref{optquo}$ that $\T$ acts on $J_{\p}$ in a manner that respects
the quotient map $J_0(N)\rightarrow J_{\p}$ so we obtain an action of
$\T\otimes_{\Z} \Z_{p}$ on 
the Tate modules $T_p(J_0(N))$ and $T_p(J_{\p})$ as $G_{\Q}:=\Gal(\Qbar/\Q)$-modules.
By 8.7 and 8.15 of \cite{mat}, there is a canonical isomorphism
\begin{align}
	\T\otimes_{\Z} \Z_p \simeq \prod_{\m\in{\rm MaxSpec}(\T\otimes_{\Z}\Z_p)} (\T\otimes_{\Z}\Z_p)_{\m}.
\end{align}

\begin{claim}
The action of $\T\otimes_{\Z} \Z_{p}$ on $T_p(J_{\p})$ factors through 
$\T_{\p}:=(\T\otimes_{\Z}\Z_p)_{\p}$, so the induced action of $\T\otimes_{\Z}\Z_p$
on $J_{\p}[p]$ is through the quotient $\T_{\p}$.
\end{claim}

\begin{proof}
First note that the functoriality of the idempotent decomposition of $\T\otimes \Q$ implies that
$$V_p(J_{\p}) = \prod_{\substack{\q \in \Spec(\T \otimes \Q)\\  \q \subset \p}} V_p(J_{\q}) =  
\prod_{\substack{\q \in \Spec(\T \otimes \Q)\\  \q \subset \p}}V_p(J)_{\q},$$ 
where the objects $V_p(J)_{\q}$ denote the  localizations of the 
$\T \otimes \Q$-module $V_p(J)$ at primes $\q$. (We will 
use the same letter $\q$ to denote both a prime ideal of $\T 
\otimes \Q$ and its inverse image in $\T$). From this we see 
that the action of $\T$ on $V_p(J_{\p})$ factors through 
$\prod_{\q \subset \p} (\T \otimes \Q)_{\q}$ and since every element of $\T - \p$ 
is mapped to a unit of $\prod_{\q \subset \p} (\T\otimes \Q)_{\q}$ (since $\q \subset \p$), the map 
$\T \rightarrow \prod_{\q \subset \p} (\T\otimes \Q)_{\q}$ factors through the 
localization $\T_{\p}$. The claim 
now follows after tensoring with $\Z_p$ and noting that $T_p(J_{\p})\subseteq V_p(J_{\p})$.
\end{proof}



We now prove:

\begin{lemma}\label{filter}
	Let $\I\subseteq \T$ be the Eisenstein ideal and $\p\supset \I$ be any prime of $\T$ having residue
	characteristic $p\neq N$.
	Then $\p^r$ kills $J_{\p}[p]$ for some $r>0$.
\end{lemma}

\begin{proof}
	The action of $\T\otimes_{\Z}\Z_p$ on 
	$J_{\p}[p]$ factors through $\T_{\p}/p \T_{\p}$.  As $\T\otimes\Z_p$ is a finite $\Z_p$-module, we see that
	$\T_{\p}/p\T_{\p}$ is a finite $\F_p$-module, and consequently is a finite local ring. 
	It follows that its maximal ideal is nilpotent.
	Thus, for some $r>0$ we have $\p^r \subseteq p \T_{\p}$, so $\p^r J_{\p}[p]=0$
	as claimed.	
\end{proof}

It follows from Lemma \ref{filter} that we have a filtration of $\T_{\p}[G_{\Q}]$ modules
\begin{align}
	J_{\p}[p] \supseteq \p J_{\p}[p] \supseteq \p^2 J_{\p}[p] \supseteq \ldots \supseteq \p^r J_{\p}[p]=0.
\end{align}
Observe that each quotient $\p^i J_{\p}[p]/\p^{i+1} J_{\p}[p]$ is killed by $\p$.  We will now show that
this implies that the quotients are all admissible.  
Recall from \cite[Corollary 1.6]{maz} that admissibility of a group scheme over $\Z$ can be checked on the corresponding
Galois module of $\Qbar$ points.  It then follows that $J_{\p}[p]$ is admissible, by general results on Jordan--H{\"o}lder series.

\begin{lemma}\label{admis}
	Let $G$ be a finite discrete $G_{\Q}$-module of $p$-power order on which $\T$ acts and 
	let $\p\supseteq \I$ be a prime of $\T$ having residual characteristic $p\neq N$ and containing the Eisenstein ideal.
	Assume that for any prime $\ell \ndiv Np$ 
	the inertia group $I_{\ell}$ at $\ell$ acts trivially on $G$, and hence that we have an action
	of $\Frob_{\ell}\in\Gal(\overline{\F}_{\ell}/\F_{\ell})$ on $G$.  Suppose that the Eichler-Shimura relation
	$$\Frob_{\ell}^2-T_{\ell}\Frob_{\ell}+\ell=0$$ holds on $G$ for all such $\ell$.
	If $\p$ kills $G$ then $G$ 
	has a filtration by admissible closed subgroups with successive quotients $\Z/p\Z$ or $\mu_p$.
\end{lemma}

\begin{proof}
	Let $\Gamma$ be the discrete finite quotient of $\Gal(\Qbar/\Q)$ through 
	which the Galois action on $G$ factors.  By hypothesis, for $l\ndiv Np$ we have that
	$\Frob_{\ell}$ acts on $G$ and
	\begin{align}
		\Frob_{\ell}^2-T_{\ell} \Frob_{\ell}+\ell = 0	
	\end{align}  
	on $G$.  But as $\p\supseteq \I$ kills $G$ and $T_{\ell}\equiv \ell+1\bmod \I$, we see that $\Frob_{\ell}$
	acts on $G$ with eigenvalues contained in $\{1,\ell\}$.  Now any $\sigma \in \Gal(\Qbar/\Q)$
	also acts on $G^{\dual}=\Hom(G,\mu_p)$ via $f^{\sigma} (g)=\sigma f(g^{\sigma^{-1}})$.
	Observe that if $g$ is an eigenvector of $\sigma=\Frob_{\ell}$ with eigenvalue $\ell$
	then $f^{\sigma}(g)=\sigma f(\ell^{-1} g)=\ell f(\ell^{-1}g)=f(g)$, so that $f(g)$ is an eigenvector
	with eigenvalue 1.  Similarly, if $g$ has eigenvalue $1$ then $f(g)$ has eigenvalue $\ell$.
	It follows that the eigenvalues $\{1,\ell\}$ of $\Frob_{\ell}$ on $G\times G^{\dual}$ occur with the same multiplicity.
	Thus, the characteristic polynomial of $\Frob_{\ell}$ on $G\times G^{\dual}$ over $\F_p$ is $(X-1)^d (X-\ell)^d$
	for some $d$ (which is the dimension of $G$ over $\F_p$).
	
	Consider now the $\F_p[G_{\Q}]$-module $(\Z/p\Z)^d\times \mu_p^d$.  
	By extending $\Gamma$ if necessary (and still preserving the finiteness of $\Gamma$) 
	we can regard $(\Z/p\Z)^d\times \mu_p^d$ as a finite discrete $\Gamma$-module.
	Since $\Frob_{\ell}$ acts on $\mu_p$ with eigenvalue $\ell$, we see that the characteristic
	polynomial of $\Frob_{\ell}$ on $(\Z/p\Z)^d\times \mu_p^d$ is also $(X-\ell)^d (X-1)^d$.
	Now by the Tchebotarev density theorem, every $\gamma\in \Gamma$ is the image of $\Frob_{\ell}$
	for some $\ell\neq N,p$.  Thus, every $\gamma\in \Gamma$ has the same characteristic polynomial
	on the two $\F_p[\Gamma]$-modules $(\Z/p\Z)^d\times \mu_p^d$ and $G$.
	Applying the Brauer--Nesbitt theorem, we conclude that these $\Gamma$-modules have the same
	semisimplifications.  Since $(\Z/p\Z)^d\times \mu_p^d$ has a filtration {\em as a Galois module} with successive quotients
	isomorphic to $\Z/p\Z$ or $\mu_p$, so does $G$.
\end{proof}

\begin{corollary}\label{admiscor}
	Let $N,p,\p$ be as before.  Then $J_{\p}[p]$ is admissible.
\end{corollary}

\begin{proof}
	By \cite[Corollary 1.6]{maz}, we need only check that the Galois module $J_{\p}[p]$ has a $\Gal(\Qbar/\Q)$-stable
	filtration by $\mu_p$'s and $\Z/p\Z$'s.  
	Using Proposition \ref{semiab} and Remark \ref{toric}, we see that $J_{\p}$ has toric reduction at $N$ and good reduction outside $N$,
	so the inertia group at any $\ell\ndiv Np$ acts trivially on the Galois module $J_{\p}[p](\Qbar)$.  Moreover, 
	the Eichler-Shimura relations hold on $J_{\p}[p],$ as they do in $J_0(N)$ and hence on the isogeny factor $J_{\p}$.
	We may therefore apply Lemma \ref{admis} to the situation discussed after the proof of Lemma \ref{filter} to conclude that
	$J_{\p}[p]$ is admissible.
\end{proof}

\begin{corollary}
	The nonzero isogeny factor $J_{\p}$ of $J_0(N)$ has rank $0$.
\end{corollary}

\begin{proof}
	By Proposition \ref{semiab} and Remark \ref{toric}, we know that $J_{\p}$ has good reduction away from $N$
	and purely toric reduction at $N$.  As $J_{\p}[p]$ is admissible by Corollary \ref{admiscor}, we see that $J_{\p}$
	satisfies the hypotheses of Theorem \ref{rk0} and hence has rank 0.
\end{proof}










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	\bibitem{maz} Mazur, B.  Modular curves and the Eisenstein ideal.  Publ. math. I.H.E.S., {\bf 47}, (2), 1977.  pp. 33--186.
	\bibitem{mat} Matsumura, H.  \textit{Commutative Ring Theory}, Cambridge University Press, 1986.
	\bibitem{milne} Milne, J.	Abelian Varieties, in {\em Arithmetic Geometry}.
	\bibitem{mum} Mumford, D. \textit{Abelian Varieties}, Oxford University Press, 1970.
\end{thebibliography}









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