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\chapter{Component groups of optimal quotients}%
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\index{Component group}%
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\index{Optimal quotient!component groups of}%
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\label{chap:compgroups}%
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Let~$A$ be an abelian variety over the rational numbers~$\Q$. The
49
Birch and Swinnerton-Dyer
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conjecture\index{BSD conjecture!and component groups}
51
supplies a formula for the order
52
of the Shafarevich-Tate group of~$A$. A key step in computing this
53
order is to find each of the Tamagawa numbers~$c_p$ of~$A$. The
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Tamagawa numbers are defined as follows, where the definition of
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N\'eron model and component group is given below.
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\begin{definition}[Tamagawa number]
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\index{Tamagawa numbers|textit}%
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\index{Optimal quotient!Tamagawa numbers of}%
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\index{Component group!rational points of}%
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Let~$p$ be a prime,
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let $\cA$ be a N\'eron model\index{N\'eron model}
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of~$A$ over the $p$-adic integers~$\Z_p$, and let $\Phi_{A,p}$
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be the component group of~$\cA$ at~$p$. Then the
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\defn{Tamagawa number} $c_p$ of~$A$ is the order of the group
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$\Phi_{A,p}(\F_p)$ of $\F_p$-rational points
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of $\Phi_{A,p}(\Fbar_p)$.
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\end{definition}
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\begin{remark}
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We warn the reader that the Tamagawa number is defined in a different
70
way in some other papers. The definitions are equivalent.
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\end{remark}
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In this chapter we present a method for computing the Tamagawa numbers
73
$c_p$, up to a power of~$2$, under the hypothesis that~$A$ has purely
74
toric reduction at~$p$. Such~$A$ are plentiful among the modular abelian
75
varieties; for example, if~$A$ is a new optimal quotient of $J_0(N)$
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and~$p$ exactly divides~$N$, then~$A$ is purely toric at~$p$.
77
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In Sections~\ref{sec:compgrpintro}--\ref{sec:compgrpmaintheorem} we
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state and prove an explicit formula involving component groups of
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fairly general abelian varieties. Then in Section~\ref{sec:compj0n}
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we turn to quotients of modular Jacobians $J_0(N)$.
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\index{Jacobian!of $X_0(N)$}
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We give several tables and issue a conjecture and a question.
84
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The results of this chapter were inspired by a letter that Ribet\index{Ribet}
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wrote to Mestre\index{Mestre},
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in which he treats the case when~$A$ is an elliptic curve.
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\section{Main results}\label{sec:compgrpintro}
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\subsection{N\'eron models and component groups}
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Let~$A$ be an abelian variety over a finite extension~$K$ of the
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$p$-adic numbers~$\Qp$. Let~$\O$ be the ring of integers of~$K$,
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let~$\m$ be its maximal ideal, and let $k=\O/\m$ be the residue class field.
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\begin{definition}[N\'eron model]\index{N\'eron model|textit}
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A \defn{N\'{e}ron model} of~$A$ is a smooth commutative group
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scheme~$\A$ over~$\O$ such that~$A$ is its generic fiber and~$\A$
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satisfies the N\'eron mapping property: the restriction map
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$$\Hom_\O(S,\A)\lra \Hom_K(S_K,A)$$
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is bijective for all {\em smooth} schemes~$S$ over~$\O$.
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\end{definition}
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The N\'eron mapping property implies that~$\A$ is unique up to a
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unique isomorphism, so we will refer without hesitation to ``the''
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N\'eron model of~$A$.
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The closed fiber~$\A_k$ of~$\A$ is a group scheme over~$k$,
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which need not be connected; denote by~$\A_k^0$ the
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connected component containing the identity.
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There is an exact sequence
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$$0\lra\A_k^0\lra\A_k\lra\Phi_A\lra0,$$
110
where~$\Phi_A$ a finite \'{e}tale group scheme
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over~$k$. Equivalently,~$\Phi_A$ may be viewed as
112
a finite abelian group equipped
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with an action of $\Gal(\kbar/k)$.
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\begin{definition}[Component group]%
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\index{Component group|textit}%
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\label{defn:componentgroup}%
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The \defn{component group} of an abelian variety~$\A$ over a
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local field~$K$ is the group scheme $\Phi_A=A_k/A_k^0$
119
defined above.
120
\end{definition}
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\subsection{Motivating problem}
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This chapter is motivated by the problem of
124
computing the groups~$\Phi_{A,p}$ attached to quotients~$A$
125
of Jacobians of modular curves~$X_0(N)$.\index{Jacobian!of $X_0(N)$}
126
When~$A$ has semistable
127
reduction, Grothendieck\index{Grothendieck} and Mumford\index{Mumford}
128
described the component group in terms of a monodromy pairing on
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certain free abelian groups. When $A=J=J_0(N)$ is the Jacobian of
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$X_0(N)$, this pairing can be explicitly computed, hence the component
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group~$\Phi_J$ can also be computed; this has been done in many cases
132
in \cite{mazur:eisenstein} and \cite{edixhoven:eisen}.
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Suppose now that $A=A_f$ is an optimal quotient of $J_0(N)$ that is
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attached to a newform~$f$, so that the kernel of the map $\pi:J\ra A$
136
is connected. There is a natural map $\pi_*:\Phi_J\ra \Phi_A$. We
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wish to compute the image and the order of the cokernel of
138
$\pi_*$.
139
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\subsection{The main result}
141
We now state our main result more precisely, necessarily supressing
142
some of the definitions of the terms used until later. Suppose
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$\pi:J\ra A$ is an optimal quotient, with~$J$
144
a Jacobian\index{Jacobian!semistable}
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with semistable reduction and~$A$ having purely toric
146
reduction. We express the component group\index{Component group}
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of~$A$ in terms of the monodromy pairing\index{Monodromy pairing}
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associated to~$J$.
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Let $m_A=\sqrt{\deg(\theta_A)}$, where $\theta_A:A^{\vee}\ra A$ is
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induced by the canonical principal polarization\index{Canonical
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polarization} of~$J$ arising from the $\theta$-divisor. Let $X_J$ be
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the character group\index{Character group of torus} of the toric
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part\index{Toric part} of the closed fiber of the N\'eron model
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of~$J$. Let~$\cL$ be the saturation of the image of $X_A$ in $X_J$.
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The monodromy pairing\index{Monodromy pairing} induces a map
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$\alp:X_J\ra \Hom(\cL,\Z)$. Let $\Phi_X$ be the cokernel of~$\alp$
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and $m_X=[\alp(X_J):\alp(\cL)]$ be the order of the finite group
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$\alp(X_J)/\alp(\cL)$. We obtain the equality
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$$\frac{\#\Phi_A}{m_A} = \frac{\#\Phi_X}{m_X}.$$
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Using the snake lemma\index{Snake lemma}, one see that
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$\Phi_X$ is isomorphic to the image of the natural map
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$\Phi_J \ra \Phi_A$, and the above formula implies that the cokernel of
164
the map $\Phi_J \ra \Phi_A$ has order $m_A/m_X$.
165
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If the optimal quotient $J\ra A$ arises from a
167
modular form on $\Gamma_0(N)$, then the quantities $m_A$, $m_X$ and
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$\Phi_X$ can be explicitly computed, hence we can compute $\#\Phi_A$.
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\section{Optimal quotients of Jacobians}
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Let~$J$ be a Jacobian\index{Jacobian}, and let
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$\theta_J$ be the canonical principal polarization
173
arising from the $\theta$-divisor.
174
Recall that an \defn{optimal quotient}\index{Optimal quotient|textit}
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of~$J$ is an abelian variety~$A$ and a surjective
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map $\pi: J \ra A$ whose
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kernel is an abelian subvariety~$B$ of~$J$.
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Denote by $J^{\vee}$ and $A^{\vee}$ the abelian varieties dual
179
to~$J$ and~$A$, respectively.
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Upon composing the dual of~$\pi$ with $\theta_J^{\vee}=\theta_J$,
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we obtain a map
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$$A^{\vee}\xrightarrow{\pi^{\vee}} J^{\vee}\xrightarrow{\theta_J} J.$$
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\begin{proposition}
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\index{Optimal quotient!dual map is injective}
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The map $A^{\vee}\ra J$ is injective.
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\end{proposition}
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\begin{proof}
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Since $\theta_J$ is an isomorphism it suffices to prove
189
that $\pi^{\vee}$ is injective.
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Since the dual of $\pi^{\vee}$ is
191
$(\pi^{\vee})^{\vee}=\pi$ and $\pi$ is surjective,
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the map $\pi^{\vee}$ must have finite kernel.
193
Thus $A^{\vee} \ra C=\im(\pi^{\vee})$ is
194
an isogeny. Let~$G$ denote the kernel of this isogeny,
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and dualize. By \cite[\S11]{milne:abvars} we have
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the following two commutative diagrams:
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$$\xymatrix{
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G\ar[r] & A^{\vee}\[email protected]{->>}[r] \ar[dr]_{\pi^{\vee}}
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& C\ar[d]\\
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&& J^{\vee}
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}\qquad \quad \xrightarrow{\textstyle \text{dualize}} \quad \qquad
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\xymatrix{
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A & C^{\vee}\ar[l] & G^{\vee}\ar[l] \\
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& J,\ar[u]_{\vphi}\ar[ul]^{\pi}
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}$$
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where $G^{\vee}$ is the Cartier dual\index{Cartier dual} of~$G$.
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Since $G^{\vee}$ is finite, $\ker(\vphi)$ is of
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finite index in $\ker(\pi)$.
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Since $\ker(\pi)$ is an abelian variety, as a group it is divisible.
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But a divisible group has no nontrivial finite-index subgroups
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(divisibility is a property inherited by quotients, and nonzero
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finite groups are not divisible).
213
Thus $\ker(\vphi)=\ker(\pi)$, so $G^{\vee}=0$. It follows that $G=0$.
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\end{proof}
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Henceforth we will abuse notation and
216
denote the injection $A^{\vee}\ra J$ by $\pi^{\vee}$.
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The kernel of $\theta_A$ equals the intersection of
218
$A^{\vee}$ and $B=\ker(\pi)$,
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as depicted in the following diagram:
220
$$\xymatrix{
221
A^{\vee}\intersect B\ar[r]\ar[d] & B\ar[d] \\
222
A^{\vee}\[email protected]{^(->}[r]^{\pi^{\vee}}\ar[dr]_{\theta_A} & J\ar[d]^\pi\\
223
& A.
224
}$$
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Since $\theta_A$ is a polarization\index{Polarization},
227
the degree $\#\ker(\theta_A)$ of $\theta_A$ is
228
a perfect square (see \cite[Thm.~13.3]{milne:abvars}).
229
Recall that
230
the \defn{modular degree}\index{Modular degree|textit} is the integer
231
$$m_A=\sqrt{\#\ker(\theta_A)}.$$
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For an algorithm to compute $m_A$,
233
see Section~\ref{sec:moddeg} and Corollary~\ref{moduluscomp}.
234
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\section{The closed fiber of the N\'{e}ron model}%
236
\index{N\'eron model!closed fiber of}%
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\index{Closed fiber of N\'eron model|textit}%
238
Let~$K$ be a finite extension of $\Qp$ with ring of integers~$\O$
239
and residue class field~$k$.
240
Let~$A$ be an abelian variety over~$K$ and denote its
241
N\'{e}ron model\index{N\'eron model} by~$\cA$.
242
Let $\Phi_A$ be the group of connected
243
components of the closed fiber $\cA_k$. This group
244
is a finite \'{e}tale group scheme\index{\'Etale group scheme|textit}
245
over~$k$; equivalently, it is
246
a finite abelian group equipped with an action of
247
$\Gal(\kbar/k)$. There is an exact sequence of group schemes
248
$$0 \ra \cA_k^0 \ra \cA_k \ra \Phi_A \ra 0.$$
249
The group scheme $\cA_k^0$ is an extension of an abelian variety~$\cB$
250
of some dimension~$a$ by a group scheme~$\cC$; we have a diagram
251
$$\[email protected]=.3cm{
252
&0\ar[d]\\
253
&{\cT}\ar[d]\\
254
0\ar[r]&{\cC}\ar[r]\ar[d]&{\cA_k^0}\ar[r]&{\cB}\ar[r]&0\\
255
&{\cU}\ar[d]\\
256
&0}$$
257
with~$\cT$ a torus of dimension~$t$
258
and~$\cU$ a unipotent group of dimension~$u$.
259
The abelian variety~$A$ is said to have \defn{purely toric reduction}%
260
\index{Purely toric reduction}
261
if $t=\dim A$, and have \defn{semistable reduction} if $u=0$.%
262
\index{Semistable reduction}
263
\begin{definition}[Character group of torus]
264
The \defn{character group}\index{Character group of torus|textit}
265
$$X_A = \Hom_{\kbar}(\cT_{/{\kbar}},{\Gm}_{/{\kbar}})\label{defn:chargroup}$$
266
is a free abelian group of rank~$t$ contravariantly associated to~$A$.
267
\end{definition}
268
As discussed in, e.g., \cite{ribet:modreps}, if~$A$ is semistable
269
there is a \defn{monodromy pairing}
270
$X_A\cross X_{A^{\vee}}\ra \Z$ and an exact sequence
271
$$0 \ra X_{A^{\vee}} \ra \Hom(X_{A},\Z) \ra \Phi_A \ra 0.$$
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\section{Rigid uniformization}
274
\index{Rigid uniformization|textit}
275
In this section we review the
276
rigid analytic uniformization
277
of a semistable\index{Semistable reduction!and uniformization}
278
abelian variety over a finite extension~$K$ of the
279
maximal unramified extension $\Qp^{\ur}$ of $\Qp$. We use this
280
uniformization to prove
281
that if~$A$ has purely toric reduction\index{Purely toric reduction},
282
and $\phi:A^{\vee}\ra A$ is a
283
symmetric isogeny\index{Symmetric isogeny} (as defined below), then
284
$$\deg(\phi) = (\# \coker(X_A\ra X_{A^{\vee}}))^2.$$
285
We also prove some lemmas about character groups\index{Character group of torus}.
286
287
It is possible to prove the assertions we will need without recourse
288
to rigid uniformization, as Ahmed Abbes has pointed out to the author.
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\subsection{Raynaud's uniformization}%
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\label{subsec:raynaud}%
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\begin{theorem}[Raynaud\index{Raynaud}]\label{raynaud}
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If~$A$ is a semistable abelian variety, its universal
294
covering (as defined in \cite{coleman:monodromy})
295
is isomorphic to an extension~$G$ of an abelian
296
variety~$B$ with good reduction by a torus~$T$. The
297
covering map from~$G$ to~$A$ is a homomorphism, and
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its kernel is a twisted free abelian group~$\Gamma$ of finite rank.
299
\end{theorem}
300
This may be summarized by the diagram
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$$\xymatrix{
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&\Gamma\ar[d] \\
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T\ar[r] & G\ar[r]\ar[d] & B\\
304
& A,
305
}$$
306
which we call the
307
\defn{uniformization cross}\index{Uniformization cross}
308
of~$A$.
309
310
\begin{remark}
311
The group~$\Gamma$ can be identified with the character
312
group $X_{A^{\vee}}$ of the previous and latter sections.
313
\end{remark}
314
315
The uniformization cross
316
of the dual abelian variety $A^{\vee}$ is
317
$$\xymatrix{
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&\Gamma^{\vee}\ar[d] \\
319
T^{\vee}\ar[r] & G^{\vee}\ar[r]\ar[d] & B^{\vee}\\
320
& A^{\vee},
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}$$
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where $\Gamma^{\vee}=\Hom(T,\Gm)$, where $T^{\vee}=\Hom(\Gamma,\Gm)$,
323
and the morphisms $\Gamma^{\vee}\ra G^{\vee}$ and $T^{\vee}\ra G^{\vee}$
324
are the one-motif duals\index{One-motif dual}
325
of the morphisms $T\ra G$ and $\Gamma\ra G$, respectively.
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For more details see, e.g., \cite{coleman:monodromy}.
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328
To avoid confusion when considering the uniformization of more than
329
one abelian variety, we will often denote the objects $T$, $G$,
330
$\Gamma$, and~$B$ connected with $A$ by $T_A$, $G_A$, $\Gamma_A$, and
331
$B_A$, respectively.
332
333
\begin{example}[Tate curve]\index{Tate curve!uniformization of}
334
If $E/\Qp$ is an elliptic curve with split multiplicative reduction,
335
then the uniformization is $E=\Gm/q^\Z$ where $q=q(j)$ is
336
obtained by inverting the expression for~$j$ as a function of
337
$q(z)=e^{2\pi iz}$.
338
\end{example}
339
340
341
\subsection{Some lemmas}
342
Let $\pi:J\ra A$ be an optimal quotient%
343
\index{Optimal quotient!and semistable reduction},
344
assume that~$J$ has semistable\index{Semistable reduction} reduction,
345
and that~$A$ has purely
346
toric\index{Purely toric reduction} reduction.
347
\begin{lemma}\label{lem:surj}
348
The map $\Gamma_J\ra \Gamma_A$ induced by~$\pi$ is surjective.
349
\end{lemma}
350
\begin{proof}
351
Since $G_J$ is simply connected,~$\pi$ induces a map
352
$G_J\ra T_A$ and a map $\Gamma_J\ra \Gamma_A$.
353
Because~$\pi$ is surjective and $T_A$ is a
354
torus, the map $G_J\ra T_A$ is surjective.
355
Upon applying the snake lemma\index{Snake lemma} to the
356
following diagram, we obtain a surjective map from $B=\ker(\pi)$ to
357
$M=\coker(\Gamma_J\ra\Gamma_A)$:
358
$$\xymatrix{
359
& \Gamma_J\ar[r]\ar[d] & \Gamma_A\ar[r]\ar[d] & M\ar[r]\ar[d]& 0 \\
360
& G_J\ar[r]\ar[d] & T_A\ar[d]\ar[r] & 0 \\
361
B\ar[r] & J\ar[r]^{\pi}& A.
362
}$$
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Since~$\pi:J\ra A$ is an optimal quotient, the kernel~$B$ is connected.
364
Thus~$M$ must also be connected.
365
Since~$M$ is discrete it follows that $M=0$.
366
\end{proof}
367
368
\subsubsection{Abelian varieties with purely toric reduction}
369
Assume that~$A$ has purely toric reduction\index{Purely toric reduction}.
370
Then $B=0$, and the uniformization
371
cross\index{Uniformization cross} is simply
372
$$\xymatrix{\Gamma\ar[d]\\ T\ar[d] \\ A.}$$
373
\begin{definition}[Symmetric isogeny]\index{Symmetric isogeny|textit}
374
A \defn{symmetric isogeny} $\vphi:A^{\vee}\ra A$
375
is an isogeny such that the map
376
$\vphi^{\vee}:A^{\vee}\ra (A^{\vee})^{\vee}=A$
377
is equal to~$\vphi$.
378
\end{definition}
379
Let $\vphi:A^{\vee}\ra A$ be a symmetric isogeny.
380
Denote by $\vphi_t:T^{\vee}\ra T$ and $\vphi_a:\Gamma^{\vee}\ra\Gamma$ the
381
maps induced by~$\vphi$.
382
\begin{proposition}\label{prop:kerphi}
383
There is an exact sequence
384
$$0\ra\ker(\vphi_t) \ra \ker(\vphi) \ra \coker(\vphi_a)\ra 0,$$
385
and $\ker(\vphi_t)$ is the Cartier dual of $\coker(\vphi_a)$.
386
\end{proposition}
387
\begin{proof}
388
Since~$\vphi$ is an isogeny we obtain the following diagram:
389
$$\xymatrix{
390
0\ar[r]\ar[d] & \Gamma^{\vee}\ar[r]^{\vphi_a}\ar[d] & \Gamma\ar[r]\ar[d]
391
& \coker(\vphi_a)\ar[d]\\
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\ker(\vphi_t)\ar[r]\ar[d]& T^{\vee}\ar[d]\ar[r]^{\vphi_t} & T\ar[r]\ar[d] & 0\\
393
\ker(\vphi)\ar[r] & A^{\vee}\ar[r]^{\vphi} & A.}$$
394
The snake lemma\index{Snake lemma} then gives the claimed exact sequence.
395
396
For the second assertion, observe that if we take one-motif
397
duals of every object in the diagram
398
$$\xymatrix{
399
& \Gamma^{\vee}\ar[r]^{\vphi_a}\ar[d] & \Gamma\ar[d]\ar[r] & \coker(\vphi_a) \\
400
\ker(\vphi_t)\ar[r] & T^{\vee}\ar[r]^{\vphi_t} & T}$$
401
we obtain the following diagram:
402
$$\xymatrix{
403
& T & T^{\vee}\ar[l]_{\vphi_a^{\vee}} & \coker(\vphi_a)^{\vee}\ar[l]\\
404
\ker(\vphi_t)^{\vee}& \Gamma\ar[l]\ar[u] &\Gamma^{\vee}.\ar[l]^{\vphi_t^{\vee}}\ar[u]
405
}$$
406
Since $\vphi$ is symmetric, $\vphi_a^{\vee}=\vphi_t$, so
407
$$\ker(\vphi_t) = \coker(\vphi_a)^{\vee}.$$
408
\end{proof}
409
410
\begin{lemma}\label{lem:isogcoker}
411
$\#\ker(\vphi)=\#\coker(\vphi_a)^2$
412
\end{lemma}
413
\begin{proof}
414
Use the exact sequence of Proposition~\ref{prop:kerphi} together
415
with the observation that the
416
order of a finite group scheme equals the order of its Cartier dual.
417
\end{proof}
418
419
\section{The main theorem}
420
\label{sec:compgrpmaintheorem}
421
Let $\pi:J\ra A$ be an optimal quotient\index{Optimal quotient},
422
with~$J$ a Jacobian\index{Jacobian!semistable}
423
having semistable reduction and~$A$
424
an abelian variety having purely toric reduction.
425
Let $X_A$, $X_{A^{\vee}}$, and $X_J$ denote the
426
character groups\index{Character group of torus} of the toric parts
427
of the closed fibers of the abelian varieties~$A$,
428
$A^{\vee}$, and~$J$, respectively.
429
430
\subsection{Description of the component group in terms of the monodromy pairing}
431
Recall that there is a pairing
432
$X_A\cross X_{A^{\vee}}\ra \Z$ called
433
the monodromy pairing\index{Monodromy pairing}.
434
We have an exact sequence
435
$$0 \ra X_{A^{\vee}} \ra \Hom(X_{A},\Z) \ra \Phi_A \ra 0.$$
436
If~$J$ is a Jacobian\index{Jacobian!is principally polarized}
437
then~$J$ is canonically self-dual via the
438
$\theta$-polarization, so
439
the monodromy pairing on~$J$
440
can be viewed as a pairing $X_J\cross X_J \ra \Z$, and
441
there is an exact sequence
442
$$0 \ra X_{J} \ra \Hom(X_J,\Z) \ra \Phi_J \ra 0.$$
443
444
\begin{example}[Tate curve]\index{Tate curve!and monodromy pairing}%
445
\index{Tate curve!component group of}%
446
Suppose $E=\Gm/q^{\Z}$ is a Tate curve over $\Qp^{\ur}$.
447
The monodromy pairing on $X_E=q^{\Z}$ is
448
$$\langle q, q\rangle = \ord_p(q)=-\ord_p(j).$$
449
Thus $\Phi_E$ is cyclic of order $-\ord_p(j)$.
450
\end{example}
451
452
453
\subsubsection{Proof of the main theorem}
454
We now prove the main theorem.
455
Let $\pi : J\ra A$ be an optimal quotient, and let
456
$\theta:A^{\vee}\ra A$ denote the induced polarization.
457
Let $\pi_*$, $\pi^*$, $\theta_*$, and $\theta^*$ be the
458
maps induced on character groups by the various functorialities,
459
as indicated in the following two key diagrams:
460
$$\[email protected]=3pc{A^{\vee} \[email protected]{^(->}[r]^{\pi^{\vee}}\ar[dr]_{\theta}
461
& J \[email protected]{->>}[d]^{\pi}\\
462
&A}
463
\qquad\qquad\qquad
464
\[email protected]=3pc{X_A \[email protected]{^(->}[r]^{\pi^*} \ar[dr]^{\theta^*}
465
& X_J \[email protected]{->>}[d]^{\pi_*} \\
466
& X_{A^{\vee}}.\[email protected]/^1.5pc/[ul]^{\theta_*}}
467
$$
468
The surjectivity of $\pi_*$ was proved in Lemma~\ref{lem:surj}.
469
The injectivity of $\pi^*$ follows because
470
$$\theta_*\pi_*\pi^*=\theta_*\theta^*=\deg(\theta)\neq 0,$$
471
and multiplication by a nonzero integer on a free abelian
472
group is injective.
473
474
Let
475
$$\alp:X_J\ra \Hom(\pi^* X_A,\Z)$$
476
be the map defined by the monodromy pairing\index{Monodromy pairing}
477
restricted to $X_J\cross \pi^* X_A$.
478
\begin{lemma}\label{lem:twokers}
479
$\ker(\pi_*) = \ker(\alp)$
480
\end{lemma}
481
\begin{proof}
482
Suppose $x\in \ker(\pi_*)$, and let $y=\pi^* z$ with
483
$z\in X_A$. Then
484
$$\langle x, y \rangle = \langle x, \pi^* z \rangle
485
= \langle \pi_* x, z \rangle = 0,$$
486
so $x\in\ker(\alp)$.
487
Next let $x\in\ker(\alp)$.
488
Then for all $z\in X_A$,
489
$$0 = \langle x, \pi^* z\rangle = \langle \pi_* x, z\rangle,$$
490
so $\pi_* x$ is in the kernel of the
491
monodromy map
492
$$X_{A^{\vee}} \ra \Hom(X_A,\Z).$$
493
Since $X_{A^{\vee}}$ and $\Hom(X_A,\Z)$ are free of the same finite
494
rank and the cokernel is torsion, the monodromy map is injective.
495
Thus $\pi_* x=0$ and $x\in\ker(\pi_*)$.
496
\end{proof}
497
498
\begin{lemma}\label{lem:compphi}
499
There is an exact sequence
500
$$X_J \ra \Hom(\pi^* X_A,\Z) \ra \Phi_A \ra 0.$$
501
\end{lemma}
502
\begin{proof}
503
Lemma~\ref{lem:twokers} gives the following
504
commutative diagram with exact rows
505
$$\xymatrix{0\ar[r]
506
& X_J/\ker(\alp)\ar[d]^{\isom} \ar[r]
507
& {\Hom(\pi^* X_A,\Z)}\ar[r] \ar[d]^{\isom} & \coker(\alp)\ar[r]\ar[d] & 0\\
508
0\ar[r] & X_{A^{\vee}}\ar[r] & {\Hom(X_A,\Z)}\ar[r] & {\Phi_A}\ar[r] & 0.}$$
509
By Lemma~\ref{lem:twokers}, the first vertical map is an isomorphism.
510
The second is an isomorphism because it is induced by the
511
isomorphism $\pi^*:X_A\ra \pi^* X_A$. It follows that
512
$\coker(\alp)\isom \Phi_A$, as claimed.
513
\end{proof}
514
515
Let $\cL$ be the \defn{saturation} of $\pi^* X_A$ in $X_J$;
516
thus $\pi^*X_A$ is a finite-index subgroup of~$\cL$
517
and the quotient $X_J/\cL$ is torsion free.
518
For~$L$ of finite index in $\cL$,
519
define the
520
\defn{modular degree}\index{Modular degree!and character group|textit}
521
of~$L$ to be
522
$$m_L = [\alp(X_J):\alp(L)],$$
523
and the
524
\defn{component group}\index{Component group!and character group|textit}
525
of~$L$ to be
526
$$\Phi_L = \coker( X_J \ra \Hom(L,\Z)).$$
527
When $L=\cL$ and $A$ is fixed, we often slightly abuse notation and
528
write $m_X=m_\cL$ and $\Phi_X=\Phi_\cL$.
529
We think of $m_X$ and $\Phi_X$ as the character group
530
``modular degree and component group''
531
of~$A$.
532
533
\begin{lemma}\label{lem:homog}
534
Choose a subgroup~$L$ of finite index in~$\cL$.
535
The rational number
536
$\ds \frac{\#\Phi_L}{m_L}$
537
is independent of the choice of~$L$.
538
\end{lemma}
539
\begin{proof}
540
Suppose $L'$ is another finite index subgroup of~$\cL$,
541
and let $n=[L:L']$. Here~$n$ is a rational number, the lattice
542
index of~$L'$ in~$L$.
543
Since~$\alp$ is injective when restricted to $\cL$, it follows that
544
$$m_{L'} = [\alp(X_J):\alp(L')]
545
= [\alp(X_J):\alp(L)]\cdot[\alp(L):\alp(L')] = m_L\cdot n.$$
546
Similarly, $\#\Phi_{L'} = \#\Phi_L\cdot n$.
547
\end{proof}
548
549
Recall that $m_A = \sqrt{\deg(\theta)}$ and
550
$$ \Phi_A \isom \coker(X_{A^{\vee}}\ra \Hom(X_A,\Z)),$$
551
where $m_A$ is the modular degree of~$A$ and $\Phi_A$ is the
552
component group of~$A$.
553
554
\begin{theorem}\label{formula}
555
For any subgroup~$L$ of finite index in $\cL$,
556
the following relation holds:
557
$$\frac{\#\Phi_A}{m_A} = \frac{\#\Phi_L}{m_L}.$$
558
\end{theorem}
559
\begin{proof}
560
By Lemma~\ref{lem:homog} we may assume that $L=\pi^*X_A$.
561
With this choice of $L$, Lemma~\ref{lem:compphi} asserts that
562
$\Phi_L \isom \Phi_A$.
563
By Lemma~\ref{lem:twokers}, properties of the index,
564
and Lemma~\ref{lem:isogcoker} we have
565
\begin{eqnarray*}
566
m_L&=&[\alp(X_J):\alp(L)] \\
567
&=& [\pi_*(X_J):\pi_*(L)]\\
568
&=& [X_{A^{\vee}}:\pi_*(\pi^*X_A)]\\
569
&=& [X_{A^{\vee}}:\theta^* X_A]\\
570
&=& \#\coker(\theta^*) \\
571
&=& \sqrt{\deg(\theta)} = m_A.
572
\end{eqnarray*}
573
\end{proof}
574
575
\begin{proposition}\label{prop:compim}
576
$$\image(\Phi_J\ra\Phi_A) \isom \Phi_\cL.$$
577
\end{proposition}
578
\begin{proof}
579
Since $\pi^*X_A\subset \cL \subset X_J$, an application
580
of Lemma~\ref{lem:compphi} gives the following commutative diagram
581
with exact rows:
582
$$\xymatrix{
583
X_J\ar[r]\[email protected]{=}[d]& \Hom(X_J,\Z)\ar[r]\ar[d]& \Phi_J \ar[r]\ar[d]& 0\\
584
X_J\ar[r]\[email protected]{=}[d]& \Hom(\cL,\Z)\ar[r]\ar[d]& \Phi_\cL \ar[r]\ar[d] & 0\\
585
X_J\ar[r]& \Hom(\pi^*X_A,\Z)\ar[r]& \Phi_A \ar[r]& 0.
586
}$$
587
The map $\Hom(\cL,\Z)\ra\Hom(\pi^*X_A,\Z)$ is an isomorphism,
588
so the map $\Phi_\cL\ra\Phi_A$ is injective. Thus
589
$$\image(\Phi_J\ra\Phi_A) \isom \image(\Phi_J\ra \Phi_\cL).$$
590
The cokernel of $\Hom(X_J,\Z)\ra\Hom(\cL,\Z)$
591
surjects onto the cokernel of $\Phi_J\ra \Phi_\cL$.
592
Using the exact sequence
593
$$0\ra \cL \ra X_J \ra X_J/\cL \ra 0,$$
594
we find that
595
$$\coker(\Hom(X_J,\Z)\ra\Hom(\cL,\Z)) \subset \Ext^1(X_J/\cL,\Z).$$
596
Because~$\cL$ is saturated, the quotient $X_J/\cL$ is torsion free,
597
so the indicated $\Ext^1$ group vanishes.
598
Thus the map $\Phi_J\ra\Phi_\cL$ is surjective,
599
from which the proposition follows.
600
\end{proof}
601
602
The following corollary
603
follows from Theorem~\ref{formula} and Proposition~\ref{prop:compim}.
604
\begin{corollary}\label{cor:div}
605
$$\#\coker(X_J\ra X_A) = \frac{m_A}{m_\cL}.$$
606
\end{corollary}
607
\begin{remark}
608
A non-obvious consequence of this corollary is that
609
$$m_\cL \mid m_A.$$
610
\end{remark}
611
612
613
\section{Optimal quotients of $J_0(N)$}\label{sec:optquoj0n}
614
\label{sec:compj0n}\index{Optimal quotient!of $J_0(N)$}
615
We now summarize some facts about $J_0(N)$ that will be used in
616
our numerical computations. Some of these facts were discussed in
617
greater generality in the previous chapters of this thesis.
618
619
\subsection{Modular curves and semistability}
620
Let $X_0(N)$ be the modular curve\index{Modular curve}
621
associated to the subgroup $\Gamma_0(N)$ of $\sltwoz$
622
that consists of those matrices which are upper
623
triangular modulo~$N$. Initially, $X_0(N)$ is constructed as
624
a Riemann surface as the quotient
625
$$\Gamma_0(N)\backslash (\{z : z \in \C,\,\Im(z)>0\}\union\P^1(\Q)).$$
626
With some work, we find that $X_0(N)$ has a canonical
627
structure of algebraic curve over~$\Q$.
628
629
Suppose that~$p$ is a prime divisor of~$N$ such that $N/p$ is
630
coprime to~$p$. We write $p\mid\mid N$. In this situation,
631
it is well-known that the Jacobian\index{Jacobian!of $X_0(N)$}
632
$J_0(N)$ of $X_0(N)$ has semistable
633
reduction at~$p$.
634
635
\subsection{Newforms and optimal quotients}
636
The Hecke algebra\index{Hecke algebra|textit}
637
$$\T=\Z[\ldots T_n\ldots]\subset\End(J_0(N))$$
638
is a commutative ring of endomorphisms of~$J_0(N)$ of $\Z$-rank
639
equal to the dimension $J_0(N)$.
640
The character group $X_{J_0(N)}$ of $J_0(N)$ at~$p$
641
is equipped with a functorial action of~$\T$.
642
The Hecke algebra~$\T$ also acts on the complex vector space
643
$S = S_2(\Gamma_0(N),\C)$
644
of cusp forms.
645
646
647
A newform~$f$ is an eigenform normalized so that the coefficient
648
of~$q$ in the Fourier expansion of~$f$ at the cusp~$\infty$ is~$1$, and
649
such that~$f$ is not a modular form of any level $N'\mid N$, with $N'$ a
650
proper divisor of~$N$.
651
652
Let~$f$ be a newform, and associate to~$f$ the ideal $I_f$ of
653
the Hecke algebra~$\T$ of elements which
654
annihilate~$f$. Then $\O_f=\T/I_f$ is an
655
order in the ring of integers of the totally real number field
656
$K_f$ obtained by adjoining the Fourier coefficients of~$f$ to~$\Q$.
657
The quotient
658
$$A_f = J_0(N)/ I_f J_0(N)$$
659
is an optimal quotient\index{Optimal quotient} of $J_0(N)$
660
of dimension equal to $[K_f:\Q]$. It is purely
661
toric\index{Purely toric reduction} at~$p$, since $p\mid \mid N$.
662
663
\subsection{Homology and the modular degree}
664
Let $H=H_1(X_0(N),\Z)$ be the integral homology of the
665
complex algebraic curve $X_0(N)$. Integration defines a
666
$\T$-equivariant nondegenerate\index{Integration pairing}
667
pairing
668
$S \cross H \ra \C$.
669
This pairing induces a map
670
$\alp: H \ra \Hom_\C(S,\C)$.
671
672
\begin{theorem}\label{Af}
673
We have the following commutative diagram of $\T$-modules:
674
$$\xymatrix{
675
H[I_f] \[email protected]{^(->}[r]\[email protected]{^(->}[d] & H \[email protected]{->>}[r]\[email protected]{^(->}[d]
676
& \alp(H)\ar[d]\[email protected]{^(->}[d]\\
677
\Hom_\C(S,\C)[I_f]\[email protected]{^(->}[r]\[email protected]{->>}[d] & \Hom_\C(S,\C)\[email protected]{->>}[r]\[email protected]{->>}[d]
678
&\Hom_\C(S[I_f],\C)\[email protected]{->>}[d]\\
679
A_f^{\vee}(\C)\[email protected]{^(->}[r]\[email protected]/_2pc/_{\theta_A}[rr] & J(\C) \[email protected]{->>}[r] & A_f(\C) \\
680
}$$
681
\end{theorem}
682
\begin{proof}
683
This can be deduced from \cite{shimura:factors}.
684
See also Section~\ref{sec:tori}.
685
\end{proof}
686
687
\begin{corollary}\label{moduluscomp}
688
$m_A^2 = [\alp(H) : \alp(H[I_f])]$.
689
\end{corollary}
690
\begin{proof}
691
Recall that $m_A$ is by definition equal to $\sqrt{\deg(\theta_A)}$.
692
The kernel of an isogeny between complex tori is
693
isomorphic to the cokernel of the induced map
694
on lattices. The corollary now follows from
695
the diagram of Theorem~\ref{Af},
696
which indicates that the index $[\alp(H):\alp(H[I_f])]$
697
is the cokernel of the map $H[I_f]\ra \alp(H).$
698
699
For more details, see Section~\ref{sec:moddeg}.
700
\end{proof}
701
702
\subsection{Rational points of the component group (Tamagawa numbers)}
703
Let $\Frob_p:X_J\ra X_J$ denote the map induced by the
704
Frobenius automorphism.
705
We have $\Frob_p=-W_p$, where $W_p$ is the map induced
706
by the Atkin-Lehner\index{Atkin-Lehner involution} involution on $J_0(p)$.
707
Let~$f$ be a newform, $A=A_f$ the corresponding optimal
708
quotient, and $w_p$ the sign of the eigenvalue of
709
$W_p$ on $f$.
710
\begin{proposition}
711
$$\Phi_A(\Fp)
712
= \begin{cases}
713
\Phi_A(\Fpbar) & \text{if $w_p=-1$},\\
714
\Phi_A(\Fpbar)[2] & \text{if $w_p=1$.}
715
\end{cases}$$
716
\end{proposition}
717
\begin{proof}
718
If $w_p=-1$, then $\Frob_p=1$ and the $\Gal(\Fpbar/\Fp)$-action
719
of $\Phi_A(\Fpbar)$ is trivial. In this case
720
$\Phi(\Fp)=\Phi(\Fpbar)$.
721
Next suppose $w_p=1$. Recall that we have an exact sequence
722
$$0\ra X_{A^{\vee}} \ra \Hom(X_A,\Z) \ra \Phi_A \ra 0.$$
723
Since $W_p$ acts as $+1$ on~$f$, it also acts as $+1$ on
724
each of the four modules~$A$,~$X_A$,~$\Hom(X_A,\Z)$, and~$\Phi_A$.
725
Thus $\Frob_p=-W_p$ acts as $-1$ on $\Phi_A$. Since the subgroup
726
of $2$-torsion elements of a finite abelian group equals the subgroup
727
of elements fixed under $-1$, it follows that
728
$\Phi_A(\Fp) = \Phi_A(\Fpbar)[2]$.
729
\end{proof}
730
731
{\bf WARNING:} When we extend this result to the whole
732
of $J_0(N)$, it is necessary to be exceedingly careful!
733
The action of $\Frob_p=T_p$ need
734
not be by $\pm 1$, even though it must be by an involution
735
of order~$2$. For example, the component group of
736
$J_0(65)$ at~$5$ is cyclic of order~$42$. The action
737
of $\Frob_5$ is by multiplication by $-13$. Note that
738
$(-13)^2 = 1 \pmod{42}$. The fixed points of
739
multiplication by~$-13$ is the order~$14$ subgroup
740
of $\Z/42\Z$.
741
742
\section{Computations}
743
Using the algorithms of Chapter~\ref{chap:computing},
744
we can enumerate the optimal
745
quotients $A_f$ of $J_0(N)$ and compute the modular degree $m_A$.
746
The method of graphs\index{Method of graphs}
747
(see \cite{mestre:graphs}) and
748
quaternion algebras (see \cite{kohel:hecke})\index{Quaternion algebras}
749
can be used to compute $X=X_{J_0(N)}$ with its $\T$-action
750
and the monodromy pairing\index{Monodromy pairing}. We can then
751
compute the following three modules:
752
the saturated submodule
753
$\cL=\bigcap_{t\in I_f} \ker(t)$
754
of~$X$,
755
the character group modular degree $m_X=m_\cL$,
756
and $\Phi_X=\Phi_\cL$.
757
By Theorem~\ref{formula} we obtain
758
$$\#\Phi_A = \#\Phi_X \cdot \frac{m_A}{m_X}.$$
759
760
Using this method, we have computed $\#\Phi_A$ in a number of cases.
761
We give tables that report on some of these computations in
762
Secton~\ref{sec:compgrptables}.
763
In the next section we discuss a conjecture and a question,
764
which were both suggested by our numerical computations.
765
766
\subsection{Conjectures and questions}%
767
\label{sec:compgroupconjectures}%
768
\index{Conjecture!about modular degree}%
769
Suppose that $N=pM$ with $(p,M)=1$.
770
Let
771
$$H_{\new} =
772
\ker\,\Bigl( H_1(X_0(N),\Z)\lra
773
H_1(X_0(M),\Z)\oplus H_1(X_0(M),\Z)\Bigr),$$
774
where the map is induced by the two natural
775
degeneracy maps $X_0(N)\ra X_0(M)$.
776
777
The Hecke algebra\index{Hecke algebra}~$\T$ acts on $H_{\new}$,
778
and also on the submodule $H_{\new}[I_f]$ of those elements that
779
are annihilated by $I_f$.
780
Integration defines a map
781
$\alp: H_{\new}\ra \Hom(S[I_f],\C)$.
782
Define the $p$-new homology modular degree $m_H$ by
783
$$m_H^2 = [\alp(H_{\new}) : \alp(H_{\new}[I_f])].$$
784
We expect that there is a very close relationship
785
between $m_X$ and $m_H$.
786
\begin{question}
787
Is $m_X$ equal to $m_H$?
788
\end{question}
789
790
The following conjecture offers
791
a refinement of some of the results of \cite{mazur:eisenstein}.
792
\index{Conjecture!refined Eisenstein}
793
\begin{conjecture}[Refined Eisenstein conjecture]\label{conj:iso}
794
Let~$p$ be a prime and let $f_1,\ldots,f_n$ be a set of
795
representatives for the Galois-conjugacy classes of newforms
796
in $S_2(\Gamma_0(p))$. Let $A_1,\ldots, A_n$ be the optimal
797
quotients associated to $f_1,\ldots,f_n$, respectively.
798
Then for each~$i$, $i=1,\ldots,n$, we have
799
$$\#A_i(\Q)_{\tor}=\#\Phi_{A_i}(\Fpbar)=\#\Phi_{A_i}(\Fp).$$
800
Furthermore,
801
$$\#\Phi_{J_0(p)}(\Fpbar)= \prod_{i=1}^d \#\Phi_{A_i}(\Fpbar).$$
802
\end{conjecture}
803
804
We have verified Conjecture~\ref{conj:iso} for all $p\leq 757$,
805
and, up to a power of~$2$, for all $p< 2000$.
806
807
\begin{remark}
808
It is tempting to guess that, e.g., the natural map
809
$$\Phi_{J_0(113)}(\Fpbar)\ra \prod_{i=1}^4 \Phi_{A_i}(\Fpbar)$$
810
is an isomorphism. Two of the $\Phi_{A_i}(\Fpbar)$ have
811
order~$2$, so the product $\prod \Phi_{A_i}(\Fpbar)$ can not
812
be a cyclic group. However, the groups
813
$\Phi_{J_0(p)}(\Fpbar)$ are known to be
814
cyclic for all primes~$p$.
815
\end{remark}
816
817
\subsection{Tables}\label{sec:compgrptables}%
818
We have computed component groups of many optimal quotients
819
$A_f$ of $J_0(N)$.
820
In this section we provide tables, which hint at the data
821
we have gathered. Our notation for optimal quotients
822
is described in Section~\ref{sec:optquo-notation}.
823
See also Table~\ref{table:shacompgps}.
824
825
\subsubsection{Table~\ref{tbl:lowlevel}: Component groups at low level}
826
Table~\ref{tbl:lowlevel} gives the component groups of the
827
quotients $A_f$ of $J_0(N)$ for $N\leq 106$.
828
The column labeled $d$ contains the
829
dimensions of the $A_f$,
830
and the column labeled $\#\Phi_{A,p}$ contains a list
831
of the orders of the component groups of $A_f$,
832
one for each divisor~$p$ of~$N$, ordered by increasing~$p$.
833
An entry of ``?'' indicates that $p^2\mid N$, so our algorithm
834
does not apply.
835
A component group order is starred if the $\Gal(\Fpbar/\Fp)$-action is
836
nontrivial.
837
838
839
\subsubsection{Table~\ref{table:big1}--\ref{table:big2}: Big component groups}
840
Using the algorithms described in Section~\ref{sec:ratpartformula},
841
we computed the rational numbers $L(A,1)/\Omega_A$
842
for every optimal quotient~$A$ that is attached to a newform
843
of level $\leq 1500$.
844
There are exactly~$5$ optimal quotients~$A$ such that the numerator of
845
$L(A,1)/\Omega_A$ is nonzero and divisible by a prime $>10^9$.
846
The Birch and Swinnerton-Dyer conjecture%
847
\index{BSD conjecture!predicts large component groups}
848
predicts that these large
849
prime divisors must divide either $\#\Phi_A$ or
850
the Shafarevich-Tate\index{Shafarevich-Tate group}
851
group of~$A$. This is the case, as Table~\ref{table:big2} shows.
852
853
854
\subsubsection{Table~\ref{table:compj0n}: Quotients of $J_0(N)$}
855
Table~\ref{table:compj0n} contains all of the invariants involved
856
in the computation of component groups for
857
each of the newform optimal quotients of levels $65$, $66$, $68$, and $69$.
858
859
\subsubsection{Table~\ref{table:compprime}:
860
Quotients of $J_0(p)^-$}
861
We computed
862
the quantities $m_A$, $m_X$ and $\Phi_X$ for each abelian
863
variety $A=A_f$ associated to a newform of prime level~$p$ with
864
$p\leq 757$. The results are as follows:
865
\begin{enumerate}
866
\item In all cases $m_A=m_X$, so the map $\Phi_J\ra \Phi_A$
867
is surjective.
868
\item $\Phi_A=1$ whenever the sign of the Atkin-Lehner involution
869
$w_p$ on~$A$ is~$1$.
870
\item $\prod \#\Phi_A(\Fpbar) = \#\Phi_J(\Fpbar)$
871
\end{enumerate}
872
Table~\ref{table:compprime}
873
lists those~$A$ of level $\leq 631$ for which $w_p=-1$, along with
874
the order of the corresponding component group.
875
876
\begin{table}
877
\ssp
878
\begin{center}
879
\caption{Component groups at low level\label{tbl:lowlevel}}
880
\end{center}%
881
\vspace{-.3in}%
882
\index{Table of!component groups at low level}%
883
\index{Component group!table of}%
884
$$
885
\begin{array}{lcl}
886
A & \, d \, & \, \#\Phi_{A,p}\, \\
887
\vspace{-2ex}\\
888
{\bf 11A} & 1 & 5\\
889
{\bf 14A} & 1 & 6^*,3\\
890
{\bf 15A} & 1 & 4^*,4\\
891
{\bf 17A} & 1 & 4\\
892
\vspace{-2ex}\\
893
{\bf 19A} & 1 & 3\\
894
{\bf 20A} & 1 & ?,2^*\\
895
{\bf 21A} & 1 & 4,2^*\\
896
{\bf 23A} & 2 & 11\\
897
\vspace{-2ex}\\
898
{\bf 24A} & 1 & ?,2^*\\
899
{\bf 26A} & 1 & 3^*,3\\
900
{\bf 26B} & 1 & 7,1^*\\
901
{\bf 27A} & 1 & ?\\
902
\vspace{-2ex}\\
903
{\bf 29A} & 2 & 7\\
904
{\bf 30A} & 1 & 4^*,3,1^*\\
905
{\bf 31A} & 2 & 5\\
906
{\bf 32A} & 1 & ?\\
907
\vspace{-2ex}\\
908
{\bf 33A} & 1 & 6^*,2\\
909
{\bf 34A} & 1 & 6,1^*\\
910
{\bf 35A} & 1 & 3^*,3\\
911
{\bf 35B} & 2 & 8,4^*\\
912
\vspace{-2ex}\\
913
{\bf 36A} & 1 & ?,?\\
914
{\bf 37A} & 1 & 1^*\\
915
{\bf 37B} & 1 & 3\\
916
{\bf 38A} & 1 & 9^*,3\\
917
\vspace{-2ex}\\
918
{\bf 38B} & 1 & 5,1^*\\
919
{\bf 39A} & 1 & 2^*,2\\
920
{\bf 39B} & 2 & 14,2^*\\
921
{\bf 40A} & 1 & ?,2\\
922
\vspace{-2ex}\\
923
{\bf 41A} & 3 & 10\\
924
{\bf 42A} & 1 & 8,2^*,1^*\\
925
{\bf 43A} & 1 & 1^*\\
926
{\bf 43B} & 2 & 7\\
927
\vspace{-2ex}\\
928
{\bf 44A} & 1 & ?,1^*\\
929
{\bf 45A} & 1 & ?,1^*\\
930
{\bf 46A} & 1 & 10^*,1\\
931
{\bf 47A} & 4 & 23\\
932
\vspace{-2ex}\\
933
{\bf 48A} & 1 & ?,2\\
934
{\bf 49A} & 1 & ?\\
935
{\bf 50A} & 1 & 1^*,?\\
936
{\bf 50B} & 1 & 5,?\\
937
\end{array}\quad
938
\begin{array}{lcl}
939
A & \, d \, & \, \#\Phi_{A,p}\, \\
940
\vspace{-2ex}\\
941
{\bf 51A} & 1 & 3,1^*\\
942
{\bf 51B} & 2 & 16^*,4\\
943
{\bf 52A} & 1 & ?,2^*\\
944
{\bf 53A} & 1 & 1^*\\
945
\vspace{-2ex}\\
946
{\bf 53A} & 1 & 1^*\\
947
{\bf 53B} & 3 & 13\\
948
{\bf 54A} & 1 & 3^*,?\\
949
{\bf 54B} & 1 & 3,?\\
950
\vspace{-2ex}\\
951
{\bf 55A} & 1 & 2,2^*\\
952
{\bf 55B} & 2 & 14^*,2\\
953
{\bf 56A} & 1 & ?,1\\
954
{\bf 56B} & 1 & ?,1^*\\
955
\vspace{-2ex}\\
956
{\bf 57A} & 1 & 2^*,1^*\\
957
{\bf 57B} & 1 & 2,2^*\\
958
{\bf 57C} & 1 & 10,1^*\\
959
{\bf 58A} & 1 & 2^*,1^*\\
960
\vspace{-2ex}\\
961
{\bf 58B} & 1 & 10,1^*\\
962
{\bf 59A} & 5 & 29\\
963
{\bf 61A} & 1 & 1^*\\
964
{\bf 61B} & 3 & 5\\
965
\vspace{-2ex}\\
966
{\bf 62A} & 1 & 4,1^*\\
967
{\bf 62B} & 2 & 66^*,3\\
968
{\bf 63A} & 1 & ?,1^*\\
969
{\bf 63B} & 2 & ?,3\\
970
\vspace{-2ex}\\
971
{\bf 64A} & 1 & ?\\
972
{\bf 65A} & 1 & 1^*,1^*\\
973
{\bf 65B} & 2 & 3^*,3\\
974
{\bf 65C} & 2 & 7,1^*\\
975
\vspace{-2ex}\\
976
{\bf 66A} & 1 & 2^*,3,1^*\\
977
{\bf 66B} & 1 & 4,1^*,1^*\\
978
{\bf 66C} & 1 & 10,5,1\\
979
{\bf 67A} & 1 & 1\\
980
\vspace{-2ex}\\
981
{\bf 67B} & 2 & 1^*\\
982
{\bf 67C} & 2 & 11\\
983
{\bf 68A} & 2 & ?,2^*\\
984
{\bf 69A} & 1 & 2,1^*\\
985
\vspace{-2ex}\\
986
{\bf 69B} & 2 & 22^*,2\\
987
{\bf 70A} & 1 & 4,2^*,1^*\\
988
{\bf 71A} & 3 & 5\\
989
{\bf 71B} & 3 & 7\\
990
\end{array}\quad
991
\begin{array}{lcl}
992
A & \, d \, & \, \#\Phi_{A,p}\, \\
993
\vspace{-2ex}\\
994
{\bf 72A} & 1 & ?,?\\
995
{\bf 73A} & 1 & 2\\
996
{\bf 73B} & 2 & 1^*\\
997
{\bf 73C} & 2 & 3\\
998
\vspace{-2ex}\\
999
{\bf 74A} & 2 & 9^*,3\\
1000
{\bf 74B} & 2 & 95,1^*\\
1001
{\bf 75A} & 1 & 1^*,?\\
1002
{\bf 75B} & 1 & 1,?\\
1003
\vspace{-2ex}\\
1004
{\bf 75C} & 1 & 5,?\\
1005
{\bf 76A} & 1 & ?,1^*\\
1006
{\bf 77A} & 1 & 2^*,1^*\\
1007
{\bf 77B} & 1 & 3^*,2\\
1008
\vspace{-2ex}\\
1009
{\bf 77C} & 1 & 6,3^*\\
1010
{\bf 77D} & 2 & 2,2^*\\
1011
{\bf 78A} & 1 & 16^*,5^*,1\\
1012
{\bf 79A} & 1 & 1^*\\
1013
\vspace{-2ex}\\
1014
{\bf 79B} & 5 & 13\\
1015
{\bf 80A} & 1 & ?,2\\
1016
{\bf 80B} & 1 & ?,2^*\\
1017
{\bf 81A} & 2 & ?\\
1018
\vspace{-2ex}\\
1019
{\bf 82A} & 1 & 2^*,1^*\\
1020
{\bf 82B} & 2 & 28,1^*\\
1021
{\bf 83A} & 1 & 1^*\\
1022
{\bf 83B} & 6 & 41\\
1023
\vspace{-2ex}\\
1024
{\bf 84A} & 1 & ?,1^*,2^*\\
1025
{\bf 84B} & 1 & ?,3,2\\
1026
{\bf 85A} & 1 & 2^*,1\\
1027
{\bf 85B} & 2 & 2^*,1^*\\
1028
\vspace{-2ex}\\
1029
{\bf 85C} & 2 & 6,1^*\\
1030
{\bf 86A} & 2 & 21^*,3\\
1031
{\bf 86B} & 2 & 55,1^*\\
1032
{\bf 87A} & 2 & 5,1^*\\
1033
\vspace{-2ex}\\
1034
{\bf 87B} & 3 & 92^*,4\\
1035
{\bf 88A} & 1 & ?,1^*\\
1036
{\bf 88B} & 2 & ?,2^*\\
1037
{\bf 89A} & 1 & 1^*\\
1038
\vspace{-2ex}\\
1039
{\bf 89B} & 1 & 2\\
1040
{\bf 89C} & 5 & 11\\
1041
{\bf 90A} & 1 & 2^*,?,3\\
1042
{\bf 90B} & 1 & 6,?,1^*\\
1043
\end{array}\quad
1044
\begin{array}{lcl}
1045
A & \, d \, & \, \#\Phi_{A,p}\, \\
1046
\vspace{-2ex}\\
1047
{\bf 90C} & 1 & 4,?,1\\
1048
{\bf 91A} & 1 & 1^*,1^*\\
1049
{\bf 91B} & 1 & 1,1\\
1050
{\bf 91C} & 2 & 7,1^*\\
1051
\vspace{-2ex}\\
1052
{\bf 91D} & 3 & 4^*,8\\
1053
{\bf 92A} & 1 & ?,1^*\\
1054
{\bf 92B} & 1 & ?,1\\
1055
{\bf 93A} & 2 & 4^*,1^*\\
1056
\vspace{-2ex}\\
1057
{\bf 93B} & 3 & 64,2^*\\
1058
{\bf 94A} & 1 & 2,1^*\\
1059
{\bf 94B} & 2 & 94^*,1\\
1060
{\bf 95A} & 3 & 10,2^*\\
1061
\vspace{-2ex}\\
1062
{\bf 95B} & 4 & 54^*,6\\
1063
{\bf 96A} & 1 & ?,2\\
1064
{\bf 96B} & 1 & ?,2^*\\
1065
{\bf 97A} & 3 & 1^*\\
1066
\vspace{-2ex}\\
1067
{\bf 97B} & 4 & 8\\
1068
{\bf 98A} & 1 & 2^*,?\\
1069
{\bf 98B} & 2 & 14,?\\
1070
{\bf 99A} & 1 & ?,1^*\\
1071
\vspace{-2ex}\\
1072
{\bf 99B} & 1 & ?,1\\
1073
{\bf 99C} & 1 & ?,1^*\\
1074
{\bf 99D} & 1 & ?,1^*\\
1075
{\bf 100A} & 1 & ?,?\\
1076
\vspace{-2ex}\\
1077
{\bf 101A} & 1 & 1^*\\
1078
{\bf 101B} & 7 & 25\\
1079
{\bf 102A} & 1 & 2^*,2^*,1^*\\
1080
{\bf 102B} & 1 & 6^*,6,1^*\\
1081
\vspace{-2ex}\\
1082
{\bf 102C} & 1 & 8,4,1\\
1083
{\bf 103A} & 2 & 1^*\\
1084
{\bf 103B} & 6 & 17\\
1085
{\bf 104A} & 1 & ?,1^*\\
1086
\vspace{-2ex}\\
1087
{\bf 104B} & 2 & ?,2\\
1088
{\bf 105A} & 1 & 1,1,1\\
1089
{\bf 105B} & 2 & 10^*,2^*,2\\
1090
{\bf 106A} & 1 & 4^*,1^*\\
1091
\vspace{-2ex}\\
1092
{\bf 106B} & 1 & 5^*,1\\
1093
{\bf 106C} & 1 & 24,1^*\\
1094
{\bf 106D} & 1 & 3,1^*\\
1095
&&\\
1096
\end{array}$$
1097
\end{table}
1098
1099
1100
\begin{table}
1101
\ssp
1102
\caption{Big $L(A,1)/\Omega_A$\label{table:big1}}
1103
\index{Table of!big $L(A,1)/\Omega_A$}
1104
\index{Component group!table of}
1105
$$\begin{array}{lccc}
1106
A & \text{dim} &
1107
N & \text{\qquad $L(A,1)/\Omega_A\cdot \text{Manin constant}$\qquad }\\
1108
\vspace{-2ex} && & \\
1109
\text{\bf 1154E} & 20&2\cdot 577 & 2^?\cdot 85495047371/17^2\\
1110
\text{\bf 1238G} & 19& 2\cdot 619 & 2^?\cdot 7553329019/5\cdot 31\\
1111
\text{\bf 1322E} & 21& 2\cdot 661 & 2^?\cdot 57851840099/331\\
1112
\text{\bf 1382D} & 20& 2\cdot 691 & 2^?\cdot 37 \cdot 1864449649 /173\\
1113
\text{\bf 1478J} & 20
1114
& 2\cdot 739 & 2^?\cdot 7\cdot 29\cdot 1183045463 / 5\cdot 37\\
1115
\end{array}$$
1116
\end{table}
1117
1118
\begin{table}
1119
\ssp
1120
\caption{Big component groups\label{table:big2}}
1121
\index{Table of!big component groups}
1122
\index{Component group!table of}
1123
$$\begin{array}{lcccccc}
1124
A &p & w & \#\Phi_X & m_X & \#\Phi_A(\Fpbar) \\
1125
\vspace{-1ex} && & & & & \\
1126
\text{\bf 1154E} &2 & - & 17^2 & 2^{24}
1127
& 2^?\cdot 17^2 \cdot 85495047371 \\
1128
&577& + & 1 & 2^{26}\cdot85495047371
1129
& 2^? \\
1130
\vspace{-1ex}&&&&&\\
1131
\text{\bf 1238G} & 2 &- & 5\cdot31 & 2^{26} &2^?\cdot 5\cdot31\cdot7553329019 \\
1132
& 619 &+ & 1 & 2^{28}\cdot 7553329019 & 2^? \\
1133
\vspace{-1ex}&&&&&\\
1134
\text{\bf 1322E} & 2 & - & 331 & 2^{28} & 2^?\cdot 331 \cdot 57851840099\\
1135
& 661& + & 1 & 2^{32}\cdot 57851840099 & 2^?\\
1136
\vspace{-1ex}&&&&&\\
1137
\text{\bf 1382D} & 2 & - & 173 & 2^{29} & 2^?\cdot 37\cdot173\cdot 1864449649 \\
1138
& 691 & + & 1 & 2^{31}\cdot 37\cdot 1864449649 & 2^?\\
1139
\vspace{-1ex}&&&&&\\
1140
\text{\bf 1478J} & 2 & - & 5\cdot37 &2^{31}
1141
& 2^?\cdot5\cdot 7\cdot29\cdot37\cdot1183045463 \\
1142
& 739 & + & 1 &2^{33}\cdot 7\cdot 29\cdot 1183045463
1143
& 2^? \\
1144
\end{array}$$
1145
\end{table}
1146
1147
1148
\begin{table}
1149
\ssp
1150
\caption{Component groups of quotients of $J_0(N)$\label{table:compj0n}}%
1151
\index{Table of!component groups of quotients}%
1152
\index{Component group!table of}%
1153
$$\begin{array}{lccccccc}
1154
A & \text{dim} & p & w_p & \#\Phi_X & m_X & m_A & \#\Phi_A \\
1155
\vspace{-1ex} && & & & & & \\
1156
\text{\bf 65A} & 1& 5 & +& 1 &2 & 2 & 1\\
1157
&& 13 &+& 1 & 2 & & 1\\
1158
1159
\text{\bf 65B} &2& 5 &+& 3 & 2^2 & 2^2 & 3\\
1160
&& 13 &- & 3 & 2^2 & & 3\\
1161
1162
\text{\bf 65C}&2 & 5 &-& 7 & 2^2& 2^2 &7 \\
1163
&& 13 &+ & 1 & 2^2& & 1\\
1164
1165
\vspace{-1ex} && & & & & & \\
1166
\text{\bf 66A}&1& 2 &+ & 1 &2 & 2^2&2 \\
1167
&& 3 &- & 3 &2^2 & & 3\\
1168
&& 11 &+ & 1 &2^2 & &1 \\
1169
1170
\text{\bf 66B}&1& 2 &- & 2 &2 & 2^2& 2^2\\
1171
&& 3 &+ & 1 &2^2& & 1\\
1172
&& 11 &+ & 1 &2^2 & & 1\\
1173
1174
\text{\bf 66C}&1& 2 & -& 1 & 2& 2^2\cdot 5& 2\cdot5\\
1175
&& 3 &- & 1 & 2^2 & & 5\\
1176
&& 11 &- & 1 & 2^2\cdot5 & &1 \\
1177
1178
\vspace{-1ex} && & & & & & \\
1179
\text{\bf 68A}&2&17&+&2 &2\cdot3 &2\cdot3 & 2 \\
1180
1181
\vspace{-1ex} && & & & & & \\
1182
\text{\bf 69A} &1&3 &-&2 &2 & 2& 2\\
1183
&&23 &+& 1&2 & & 1\\
1184
1185
\text{\bf 69B} &2&3 &+&2 &2 &2\cdot11& 2\cdot11 \\
1186
&&23 &-&2 &2\cdot11 && 2 \\
1187
1188
\end{array}$$
1189
\end{table}
1190
1191
1192
\begin{table}
1193
\ssp
1194
\caption{Component groups of quotients of $J_0(p)^{-}$
1195
\index{Table of!component groups at prime level}
1196
\index{Component group!table of}
1197
\label{table:compprime}}%
1198
\vspace{-.3in}%
1199
$$
1200
\begin{array}{lcc}
1201
\vspace{-2ex}\\
1202
A & d & \#\Phi_A \\
1203
\vspace{-2ex}\\
1204
{\bf 11A}&1&5\\
1205
{\bf 17A}&1&2^2\\
1206
{\bf 19A}&1&3\\
1207
{\bf 23A}&2&11\\
1208
\vspace{-2ex}& &\\
1209
{\bf 29A}&2&7\\
1210
{\bf 31A}&2&5\\
1211
{\bf 37B}&1&3\\
1212
{\bf 41A}&3&2\cdot5\\
1213
\vspace{-2ex}& &\\
1214
{\bf 43B}&2&7\\
1215
{\bf 47A}&4&23\\
1216
{\bf 53B}&3&13\\
1217
{\bf 59A}&5&29\\
1218
\vspace{-2ex}& &\\
1219
{\bf 61B}&3&5\\
1220
{\bf 67A}&1&1\\
1221
{\bf 67C}&2&11\\
1222
{\bf 71A}&3&5\\
1223
\vspace{-2ex}& &\\
1224
{\bf 71B}&3&7\\
1225
{\bf 73A}&1&2\\
1226
{\bf 73C}&2&3\\
1227
{\bf 79B}&5&13\\
1228
\vspace{-2ex}& &\\
1229
{\bf 83B}&6&41\\
1230
{\bf 89B}&1&2\\
1231
{\bf 89C}&5&11\\
1232
{\bf 97B}&4&2^3\\
1233
\vspace{-2ex}& &\\
1234
{\bf 101B}&7&5^2\\
1235
{\bf 103B}&6&17\\
1236
{\bf 107B}&7&53\\
1237
{\bf 109A}&1&1\\
1238
\vspace{-2ex}& &\\
1239
{\bf 109C}&4&3^2\\
1240
{\bf 113A}&1&2\\
1241
{\bf 113B}&2&2\\
1242
{\bf 113D}&3&7\\
1243
\vspace{-2ex}& &\\
1244
{\bf 127B}&7&3\cdot7\\
1245
{\bf 131B}&10&5\cdot13\\
1246
{\bf 137B}&7&2\cdot17\\
1247
{\bf 139A}&1&1\\
1248
\vspace{-2ex}& &\\
1249
{\bf 139C}&7&23\\
1250
{\bf 149B}&9&37\\
1251
{\bf 151B}&3&1\\
1252
{\bf 151C}&6&5^2\\
1253
\end{array}\,\,
1254
\begin{array}{lcc}
1255
\vspace{-2ex}\\
1256
A & d& \#\Phi_A \\
1257
\vspace{-2ex}\\
1258
{\bf 157B}&7&13\\
1259
{\bf 163C}&7&3^3\\
1260
{\bf 167B}&12&83\\
1261
{\bf 173B}&10&43\\
1262
\vspace{-2ex} &\\
1263
{\bf 179A}&1&1\\
1264
{\bf 179C}&11&89\\
1265
{\bf 181B}&9&3\cdot5\\
1266
{\bf 191B}&14&5\cdot19\\
1267
\vspace{-2ex} &\\
1268
{\bf 193C}&8&2^4\\
1269
{\bf 197C}&10&7^2\\
1270
{\bf 199A}&2&1\\
1271
{\bf 199C}&10&3\cdot11\\
1272
\vspace{-2ex} &\\
1273
{\bf 211A}&2&5\\
1274
{\bf 211D}&9&7\\
1275
{\bf 223C}&12&37\\
1276
{\bf 227B}&2&1\\
1277
\vspace{-2ex} &\\
1278
{\bf 227C}&2&1\\
1279
{\bf 227E}&10&113\\
1280
{\bf 229C}&11&19\\
1281
{\bf 233A}&1&2\\
1282
\vspace{-2ex} &\\
1283
{\bf 233C}&11&29\\
1284
{\bf 239B}&17&7\cdot17\\
1285
{\bf 241B}&12&2^2\cdot5\\
1286
{\bf 251B}&17&5^3\\
1287
\vspace{-2ex} &\\
1288
{\bf 257B}&14&2^6\\
1289
{\bf 263B}&17&131\\
1290
{\bf 269C}&16&67\\
1291
{\bf 271B}&16&3^2\cdot5\\
1292
\vspace{-2ex} &\\
1293
{\bf 277B}&3&1\\
1294
{\bf 277D}&9&23\\
1295
{\bf 281B}&16&2\cdot5\cdot7\\
1296
{\bf 283B}&14&47\\
1297
\vspace{-2ex} &\\
1298
{\bf 293B}&16&73\\
1299
{\bf 307A}&1&1\\
1300
{\bf 307B}&1&1\\
1301
{\bf 307C}&1&1\\
1302
\vspace{-2ex} &\\
1303
{\bf 307D}&1&1\\
1304
{\bf 307E}&2&3\\
1305
{\bf 307F}&9&17\\
1306
{\bf 311B}&22&5\cdot31\\
1307
\end{array}\,\,
1308
\begin{array}{lcc}
1309
\vspace{-2ex}\\
1310
A & d &\#\Phi_A \\
1311
\vspace{-2ex}\\
1312
{\bf 313A}&2&1\\
1313
{\bf 313C}&12&2\cdot13\\
1314
{\bf 317B}&15&79\\
1315
{\bf 331D}&16&5\cdot11\\
1316
\vspace{-2ex} &\\
1317
{\bf 337B}&15&2^2\cdot7\\
1318
{\bf 347D}&19&173\\
1319
{\bf 349B}&17&29\\
1320
{\bf 353A}&1&2\\
1321
\vspace{-2ex} &\\
1322
{\bf 353B}&3&2\\
1323
{\bf 353D}&14&2\cdot11\\
1324
{\bf 359D}&24&179\\
1325
{\bf 367B}&19&61\\
1326
\vspace{-2ex} &\\
1327
{\bf 373C}&17&31\\
1328
{\bf 379B}&18&3^2\cdot7\\
1329
{\bf 383C}&24&191\\
1330
{\bf 389A}&1&1\\
1331
\vspace{-2ex} &\\
1332
{\bf 389E}&20&97\\
1333
{\bf 397B}&2&1\\
1334
{\bf 397C}&5&11\\
1335
{\bf 397D}&10&3\\
1336
\vspace{-2ex} &\\
1337
{\bf 401B}&21&2^2\cdot5^2\\
1338
{\bf 409B}&20&2\cdot17\\
1339
{\bf 419B}&26&11\cdot19\\
1340
{\bf 421B}&19&5\cdot7\\
1341
\vspace{-2ex} &\\
1342
{\bf 431B}&1&1\\
1343
{\bf 431D}&3&1\\
1344
{\bf 431F}&24&5\cdot43\\
1345
{\bf 433A}&1&1\\
1346
\vspace{-2ex} &\\
1347
{\bf 433B}&3&1\\
1348
{\bf 433D}&16&2^2\cdot3^2\\
1349
{\bf 439C}&25&73\\
1350
{\bf 443C}&1&1\\
1351
\vspace{-2ex} &\\
1352
{\bf 443E}&22&13\cdot17\\
1353
{\bf 449B}&23&2^4\cdot7\\
1354
{\bf 457C}&20&2\cdot19\\
1355
{\bf 461D}&26&5\cdot23\\
1356
\vspace{-2ex} &\\
1357
{\bf 463B}&22&7\cdot11\\
1358
{\bf 467C}&26&233\\
1359
{\bf 479B}&32&239\\
1360
{\bf 487A}&2&1\\
1361
\end{array}\,\,
1362
\begin{array}{lcc}
1363
\vspace{-2ex}&\\
1364
A & d & \#\Phi_A \\
1365
\vspace{-2ex}&\\
1366
{\bf 487B}&2&3\\
1367
{\bf 487C}&3&1\\
1368
{\bf 487D}&16&3^3\\
1369
{\bf 491C}&29&5\cdot7^2\\
1370
\vspace{-2ex} &\\
1371
{\bf 499C}&23&83\\
1372
{\bf 503B}&1&1\\
1373
{\bf 503C}&1&1\\
1374
{\bf 503D}&3&1\\
1375
\vspace{-2ex} &\\
1376
{\bf 503F}&26&251\\
1377
{\bf 509B}&28&127\\
1378
{\bf 521B}&29&2\cdot5\cdot13\\
1379
{\bf 523C}&26&3\cdot29\\
1380
\vspace{-2ex} &\\
1381
{\bf 541B}&24&3^2\cdot5\\
1382
{\bf 547C}&25&7\cdot13\\
1383
{\bf 557B}&1&1\\
1384
{\bf 557D}&26&139\\
1385
\vspace{-2ex} &\\
1386
{\bf 563A}&1&1\\
1387
{\bf 563E}&31&281\\
1388
{\bf 569B}&31&2\cdot71\\
1389
{\bf 571A}&1&1\\
1390
\vspace{-2ex} &\\
1391
{\bf 571B}&1&1\\
1392
{\bf 571C}&2&1\\
1393
{\bf 571D}&2&1\\
1394
{\bf 571F}&4&1\\
1395
\vspace{-2ex} &\\
1396
{\bf 571I}&18&5\cdot19\\
1397
{\bf 577A}&2&3\\
1398
{\bf 577B}&2&1\\
1399
{\bf 577C}&3&1\\
1400
\vspace{-2ex} &\\
1401
{\bf 577D}&18&2^4\\
1402
{\bf 587C}&31&293\\
1403
{\bf 593B}&1&2\\
1404
{\bf 593C}&2&1\\
1405
\vspace{-2ex} &\\
1406
{\bf 593E}&27&2\cdot37\\
1407
{\bf 599C}&37&13\cdot23\\
1408
{\bf 601B}&29&2\cdot5^2\\
1409
{\bf 607D}&31&101\\
1410
\vspace{-2ex} &\\
1411
{\bf 613C}&27&3\cdot17\\
1412
{\bf 617B}&28&2\cdot7\cdot11\\
1413
{\bf 619B}&30&103\\
1414
{\bf 631B}&32&3\cdot5\cdot7\\
1415
\end{array}$$
1416
\end{table}
1417
1418
1419