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Author: William A. Stein
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19\begin{document}
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23\par\noindent
24{Preprint (\today), Version 0.5}
25\vspace{15ex}
26
27\par\noindent
28{\bf \LARGE Component groups of optimal quotients\\
29of Jacobians}\\
30\vspace{3ex}
31\par\noindent
32{\large W.A. Stein}\\
33{\small Department of Mathematics, University of California, Berkeley,
34CA 94720, USA}
35
36\section*{Introduction}
37Let $A$ be an abelian variety over a finite extension $K$ of the
38$p$-adic numbers $\Qp$.  Let $\O$ be the ring of integers of $K$,
39$\m$ its maximal ideal and $k=\O/\m$ the residue class field.
40The N\'{e}ron model of $A$ is a smooth commutative group scheme $\A$
41over $\O$ such that $A$ is its generic fiber and satisfying the
42property:\\
43the restriction map
44$$\Hom_\O(S,\A)\lra \Hom_K(S/K,A)$$
45is bijective for all schemes $S$ over $\O$.
46The special fiber $\A_k$ is a group scheme over $k$,
47which need not be connected.  Denote by
48$\A_k^0$ the connected component containing the identity.
49There is an exact sequence
50     $$0\lra\A_k^0\lra\A_k\lra\Phi_A\lra0$$
51with $\Phi_A$ a finite \'{e}tale group scheme over $k$,
52i.e., a finite abelian group equipped with an action of
53$\Gal(\kbar/k)$.
54
55In this paper we study the group $\Phi_A$
56with particular emphasis on quotients $A$ of Jacobians
57of modular curves $X_0(N)$.  When $A$ has semistable reduction
58Grothendieck described the component group in terms
59of a monodromy pairing on certain free abelian groups.
60When $A=J$ is the Jacobian of $X_0(N)$, this pairing
61can be explicitly computed, hence so can $\Phi_J$,
62as has been done in many cases in
63\cite{mazur:eisenstein} and \cite{edixhoven:eisen}.
64Suppose now that $A$ is a simple quotient of $J$ and
65that the kernel of the map $J\ra A$ is connected.
66There is a natural map $\Phi_J\ra \Phi_A$.  In this
67paper we give a formula which can be used to compute
68the image and the order of the cokernel.
69
70We now state our main result in more precise language.
71Suppose $\pi:J\ra A$ is an optimal quotient,
72with $J$ a semistable Jacobian and $A$ purely toric.
73We express the component group of $A$ in terms of
74the monodromy pairing associated to $J$.
75Let $m_A=\sqrt{\deg(\theta_A)}$ where
76$\theta_A:A'\ra A$ is induced by
77the canonical principal polarization of $J$.
78Let $X_J$ be the character group of the toric part
79of the special fiber of $J$.  Let $\L$ be the saturation
80of the image of $X_A$ in $X_J$.  The monodromy pairing
81defines a map $\alp:X_J\ra \Hom(\L,\Z)$.
82Let $\Phi_X$ be the cokernel of $\alp$ and
83$m_X=[\alp(X_J):\alp(\L)]$ be the order of the finite
84group $\alp(X_J)/\alp(\L)$.   We prove that
85   $$\frac{|\Phi_A|}{m_A} = \frac{|\Phi_X|}{m_X}.$$
86More precisely, $\Phi_X$ is the image of $\Phi_J \ra \Phi_A$
87and the cokernel of $\Phi_J \ra \Phi_A$ has order $m_A/m_X$.
88If the optimal quotient $J\ra A$ arises from a modular
89form on $\Gamma_0(N)$,
90then the quantities $m_A$, $m_X$ and $\Phi_X$ can
91be explicitly computed, hence so can $|\Phi_A|$.
92Having done this, we present some tables and conjectures
93which they suggest.
94
95{\bf Acknowledgement: }
96I am deeply grateful for conversations with
97A. Agashe, R. Coleman, B. Edixhoven, D. Lorenzini,
98B. Mazur, L. Merel, K. Ribet, and S. Takahashi.
99
100\section{Optimal quotients of jacobians}
101Let $J$ be a Jacobian equipped with its canonical principal
102polarization $\theta_J$.
103An {\bf optimal quotient} of $J$ is an
104abelian variety $A$ and a surjective
105map $\pi: J \ra A$ whose
106kernel is an abelian subvariety $B$ of $J$.
107Denote by $J'$ and $A'$ the abelian varieties dual to $J$ and $A$,
108respectively.
109Dualizing $\pi$ and composing with $\theta_J'=\theta_J$
110we obtain a map
111    $A'\xrightarrow{\pi'} J'\xrightarrow{\theta_J} J$.
112\begin{proposition}
113$A'\ra J$ is injective.
114\end{proposition}
115\begin{proof}
116Since $\theta_J$ is an isomorphism it suffices to prove
117that $\pi'$ is injective.
118Since the dual of $\pi'$ is $(\pi')'=\pi$ and $\pi$ is surjective,
119$\pi'$ must have finite kernel.
120Thus $A' \ra C=\im(\pi')$ is
121an isogeny.  Let $G$ denote the kernel,
122and dualize. By \cite[\S11]{milne:abvars} we have have
123$$\xymatrix{ 124 G\ar[r] & A'\[email protected]{->>}[r] \ar[dr]_{\pi'} 125 & C\ar[d]\\ 126 && J' 127}\qquad \quad \xrightarrow{\textstyle \text{dualize}} \quad \qquad 128\xymatrix{ 129 A & C\ar[l] & G'\ar[l] \\ 130 & J\ar[u]_{\vphi}\ar[ul]^{\pi} 131}$$
132with $G'$ the Cartier dual of $G$.
133Since $G'$ is finite, $\ker(\vphi)\subset\ker(\pi)$
134is of finite index.
135Since $\ker(\pi)$ is an abelian  variety it is divisible.
136Thus $\ker(\vphi)=\ker(\pi)$ so $G'=0$ and hence $G=0$.
137\end{proof}
138We denote the map $A'\ra J$ by $\pi'$.
139The kernel of $\theta_A$ measures the intersection of
140$A'$ and $B=\ker(\theta_A)$ inside of $J$
141as shown in the following diagram.
142$$\xymatrix{ 143A'\intersect B\ar[r]\ar[d] & B\ar[d] \\ 144 A'\[email protected]{^(->}[r]^{\pi'}\ar[dr]_{\theta_A} & J\ar[d]^\pi\\ 145 & A 146}$$
147
148Because $\theta_A$ is a polarization, $|\ker(\theta_A)|$ is
149a square \cite[Theorem 13.3]{milne:abvars}.   The
150{\bf congruence modulus} is the integer
151  $$m_A=\sqrt{|\ker(\theta_A)|}.$$
152
153\section{The special fiber of the N\'{e}ron model}
154Let $K$ be a finite extension of $\Qp$ with ring of integers $\O$
155and residue class field $k$.
156Let $A/K$ be an abelian variety and denote its N\'{e}ron model
157by $\cA$.
158Let $\Phi_A$ be the group of connected components of
159the special fiber $\cA_k$. This group
160is a finite \'{e}tale group scheme over $k$, i.e.,
161a finite abelian group equipped with an action of
162$\Gal(\kbar/k)$. There is an exact sequence of group schemes
163    $$0 \ra \cA_k^0 \ra \cA_k \ra \Phi_A \ra 0.$$
164The group $\cA_k^0$ is an extension of an abelian variety
165$\cB$ of dimension $a$ by the product of a torus $\cT$ of
166dimension $t$ and of a unipotent group $\cU$ of dimension $u$:
167$$0\ra \cU\cross \cT \ra \cA_k^0 \ra \cB \ra 0.$$
168The abelian variety $A$ is said to have {\bf purely toric} reduction
169if $t=\dim A$, and is {\bf semistable} if $u=0$.
170The character group $X_A = \Hom(\cT,\Gm)$ is a free abelian
171group of rank $t$ contravariantly associated to $A$.
172If $A$ is semistable there is a monodromy pairing
173$X_A\cross X_{A'}\ra \Z$ and an exact sequence
174  $$0 \ra X_{A'} \ra \Hom(X_{A},\Z) \ra \Phi_A \ra 0.$$
175\comment{
176Suppose again that $J\ra A$ is a symmetric optimal quotient, that $J$
177has semistable reduction and $A$ has purely toric reduction.
178Since $J$ is canonically self-dual the monodromy pairing
179defines a map $X=X_J \ra \Hom(X,\Z)$.  By functoriality there
180is a map $X\ra X_{A'}$.
181Let $\alpha:X\ra \Hom(X_{A'},\Z)$ be the resulting map.
182Let $L \subset X$ be the saturation of the image of $X_{A}$
183in $X$, i.e.,the image of $X_{A}$ in $L$ has finite index
184and $X/L$ is torsion free.  The {\bf character group congruence modulus}
185of the optimal quotient $J\ra A$ is the integer
186    $$m_x=[\alpha(X):\alpha(L)].$$
187We now state our main result.
188  $$|\Phi_A| = |\image(\Phi_J\ra\Phi_A)|\cdot \frac{m_\theta}{m_x}$$
189This is an expression for the cokernel of the map $\Phi_J\ra \Phi_A$
190as the quotient of two congruence moduli.}
191
192\section{Rigid uniformization}
193In this section we review the rigid analytic uniformization of
194a semistable abelian variety over a finite extension $K$ of the
195maximal unramified extension $\Qp^{\ur}$ of $\Qp$.   We use this
196to prove that if $A$ is purely toric, and $\phi:A'\ra A$ is an
197isogeny, then
198  $$\deg(\phi) = |\coker(X_A\ra X_{A'})|^2.$$
199We also prove a few lemmas about the character groups $X_A$.
200
201
202  \subsection{Raynaud-van der Put uniformization}\label{subsec:raynaud}
203
204\begin{theorem}[Raynaud, van der Put]\label{raynaud}
205If $A$ is a semistable Abelian variety, its universal
206covering is isomorphic to an extension $G$ of an abelian
207variety $B$ with good reduction by a torus $T$, the
208covering map from $G$ to $A$ is a homomorphism and
209its kernel is a twisted free Abelian group $\Gamma$ of finite rank.
210\end{theorem}
211This may be summarized by the following diagram,
212$$\xymatrix{ 213 &\Gamma\ar[d] \\ 214 T\ar[r] & G\ar[r]\ar[d] & B\\ 215 & A 216}$$
217which we call the {\bf uniformization cross} of $A$.
218The group $\Gamma$ can be identified with the character
219group $X_A$ of the previous section.
220The uniformization cross of the dual abelian variety
221$A'$ is
222$$\xymatrix{ 223 &\Gamma'\ar[d] \\ 224 T'\ar[r] & G'\ar[r]\ar[d] & B'\\ 225 & A' 226}$$
227where $\Gamma'=\Hom(T,\Gm)$ and $T'=\Hom(\Gamma,\Gm)$
228and the morphisms $\Gamma'\ra G'$ and $T'\ra G'$ are the
229one-motive duals of the morphisms $T\ra G$ and $\Gamma\ra G$,
230respectively.
231For more details see \cite{coleman:monodromy}.
232
233\begin{example}[Tate curve]
234If $E/\Qp$ is an elliptic curve with multiplicative reduction
235then the uniformization is $E=\Gm/q^\Z$ where $q=q(j)$ is
236obtained by inverting the expression for $j$ as a function of
237$q(z)=e^{2\pi iz}$.
238\end{example}
239
240
241  \subsection{Some lemmas}
242Let $\pi:J\ra A$ be an optimal quotient,
243with $J$ semistable and $A$ purely toric.
244\begin{lemma}\label{lem:surj}
245The map $\Gamma_J\ra \Gamma_A$ induced by $\pi$ is surjective.
246\end{lemma}
247\begin{proof}
248Because $G_J$ is simply connected,
249$\pi$ induces a map
250$G_J\ra T_A$ and a map $\Gamma_J\ra \Gamma_A$.
251Because $\pi$ is surjective and $T(A)$ is a
252torus, the map $G_J\ra T_A$ is surjective.
253The snake lemma applied to the following diagram gives
254a surjective map from $B=\ker(\pi)$ to
255$M=\coker(\Gamma_J\ra\Gamma_A)$.
256$$\xymatrix{ 257 & \Gamma_J\ar[r]\ar[d] & \Gamma_A\ar[r]\ar[d] & M\ar[r]\ar[d]& 0 \\ 258 & G_J\ar[r]\ar[d] & T_A\ar[d]\ar[r] & 0 \\ 259B\ar[r] & J\ar[r]^{\pi}& A 260}$$
261Because $\pi$ is optimal, $B$ is connected so $M$ must also be connected.
262Since $M$ is discrete it follows that $M=0$.
263\end{proof}
264
265\subsection{Purely toric abelian varieties}
266Assume that $A$ is purely toric. Then
267$B=0$, and  the uniformization cross becomes
268     $$\xymatrix{\Gamma\ar[d]\\ T\ar[d] \\ A}$$
269Let $\vphi:A'\ra A$ be a {\bf symmetric isogeny}, i.e.,
270$\vphi':A'\ra (A')'=A$ is equal to $\vphi$.
271Denote by $\vphi_t:T'\ra T$ and $\vphi_a:\Gamma'\ra\Gamma$ the
272induced maps.
273\begin{proposition}\label{prop:kerphi}
274There is an exact sequence
275$$0\ra\ker(\vphi_t) \ra \ker(\vphi) \ra \coker(\vphi_a)\ra 0,$$
276and $\ker(\vphi_t)$ is dual to $\coker(\vphi_a)$.
277\end{proposition}
278\begin{proof}
279Since $\vphi$ is an isogeny we have the following diagram:
280$$\xymatrix{ 281 0\ar[r]\ar[d] & \Gamma'\ar[r]\ar[d] & \Gamma\ar[r]\ar[d] 282 & \coker(\vphi_a)\ar[d]\\ 283 \ker(\vphi_t)\ar[r]\ar[d]& T'\ar[d]\ar[r] & T\ar[r]\ar[d] & 0\\ 284 \ker(\vphi)\ar[r] & A'\ar[r]^{\vphi} & A}$$
285The snake lemma then gives the claimed exact sequence.
286For the second assertion observe that the one-motive dual of the diagram
287$$\xymatrix{ 288 & \Gamma'\ar[r]\ar[d] & \Gamma\ar[d]\ar[r] & \coker(\vphi_a) \\ 289 \ker(\vphi_t)\ar[r] & T'\ar[r]^{\vphi} & T}$$
290is the diagram
291$$\xymatrix{ 292 & T & T'\ar[l]_{\vphi'} & \coker(\vphi_a)'\ar[l]\\ 293 \ker(\vphi_t)'& \Gamma\ar[l]\ar[u] &\Gamma'\ar[l]\ar[u] 294}$$
295Since $\vphi$ is symmetric, $\vphi'=\vphi$ and so
296     $$\ker(\vphi_t) = \coker(\vphi_a)'.$$
297\end{proof}
298
299\begin{lemma}\label{lem:isogcoker}
300$|\ker(\vphi)|=|\coker(\vphi_a)|^2$
301\end{lemma}
302\begin{proof}
303The order of a finite group scheme equals the order of its
304dual.
305\end{proof}
306
307\section{The main theorem}
308Let $\pi:J\ra A$ be an optimal quotient,
309with $J$ a semistable Jacobian and $A$ purely toric.
310Let $X_A$, $X_{A'}$, and $X_J$ denote the
311character groups of the toric parts of the
312special fibers.
313
314\subsection{Monodromy description of the component group}
315There is a pairing
316$X_A\cross X_{A'}\ra \Z$ called
317the monodromy pairing.  We have an exact sequence
318  $$0 \ra X_{A'} \ra \Hom(X_{A},\Z) \ra \Phi_A \ra 0.$$
319If $J$ is a Jacobian then $J$ is canonically self-dual so
320the monodromy pairing on $J$
321can be viewed as a pairing $X_J\cross X_J \ra \Z$ and
322there is an exact sequence
323  $$0 \ra X_{J} \ra \Hom(X_J,\Z) \ra \Phi_J \ra 0.$$
324
325\begin{example}[Tate curve]
326Suppose $E=\Gm/q^{\Z}$ is a Tate curve.
327The monodromy pairing on $X_E=q^{\Z}$ is
328$$\langle q, q\rangle = \ord_p(q)=-\ord_p(j).$$
329Thus $\Phi_E$ is cyclic of order $-\ord_p(j)$.
330\end{example}
331
332
333  \subsection{Proof of the main theorem}
334We now prove the main theorem.
335The key diagrams are
336$$\[email protected]=3pc{A' \[email protected]{^(->}[r]^{\pi'}\ar[dr]_{\theta} 337 & J \[email protected]{->>}[d]^{\pi}\\ 338 &A} 339\qquad\qquad\qquad 340 \[email protected]=3pc{X_A \[email protected]{^(->}[r]^{\pi^*} \ar[dr]^{\theta^*} 341 & X_J \[email protected]{->>}[d]^{\pi_*} \\ 342 & X_{A'}\[email protected]/^1.5pc/[ul]^{\theta_*}} 343$$
344The surjectivity of $\pi_*$ was proved in Lemma~\ref{lem:surj}.
345The injectivity of $\pi^*$ follows because
346$$\theta_*\pi_*\pi^*=\theta_*\theta^*=\deg(\theta)\neq 0,$$
347and multiplication by $\deg(\theta)$ on a free abelian
348group is injective.
349
350Let
351 $$\alp:X_J\ra \Hom(\pi^* X_A,\Z)$$
352be the map defined by the monodromy pairing restricted
353to $X_J\cross \pi^* X_A$.
354\begin{lemma}\label{lem:twokers}
355$\ker(\pi_*) = \ker(\alp)$
356\end{lemma}
357\begin{proof}
358Suppose $x\in \ker(\pi_*)$ and let $y=\pi^* z$ with
359$z\in X_A$.  Then
360$$\langle x, y \rangle = \langle x, \pi^* z \rangle 361 = \langle \pi_* x, z \rangle = 0$$
362so $x\in\ker(\alp)$.
363Next let $x\in\ker(\alp)$.
364Then for all $z\in X_A$,
365$$0 = \langle x, \pi^* z\rangle = \langle \pi_* x, z\rangle$$
366so $\pi_* x$ is in the kernel of the
367monodromy map
368$$X_{A'} \ra \Hom(X_A,\Z).$$
369Since $X_{A'}$ and $\Hom(X_A,\Z)$ are free of the same rank
370and the cokernel is torsion, the monodromy map is injective.
371Thus $\pi_* x=0$ and $x\in\ker(\pi_*)$.
372\end{proof}
373
374\begin{lemma}\label{lem:compphi}
375There is an exact sequence
376$$X_J \ra \Hom(\pi^* X_A,\Z) \ra \Phi_A \ra 0.$$
377\end{lemma}
378\begin{proof}
379Lemma~\ref{lem:twokers} gives the following
380commutative diagram with exact rows
381$$\xymatrix{0\ar[r] 382 & X_J/\ker(\alp)\ar[d]^{\isom} \ar[r] 383 & {\Hom(\pi^* X_A,\Z)}\ar[r] \ar[d]^{\isom} & \coker(\alp)\ar[r]\ar[d] & 0\\ 384 0\ar[r] & X_{A'}\ar[r] & {\Hom(X_A,\Z)}\ar[r] & {\Phi_A}\ar[r] & 0}$$
385By Lemma~\ref{lem:twokers}, the first vertical map is an isomorphism.
386The second is an isomorphism because it is induced by the
387isomorphism $\pi^*:X_A\ra \pi^* X_A$.  It follows that
388$\coker(\alp)\isom \Phi_A$, as claimed.
389\end{proof}
390
391Let $\L$ be the {\bf saturation} of $\pi^* X_A$ in $X_J$, i.e.,
392$[\L:\pi^*X_A]$ is finite and $X_J/\L$ is torsion free.
393Suppose $L$ is of finite index in $\L$.
394Define the {\bf congruence modulus} of $L$
395   $$m_L = [\alp(X_J):\alp(L)]$$
396and the {\bf component group} by
397  $$\Phi_L = \coker( X_J \ra \Hom(L,\Z)).$$
398When $L=\L$ we often set $m_X=m_\L$ and $\Phi_X=\Phi_\L$
399and think of $m_X$ and $\Phi_X$ as the character group
400congruence modulus and component group.''
401
402\begin{lemma}\label{lem:homog}
403The rational number $\ds \frac{|\Phi_L|}{m_L}$ does not
404depend on the choice of $L$.
405\end{lemma}
406\begin{proof}
407If $L'$ is another choice let $n=[L:L']\in\Q$.
408Then since $\alp$ is injective when restricted to $\L$,
409 $$m_{L'} = [\alp(X_J):\alp(L')] 410 = [\alp(X_J):\alp(L)]\cdot[\alp(L):\alp(L')] = m_L\cdot n$$
411and similarly $|\Phi_{L'}| = |\Phi_L|\cdot n$.
412\end{proof}
413
414Recall that we defined
415\begin{eqnarray*}
416  m_A &=& \sqrt{\deg(\theta)}\\
417  \Phi_A &=& \coker(X_{A'}\ra \Hom(X_A,\Z))
418\end{eqnarray*}
419
420\begin{theorem}\label{formula}
421For any $L$ of finite index in $\L$
422the following relation holds:
423$$\frac{|\Phi_A|}{m_A} = \frac{|\Phi_L|}{m_L}.$$
424\end{theorem}
425\begin{proof}
426By Lemma~\ref{lem:homog} we may assume that $L=\pi^*X_A$.
427With this choice of $L$, Lemma~\ref{lem:compphi} says that
428$\Phi_L \isom \Phi_A$.
429By Lemma~\ref{lem:twokers}, properties of the index,
430and Lemma~\ref{lem:isogcoker} we have
431\begin{eqnarray*}
432m_L&=&[\alp(X_J):\alp(L)] \\
433   &=& [\pi_*(X_J):\pi_*(L)]\\
434   &=& [X_{A'}:\pi_*(\pi^*X_A)]\\
435   &=& [X_{A'}:\theta^* X_A]\\
436   &=& \coker(\theta^*) \\
437   &=& \sqrt{\deg(\theta)} = m_A.
438\end{eqnarray*}
439\end{proof}
440
441\begin{proposition}\label{prop:compim}
442$$\image(\Phi_J\ra\Phi_A) \isom \Phi_\L.$$
443\end{proposition}
444\begin{proof}
445Because $\pi^*X_A\subset \L \subset X_J$, by
446Lemma~\ref{lem:compphi} we obtain a commutative diagram
447with exact rows
448$$\xymatrix{ 449 X_J\ar[r]\[email protected]{=}[d]& \Hom(X_J,\Z)\ar[r]\ar[d]& \Phi_J \ar[r]\ar[d]& 0\\ 450 X_J\ar[r]\[email protected]{=}[d]& \Hom(\L,\Z)\ar[r]\ar[d]& \Phi_\L \ar[r]\ar[d] & 0\\ 451 X_J\ar[r]& \Hom(\pi^*X_A,\Z)\ar[r]& \Phi_A \ar[r]& 0 452}$$
453The map $\Hom(\L,\Z)\ra\Hom(\pi^*X_A,\Z)$ is an isomorphism
454so $\Phi_\L\ra\Phi_A$ is injective, hence
455 $$\image(\Phi_J\ra\Phi_A) \isom \image(\Phi_J\ra \Phi_\L).$$
456The cokernel of $\Hom(X_J,\Z)\ra\Hom(\L,\Z)$
457surjects onto the cokernel of $\Phi_J\ra \Phi_\L$.
458Using the exact sequence
459$$0\ra \L \ra X_J \ra X_J/\L \ra 0,$$
460we find that
461$$\coker(\Hom(X_J,\Z)\ra\Hom(\L,\Z)) \subset \Ext^1(X_J/\L,\Z)=0,$$
462where $\Ext^1$ vanishes because $\L$ is saturated
463so that $X_J/\L$ is torsion free.  Thus the cokernel of
464$\Phi_J\ra\Phi_\L$ is $0$, from which the proposition follows.
465\end{proof}
466
467The following corollary
468follows from Theorem~\ref{formula} and Proposition~\ref{prop:compim}.
469\begin{corollary}\label{cor:div}
470$$|\coker(X_J\ra X_A)| = \frac{m_A}{m_\L}.$$
471As a consequence, $m_\L | m_A.$
472\end{corollary}
473
474\section{Optimal quotients of $J_0(N)$}
475Let $X_0(N)/\Q$ be the modular curve associated to the congruence
476subgroup $\Gamma_0(N)\subset\sltwoz$ of matrices which are upper
477triangular modulo $N$.  Let $p$ be a prime divisor of $N$ which is
478coprime to $M=N/p$.  The Jacobian $J=J_0(N)$ of $X_0(N)$ has semistable
479reduction at $p$.   The Hecke algebra
480$$\T=\Z[\ldots T_n\ldots]\subset\End(J)$$
481is a commutative ring of endomorphisms of $J$ of $\Z$-rank $=\dim J$.
482The character group $X_J$ is equipped with a
483functorial action of $\T$.
484The Hecke algebra $\T$ also act on the cusp
485forms $$S = S_2(\Gamma_0(N),\C).$$
486A newform $f$ is an eigenform normalized so that the coefficient
487of $q$ in the expansion of $f$ at the cusp $\infty$ is $1$, and
488such that $f$ does not occur at any level $N'\mid N$ with $N'\neq N$.
489If $f$ is a newform, let $I_f$ be the ideal in $\T$ of
490elements which annihilate $f$.  Then $\O_f=\T/I_f$ is an
491order in the ring of integers of the totally real number field
492$K_f$ obtained by adjoining the Fourier coefficients of $f$ to $\Q$.
493The quotient
494  $$A_f = J_0(N)/ I_f J_0(N)$$
495 is a purely toric optimal quotient of dimension $[K_f:\Q]$.
496
497Let $H=H_1(X_0(N),\Z)$ be the integral homology of the
498complex algebraic curve $X_0(N)$.  Integration defines a
499$\T$-equivariant nondegenerate
500pairing  $S \cross H \ra \C$ which we view as a map
501$\alp: H \ra \Hom_\C(S,\C)$.
502
503\begin{theorem}\label{Af}
504We have the following commutative diagram of $\T$-modules:
505$$\xymatrix{ 506 H[I_f] \[email protected]{^(->}[r]\[email protected]{^(->}[d] & H \[email protected]{->>}[r]\[email protected]{^(->}[d] 507 & \alp(H)\ar[d]\[email protected]{^(->}[d]\\ 508\Hom_\C(S,\C)[I_f]\[email protected]{^(->}[r]\[email protected]{->>}[d] & \Hom_\C(S,\C)\[email protected]{->>}[r]\[email protected]{->>}[d] 509 &\Hom_\C(S[I_f],\C)\[email protected]{->>}[d]\\ 510 A_f'(\C)\[email protected]{^(->}[r]\[email protected]/_2pc/_{\theta_A}[rr] & J(\C) \[email protected]{->>}[r] & A_f(\C) \\ 511}$$
512\end{theorem}
513\begin{proof}
514This can be deduced from \cite{shimura:factors}.
515\end{proof}
516
517\begin{corollary}\label{moduluscomp}
518$m_A^2 = [\alp(H) : \alp(H[I_f])]$.
519\end{corollary}
520\begin{proof}
521Recall that $m_A$ is defined to be $\sqrt{\deg(\theta_A)}$.
522The kernel of an isogeny of complex tori is
523isomorphic to the cokernel of the induced map
524on lattices.  The corollary now follows from
525the diagram of Theorem~\ref{Af}
526which indicates that the index $[\alp(H):\alp(H[I_f])]$
527is the cokernel of the map $H[I_f]\ra \alp(H).$
528\end{proof}
529
530Let $\Frob_p:X_J\ra X_J$ denote the map induced by Frobenius.
531One has $\Frob_p=-W_p$, where $W_p$ is the map induced
532by the Atkin-Lehner involution on $J_0(p)$.
533Let $f$ be a newform, $A=A_f$ the corresponding optimal
534quotient, and $w_p$ the sign of the eigenvalue of
535$W_p$ on $f$.
536\begin{proposition}
537$$\Phi_A(\Fp) 538 = \begin{cases} 539 \Phi_A(\Fpbar) & \text{if w_p=-1},\\ 540 \Phi_A(\Fpbar)[2] & \text{if w_p=1.} 541 \end{cases}$$
542\end{proposition}
543\begin{proof}
544If $w_p=-1$, then $\Frob_p=1$ and the $\Gal(\Fpbar/\Fp)$-action
545of $\Phi_A(\Fpbar)$ is trivial.  Thus in this case,
546$\Phi(\Fp)=\Phi(\Fpbar)$.
547Next suppose $w_p=1$.  We have an exact sequence
548   $$0\ra X_{A'} \ra \Hom(X_A,\Z) \ra \Phi_A \ra 0.$$
549Since $W_p$ acts as $+1$ on $f$ it acts as $+1$ on
550$A$, on $X_A$, on $\Hom(X_A,\Z)$, and on $\Phi_A$.
551Thus $\Frob_p=-W_p$ acts as $-1$ on $\Phi_A$.  The $2$-torsion
552in a finite abelian group equals the fixed points under $-1$.
553\end{proof}
554
555\subsection{Computation}
556Suitable generalizations of the algorithms described in
557\cite{cremona:algs} can be used to enumerate the optimal
558quotients $A_f$ and to compute $m_A$.  These will be
559described in the author's Berkeley
560Ph.D. thesis \cite{stein:phd}.
561The method of graphs \cite{mestre:graphs} and \cite{kohel:hecke}
562can be used to compute $X=X_{J_0(N)}$ with its $\T$-action
563and the monodromy pairing.  We can then compute
564   $$\L=\bigcap_{t\in I_f} \ker(t|_X),$$
565$m_X:=m_\L$, and $\Phi_X:=\Phi_\L$.
566By Theorem~\ref{formula} we can now compute
567  $$|\Phi_A| = |\Phi_X| \cdot \frac{m_A}{m_X}.$$
568We have computed $\Phi_A$ in a number of cases.  In the
569next subsection we discuss two conjectures suggested by
570our numerical computations.
571
572 \subsection{Conjectures}
573Our numerical computations suggest the following conjectures.
574Suppose that $N=pM$ with $(p,M)=1$.
575Let
576$$H_{\new} = 577 \ker\,\Bigl( H_1(X_0(N),\Z)\lra 578 (H_1(X_0(M),\Z)\oplus H_1(X_0(M),\Z)\Bigr),$$
579where the map is induced by the two natural
580degeneracy maps $X_0(N)\ra X_0(M)$.
581The Hecke algebra $\T$ acts on $H_{\new}$,
582and on the submodule $H_{\new}[I_f]$ of elements annihilated
583by $I_f$. Integration defines a map
584$\alp: H_{\new}\ra \Hom(S[I_f],\C)$.
585Define the homology congruence modulus $m_H$ by
586$$m_H^2 = [\alp(H_{\new}) : \alp(H_{\new}[I_f])].$$
587We expect that there is a very close relationship
588between $m_X$ and $m_H$.
589\begin{conjecture}\label{conj:deg}
590Up to powers of $2$,
591  $$m_X = m_H.$$
592\end{conjecture}
593
594When $N=p$ is prime we make the following conjecture.
595\begin{conjecture}\label{conj:iso}
596Let $p$ be a prime and let $f_1,\ldots,f_n$ be a set of
597representatives of the Galois conjugacy classes for newforms
598in $S_2(\Gamma_0(p))$. Let $A_1,\ldots, A_n$ be the corresponding
599optimal quotients.  Then the natural maps
600\begin{eqnarray*}
601\Phi_{J_0(p)} &\lra& \prod_{i=1}^n \Phi_{A_i}\\
602J_0(p)(\Q) &\lra& \prod_{i=1}^n A_i(\Q)
603\end{eqnarray*}
604are isomorphisms.
605\end{conjecture}
606We offer the tables in the next section as evidence that
607an assertion such as the above two conjectures may be true.
608
609\section{Tables}
610We computed several component groups of optimal quotients
611$A_f$ of $J_0(N)$ associated to newforms $f$.
612We denote such an optimal quotient by
613\begin{center}
614{\bf  N\, isogeny-class\, dimension}
615\end{center}
616The dimension frequently determines the factor, so it
617is included in the notation.
618
619
620\subsection{Table 1: Some large component groups predicted by
621the Birch and Swinnerton-Dyer conjecture}
622Using the algorithm described in \cite{stein:vissha} we computed
623the special value $L(A,1)/\Omega$ (up to a Manin constant)
624for every optimal quotient $A=A_f$ of level $\leq 1500$.
625We found exactly five for which the numerator of
626$L(A,1)/\Omega$ is nonzero and divisible by a
627prime number $>10^9$.
628These are given below.
629$$\begin{array}{|lcc|}\hline 630 A & N & \text{\qquad L(A,1)/\Omega\cdot \text{Manin constant}\qquad }\\\hline 631\text{\bf 1154E20}&2\cdot 577 & 2^?\cdot 85495047371/17^2\\ 632\text{\bf 1238G19}& 2\cdot 619 & 2^?\cdot 7553329019/5\cdot 31\\ 633\text{\bf 1322E21}& 2\cdot 661 & 2^?\cdot 57851840099/331\\ 634\text{\bf 1382D20}& 2\cdot 691 & 2^?\cdot 37 \cdot 1864449649 /173\\ 635\text{\bf 1478J20}& 2\cdot 739 & 2^?\cdot 7\cdot 29\cdot 1183045463 / 5\cdot 37\\\hline 636\end{array}$$
637The Birch and Swinnerton-Dyer conjecture predicts that these large
638prime divisors must divide either $|\Phi_A|$ or
639the Shafarevich-Tate group of $A$.  We computed $\Phi_A$ and
640found that this was the case.
641
642$$\begin{array}{|lccccc|}\hline 643 A & p & w & |\Phi_X| & m_X & |\Phi_A| \\\hline 644\text{\bf 1154E20}&2 & - & 17^2 & 2^{24} 645 & 2^?\cdot 17^2 \cdot 85495047371 \\ 646 &577& + & 1 & 2^{26}\cdot85495047371 647 & 2^? \\ 648\vspace{-1ex}&&&&&\\ 649\text{\bf 1238G19}& 2 &- & 5\cdot31 & 2^{26} &2^?\cdot 5\cdot31\cdot7553329019 \\ 650 & 619 &+ & 1 & 2^{28}\cdot 7553329019 & 2^? \\ 651\vspace{-1ex}&&&&&\\ 652\text{\bf 1322E21}& 2 & - & 331 & 2^{28} & 2^?\cdot 331 \cdot 57851840099\\ 653 & 661& + & 1 & 2^{32}\cdot 57851840099 & 2^?\\ 654\vspace{-1ex}&&&&&\\ 655\text{\bf 1382D20}& 2 & - & 173 & 2^{29} & 2^?\cdot 37\cdot173\cdot 1864449649 \\ 656 & 691 & + & 1 & 2^{31}\cdot 37\cdot 1864449649 & 2^?\\ 657\vspace{-1ex}&&&&&\\ 658\text{\bf 1478J20}& 2 & - & 5\cdot37 &2^{31} 659 & 2^?\cdot5\cdot 7\cdot29\cdot37\cdot1183045463 \\ 660 & 739 & + & 1 &2^{33}\cdot 7\cdot 29\cdot 1183045463 661 & 2^? \\ 662\hline 663\end{array}$$
664
665
666\subsection{Table 3: Some quotients of $J_0(N)$}
667In this table we give the invariants defined above for
668the optimal quotients of levels $65$, $66$, $68$, and $69$.
669$$\begin{array}{|lccccccc|}\hline 670 A & p & w & |\Phi_X| & m_X & m_H & m_A & |\Phi_A| \\\hline 671\text{\bf 65A1} & 5 & +& 1 &2 & ? & 2 & 1\\ 672 & 13 &+& 1 & 2& ? & & 1\\ 673 674\text{\bf 65B2} & 5 &+& 3 & 2^2&? & 2^2 & 3\\ 675 & 13 &- & 3 & 2^2&? & & 3\\ 676 677\text{\bf 65C2} & 5 &-& 7 & 2^2&? & 2^2 &7 \\ 678 & 13 &+ & 1 & 2^2&? & & 1\\ 679 680\vspace{-1ex} & & & & & & & \\ 681\text{\bf 66A1}& 2 &+ & 1 &2 & ?& 2^2&2 \\ 682 & 3 &- & 3 &2^2 & ?& & 3\\ 683 & 11 &+ & 1 &2^2 & ?& &1 \\ 684 685\text{\bf 66B1}& 2 &- & 2 &2 & ?& 2^2& 2^2\\ 686 & 3 &+ & 1 &2^2& ?& & 1\\ 687 & 11 &+ & 1 &2^2 &? & & 1\\ 688 689\text{\bf 66C1}& 2 & -& 1 & 2&? & 2^2\cdot 5& 2\cdot5\\ 690 & 3 &- & 1 & 2^2&? & & 5\\ 691 & 11 &- & 1 & 2^2\cdot5&? & &1 \\ 692 693\vspace{-1ex} & & & & & & & \\ 694\text{\bf 68A2} &17&+&2 &2\cdot3 &? &2\cdot3 & 2 \\ 695 696\vspace{-1ex} & & & & & & & \\ 697\text{\bf 69A1} &3 &-&2 &2 & ?&2& 2\\ 698 &23 &+& 1&2 &? & & 1\\ 699 700\text{\bf 69B2} &3 &+&2 &2 &? &2\cdot11& 2\cdot11 \\ 701 &23 &-&2 &2\cdot11 &? && 2 \\ 702 703\hline 704\end{array}$$
705
706
707\subsection{Table 3: Some quotients of $J_0(p)$}
708Using the method of graphs and modular symbols we computed
709the quantities $m_A$, $m_L$ and $\Phi_L$ for each abelian
710variety $A=A_f$ associated to a newform of prime level
711$p\leq 757$.  The results were as follows:
712\begin{enumerate}
713\item In all cases $m_A=m_L$, so the map $\Phi_J\ra \Phi_A$
714is surjective.
715\item $\Phi_A=1$ whenever the sign of the Atkin-Lehner involution
716$w_p$ on $A$ is $1$.
717\item $\prod |\Phi_A| = |\Phi_J|$
718\end{enumerate}
719Table 1 lists those $A$ of level $\leq 631$ for which $w_p=-1$, along with
720the order of the component group.
721
722\newpage
723Table 1: Some quotients of $J_0(p)$~%
724$$725\begin{array}{|lc}\hline 726\vspace{-2ex}\\ 727A & |\Phi_A| \\ 728\vspace{-2ex}\\\hline 72911\text{A}1&5\\ 73017\text{A}1&2^2\\ 73119\text{A}1&3\\ 73223\text{A}2&11\\ 733\vspace{-2ex} &\\ 73429\text{A}2&7\\ 73531\text{A}2&5\\ 73637\text{B}1&3\\ 73741\text{A}3&2\cdot5\\ 738\vspace{-2ex} &\\ 73943\text{B}2&7\\ 74047\text{A}4&23\\ 74153\text{B}3&13\\ 74259\text{A}5&29\\ 743\vspace{-2ex} &\\ 74461\text{B}3&5\\ 74567\text{A}1&1\\ 74667\text{C}2&11\\ 74771\text{A}3&5\\ 748\vspace{-2ex} &\\ 74971\text{B}3&7\\ 75073\text{A}1&2\\ 75173\text{C}2&3\\ 75279\text{B}5&13\\ 753\vspace{-2ex} &\\ 75483\text{B}6&41\\ 75589\text{B}1&2\\ 75689\text{C}5&11\\ 75797\text{B}4&2^3\\ 758\vspace{-2ex} &\\ 759101\text{B}7&5^2\\ 760103\text{B}6&17\\ 761107\text{B}7&53\\ 762109\text{A}1&1\\ 763\vspace{-2ex} &\\ 764109\text{C}4&3^2\\ 765113\text{A}1&2\\ 766113\text{B}2&2\\ 767113\text{D}3&7\\ 768\vspace{-2ex} &\\ 769127\text{B}7&3\cdot7\\ 770131\text{B}10&5\cdot13\\ 771137\text{B}7&2\cdot17\\ 772139\text{A}1&1\\ 773\vspace{-2ex} &\\ 774139\text{C}7&23\\ 775149\text{B}9&37\\ 776151\text{B}3&1\\ 777151\text{C}6&5^2\\ 778\hline\end{array} 779\begin{array}{lc}\hline 780\vspace{-2ex}\\ 781A & |\Phi_A| \\ 782\vspace{-2ex}\\\hline 783157\text{B}7&13\\ 784163\text{C}7&3^3\\ 785167\text{B}12&83\\ 786173\text{B}10&43\\ 787\vspace{-2ex} &\\ 788179\text{A}1&1\\ 789179\text{C}11&89\\ 790181\text{B}9&3\cdot5\\ 791191\text{B}14&5\cdot19\\ 792\vspace{-2ex} &\\ 793193\text{C}8&2^4\\ 794197\text{C}10&7^2\\ 795199\text{A}2&1\\ 796199\text{C}10&3\cdot11\\ 797\vspace{-2ex} &\\ 798211\text{A}2&5\\ 799211\text{D}9&7\\ 800223\text{C}12&37\\ 801227\text{B}2&1\\ 802\vspace{-2ex} &\\ 803227\text{C}2&1\\ 804227\text{E}10&113\\ 805229\text{C}11&19\\ 806233\text{A}1&2\\ 807\vspace{-2ex} &\\ 808233\text{C}11&29\\ 809239\text{B}17&7\cdot17\\ 810241\text{B}12&2^2\cdot5\\ 811251\text{B}17&5^3\\ 812\vspace{-2ex} &\\ 813257\text{B}14&2^6\\ 814263\text{B}17&131\\ 815269\text{C}16&67\\ 816271\text{B}16&3^2\cdot5\\ 817\vspace{-2ex} &\\ 818277\text{B}3&1\\ 819277\text{D}9&23\\ 820281\text{B}16&2\cdot5\cdot7\\ 821283\text{B}14&47\\ 822\vspace{-2ex} &\\ 823293\text{B}16&73\\ 824307\text{A}1&1\\ 825307\text{B}1&1\\ 826307\text{C}1&1\\ 827\vspace{-2ex} &\\ 828307\text{D}1&1\\ 829307\text{E}2&3\\ 830307\text{F}9&17\\ 831311\text{B}22&5\cdot31\\ 832\hline\end{array} 833\begin{array}{lc}\hline 834\vspace{-2ex}\\ 835A & |\Phi_A| \\ 836\vspace{-2ex}\\\hline 837313\text{A}2&1\\ 838313\text{C}12&2\cdot13\\ 839317\text{B}15&79\\ 840331\text{D}16&5\cdot11\\ 841\vspace{-2ex} &\\ 842337\text{B}15&2^2\cdot7\\ 843347\text{D}19&173\\ 844349\text{B}17&29\\ 845353\text{A}1&2\\ 846\vspace{-2ex} &\\ 847353\text{B}3&2\\ 848353\text{D}14&2\cdot11\\ 849359\text{D}24&179\\ 850367\text{B}19&61\\ 851\vspace{-2ex} &\\ 852373\text{C}17&31\\ 853379\text{B}18&3^2\cdot7\\ 854383\text{C}24&191\\ 855389\text{A}1&1\\ 856\vspace{-2ex} &\\ 857389\text{E}20&97\\ 858397\text{B}2&1\\ 859397\text{C}5&11\\ 860397\text{D}10&3\\ 861\vspace{-2ex} &\\ 862401\text{B}21&2^2\cdot5^2\\ 863409\text{B}20&2\cdot17\\ 864419\text{B}26&11\cdot19\\ 865421\text{B}19&5\cdot7\\ 866\vspace{-2ex} &\\ 867431\text{B}1&1\\ 868431\text{D}3&1\\ 869431\text{F}24&5\cdot43\\ 870433\text{A}1&1\\ 871\vspace{-2ex} &\\ 872433\text{B}3&1\\ 873433\text{D}16&2^2\cdot3^2\\ 874439\text{C}25&73\\ 875443\text{C}1&1\\ 876\vspace{-2ex} &\\ 877443\text{E}22&13\cdot17\\ 878449\text{B}23&2^4\cdot7\\ 879457\text{C}20&2\cdot19\\ 880461\text{D}26&5\cdot23\\ 881\vspace{-2ex} &\\ 882463\text{B}22&7\cdot11\\ 883467\text{C}26&233\\ 884479\text{B}32&239\\ 885487\text{A}2&1\\ 886\hline\end{array} 887\begin{array}{lc|}\hline 888\vspace{-2ex}&\\ 889A & |\Phi_A| \\ 890\vspace{-2ex}&\\\hline 891487\text{B}2&3\\ 892487\text{C}3&1\\ 893487\text{D}16&3^3\\ 894491\text{C}29&5\cdot7^2\\ 895\vspace{-2ex} &\\ 896499\text{C}23&83\\ 897503\text{B}1&1\\ 898503\text{C}1&1\\ 899503\text{D}3&1\\ 900\vspace{-2ex} &\\ 901503\text{F}26&251\\ 902509\text{B}28&127\\ 903521\text{B}29&2\cdot5\cdot13\\ 904523\text{C}26&3\cdot29\\ 905\vspace{-2ex} &\\ 906541\text{B}24&3^2\cdot5\\ 907547\text{C}25&7\cdot13\\ 908557\text{B}1&1\\ 909557\text{D}26&139\\ 910\vspace{-2ex} &\\ 911563\text{A}1&1\\ 912563\text{E}31&281\\ 913569\text{B}31&2\cdot71\\ 914571\text{A}1&1\\ 915\vspace{-2ex} &\\ 916571\text{B}1&1\\ 917571\text{C}2&1\\ 918571\text{D}2&1\\ 919571\text{F}4&1\\ 920\vspace{-2ex} &\\ 921571\text{I}18&5\cdot19\\ 922577\text{A}2&3\\ 923577\text{B}2&1\\ 924577\text{C}3&1\\ 925\vspace{-2ex} &\\ 926577\text{D}18&2^4\\ 927587\text{C}31&293\\ 928593\text{B}1&2\\ 929593\text{C}2&1\\ 930\vspace{-2ex} &\\ 931593\text{E}27&2\cdot37\\ 932599\text{C}37&13\cdot23\\ 933601\text{B}29&2\cdot5^2\\ 934607\text{D}31&101\\ 935\vspace{-2ex} &\\ 936613\text{C}27&3\cdot17\\ 937617\text{B}28&2\cdot7\cdot11\\ 938619\text{B}30&103\\ 939631\text{B}32&3\cdot5\cdot7\\ 940\hline\end{array} 941$$
942
943
944\bibliography{biblio}
945
946\end{document}
947
948
949
950
951$$\begin{array}{|lcccccc|}\hline 952 A & p & w & |\Phi_X| & m_X & m_A & |\Phi_A| \\\hline 953\text{\bf 1154E20}&2 & - & 17^2 & 2^{24}& 2^?\cdot 85495047371 954 & 2^?\cdot 17^2 \cdot 85495047371 \\ 955 &577& + & 1 & 2^{26}\cdot85495047371 & 956 & 2^? \\ 957\vspace{-1ex}&&&&&&\\ 958\text{\bf 1238G19}& 2 &- & 5\cdot31 & 2^{26}& 2^?\cdot7553329019 &2^?\cdot 5\cdot31\cdot7553329019 \\ 959 & 619 &+ & 1 & 2^{28}\cdot 7553329019 & & 2^? \\ 960\vspace{-1ex}&&&&&&\\ 961\text{\bf 1322E21}& 2 & - & 331 & 2^{28} & 2^?\cdot57851840099 & 2^?\cdot 331 \cdot 57851840099\\ 962 & 661& + & 1 & 2^{32}\cdot 57851840099 & & 2^?\\ 963\vspace{-1ex}&&&&&&\\ 964\text{\bf 1382D20}& 2 & - & 173 & 2^{29} & 2^?\cdot37\cdot1864449649 & 2^?\cdot 37\cdot173\cdot 1864449649 \\ 965 & 691 & + & 1 & 2^{31}\cdot 37\cdot 1864449649 & & 2^?\\ 966\vspace{-1ex}&&&&&&\\ 967\text{\bf 1478J20}& 2 & - & 5\cdot37 &2^{31} 968 & 2^?\cdot7\cdot29\cdot1183045463 & 2^?\cdot5\cdot 7\cdot29\cdot37\cdot1183045463 \\ 969 & 739 & + & 1 &2^{33}\cdot 7\cdot 29\cdot 1183045463 970 & & 2^? \\ 971\hline 972\end{array}$$
973