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\chapter{Component groups of optimal quotients}
\index{Component group}
\index{Optimal quotient!component groups of}
\label{chap:compgroups}
Let~$A$ be an abelian variety over the rational numbers~$\Q$. The
Birch and Swinnerton-Dyer
conjecture\index{BSD conjecture!and component groups}
supplies a formula for the order
of the Shafarevich-Tate group of~$A$. A key step in computing this
order is to find each of the Tamagawa numbers~$c_p$ of~$A$. The
Tamagawa numbers are defined as follows, where the definition of
N\'eron model and component group is given below.
\begin{definition}[Tamagawa number]
\index{Tamagawa numbers|textit}
\index{Optimal quotient!Tamagawa numbers of}
\index{Component group!rational points of}
Let~$p$ be a prime,
let $\cA$ be a N\'eron model\index{N\'eron model}
of~$A$ over the $p$-adic integers~$\Z_p$, and let $\Phi_{A,p}$
be the component group of~$\cA$ at~$p$. Then the
\defn{Tamagawa number} $c_p$ of~$A$ is the order of the group
$\Phi_{A,p}(\F_p)$ of $\F_p$-rational points
of $\Phi_{A,p}(\Fbar_p)$.
\end{definition}
\begin{remark}
We warn the reader that the Tamagawa number is defined in a different
way in some other papers. The definitions are equivalent.
\end{remark}
In this chapter we present a method for computing the Tamagawa numbers
$c_p$, up to a power of~$2$, under the hypothesis that~$A$ has purely
toric reduction at~$p$. Such~$A$ are plentiful among the modular abelian
varieties; for example, if~$A$ is a new optimal quotient of $J_0(N)$
and~$p$ exactly divides~$N$, then~$A$ is purely toric at~$p$.
In Sections~\ref{sec:compgrpintro}--\ref{sec:compgrpmaintheorem} we
state and prove an explicit formula involving component groups of
fairly general abelian varieties. Then in Section~\ref{sec:compj0n}
we turn to quotients of modular Jacobians $J_0(N)$.
\index{Jacobian!of $X_0(N)$}
We give several tables and issue a conjecture and a question.
The results of this chapter were inspired by a letter that Ribet\index{Ribet}
wrote to Mestre\index{Mestre},
in which he treats the case when~$A$ is an elliptic curve.
\section{Main results}\label{sec:compgrpintro}
\subsection{N\'eron models and component groups}
Let~$A$ be an abelian variety over a finite extension~$K$ of the
$p$-adic numbers~$\Qp$. Let~$\O$ be the ring of integers of~$K$,
let~$\m$ be its maximal ideal, and let $k=\O/\m$ be the residue class field.
\begin{definition}[N\'eron model]\index{N\'eron model|textit}
A \defn{N\'{e}ron model} of~$A$ is a smooth commutative group
scheme~$\A$ over~$\O$ such that~$A$ is its generic fiber and~$\A$
satisfies the N\'eron mapping property: the restriction map
$$\Hom_\O(S,\A)\lra \Hom_K(S_K,A)$$
is bijective for all {\em smooth} schemes~$S$ over~$\O$.
\end{definition}
The N\'eron mapping property implies that~$\A$ is unique up to a
unique isomorphism, so we will refer without hesitation to ``the''
N\'eron model of~$A$.
The closed fiber~$\A_k$ of~$\A$ is a group scheme over~$k$,
which need not be connected; denote by~$\A_k^0$ the
connected component containing the identity.
There is an exact sequence
$$0\lra\A_k^0\lra\A_k\lra\Phi_A\lra0,$$
where~$\Phi_A$ a finite \'{e}tale group scheme
over~$k$. Equivalently,~$\Phi_A$ may be viewed as
a finite abelian group equipped
with an action of $\Gal(\kbar/k)$.
\begin{definition}[Component group]
\index{Component group|textit}
\label{defn:componentgroup}
The \defn{component group} of an abelian variety~$\A$ over a
local field~$K$ is the group scheme $\Phi_A=A_k/A_k^0$
defined above.
\end{definition}
\subsection{Motivating problem}
This chapter is motivated by the problem of
computing the groups~$\Phi_{A,p}$ attached to quotients~$A$
of Jacobians of modular curves~$X_0(N)$.\index{Jacobian!of $X_0(N)$}
When~$A$ has semistable
reduction, Grothendieck\index{Grothendieck} and Mumford\index{Mumford}
described the component group in terms of a monodromy pairing on
certain free abelian groups. When $A=J=J_0(N)$ is the Jacobian of
$X_0(N)$, this pairing can be explicitly computed, hence the component
group~$\Phi_J$ can also be computed; this has been done in many cases
in \cite{mazur:eisenstein} and \cite{edixhoven:eisen}.
Suppose now that $A=A_f$ is an optimal quotient of $J_0(N)$ that is
attached to a newform~$f$, so that the kernel of the map $\pi:J\ra A$
is connected. There is a natural map $\pi_*:\Phi_J\ra \Phi_A$. We
wish to compute the image and the order of the cokernel of
$\pi_*$.
\subsection{The main result}
We now state our main result more precisely, necessarily supressing
some of the definitions of the terms used until later. Suppose
$\pi:J\ra A$ is an optimal quotient, with~$J$
a Jacobian\index{Jacobian!semistable}
with semistable reduction and~$A$ having purely toric
reduction. We express the component group\index{Component group}
of~$A$ in terms of the monodromy pairing\index{Monodromy pairing}
associated to~$J$.
Let $m_A=\sqrt{\deg(\theta_A)}$, where $\theta_A:A^{\vee}\ra A$ is
induced by the canonical principal polarization\index{Canonical
polarization} of~$J$ arising from the $\theta$-divisor. Let $X_J$ be
the character group\index{Character group of torus} of the toric
part\index{Toric part} of the closed fiber of the N\'eron model
of~$J$. Let~$\cL$ be the saturation of the image of $X_A$ in $X_J$.
The monodromy pairing\index{Monodromy pairing} induces a map
$\alp:X_J\ra \Hom(\cL,\Z)$. Let $\Phi_X$ be the cokernel of~$\alp$
and $m_X=[\alp(X_J):\alp(\cL)]$ be the order of the finite group
$\alp(X_J)/\alp(\cL)$. We obtain the equality
$$\frac{\#\Phi_A}{m_A} = \frac{\#\Phi_X}{m_X}.$$
Using the snake lemma\index{Snake lemma}, one see that
$\Phi_X$ is isomorphic to the image of the natural map
$\Phi_J \ra \Phi_A$, and the above formula implies that the cokernel of
the map $\Phi_J \ra \Phi_A$ has order $m_A/m_X$.
If the optimal quotient $J\ra A$ arises from a
modular form on $\Gamma_0(N)$, then the quantities $m_A$, $m_X$ and
$\Phi_X$ can be explicitly computed, hence we can compute $\#\Phi_A$.
\section{Optimal quotients of Jacobians}
Let~$J$ be a Jacobian\index{Jacobian}, and let
$\theta_J$ be the canonical principal polarization
arising from the $\theta$-divisor.
Recall that an \defn{optimal quotient}\index{Optimal quotient|textit}
of~$J$ is an abelian variety~$A$ and a surjective
map $\pi: J \ra A$ whose
kernel is an abelian subvariety~$B$ of~$J$.
Denote by $J^{\vee}$ and $A^{\vee}$ the abelian varieties dual
to~$J$ and~$A$, respectively.
Upon composing the dual of~$\pi$ with $\theta_J^{\vee}=\theta_J$,
we obtain a map
$$A^{\vee}\xrightarrow{\pi^{\vee}} J^{\vee}\xrightarrow{\theta_J} J.$$
\begin{proposition}
\index{Optimal quotient!dual map is injective}
The map $A^{\vee}\ra J$ is injective.
\end{proposition}
\begin{proof}
Since $\theta_J$ is an isomorphism it suffices to prove
that $\pi^{\vee}$ is injective.
Since the dual of $\pi^{\vee}$ is
$(\pi^{\vee})^{\vee}=\pi$ and $\pi$ is surjective,
the map $\pi^{\vee}$ must have finite kernel.
Thus $A^{\vee} \ra C=\im(\pi^{\vee})$ is
an isogeny. Let~$G$ denote the kernel of this isogeny,
and dualize. By \cite[\S11]{milne:abvars} we have
the following two commutative diagrams:
$$\xymatrix{
G\ar[r] & A^{\vee}\[email protected]{->>}[r] \ar[dr]_{\pi^{\vee}}
& C\ar[d]\\
&& J^{\vee}
}\qquad \quad \xrightarrow{\textstyle \text{dualize}} \quad \qquad
\xymatrix{
A & C^{\vee}\ar[l] & G^{\vee}\ar[l] \\
& J,\ar[u]_{\vphi}\ar[ul]^{\pi}
}$$
where $G^{\vee}$ is the Cartier dual\index{Cartier dual} of~$G$.
Since $G^{\vee}$ is finite, $\ker(\vphi)$ is of
finite index in $\ker(\pi)$.
Since $\ker(\pi)$ is an abelian variety, as a group it is divisible.
But a divisible group has no nontrivial finite-index subgroups
(divisibility is a property inherited by quotients, and nonzero
finite groups are not divisible).
Thus $\ker(\vphi)=\ker(\pi)$, so $G^{\vee}=0$. It follows that $G=0$.
\end{proof}
Henceforth we will abuse notation and
denote the injection $A^{\vee}\ra J$ by $\pi^{\vee}$.
The kernel of $\theta_A$ equals the intersection of
$A^{\vee}$ and $B=\ker(\pi)$,
as depicted in the following diagram:
$$\xymatrix{
A^{\vee}\intersect B\ar[r]\ar[d] & B\ar[d] \\
A^{\vee}\[email protected]{^(->}[r]^{\pi^{\vee}}\ar[dr]_{\theta_A} & J\ar[d]^\pi\\
& A.
}$$
Since $\theta_A$ is a polarization\index{Polarization},
the degree $\#\ker(\theta_A)$ of $\theta_A$ is
a perfect square (see \cite[Thm.~13.3]{milne:abvars}).
Recall that
the \defn{modular degree}\index{Modular degree|textit} is the integer
$$m_A=\sqrt{\#\ker(\theta_A)}.$$
For an algorithm to compute $m_A$,
see Section~\ref{sec:moddeg} and Corollary~\ref{moduluscomp}.
\section{The closed fiber of the N\'{e}ron model}
\index{N\'eron model!closed fiber of}
\index{Closed fiber of N\'eron model|textit}
Let~$K$ be a finite extension of $\Qp$ with ring of integers~$\O$
and residue class field~$k$.
Let~$A$ be an abelian variety over~$K$ and denote its
N\'{e}ron model\index{N\'eron model} by~$\cA$.
Let $\Phi_A$ be the group of connected
components of the closed fiber $\cA_k$. This group
is a finite \'{e}tale group scheme\index{\'Etale group scheme|textit}
over~$k$; equivalently, it is
a finite abelian group equipped with an action of
$\Gal(\kbar/k)$. There is an exact sequence of group schemes
$$0 \ra \cA_k^0 \ra \cA_k \ra \Phi_A \ra 0.$$
The group scheme $\cA_k^0$ is an extension of an abelian variety~$\cB$
of some dimension~$a$ by a group scheme~$\cC$; we have a diagram
$$\[email protected]=.3cm{
&0\ar[d]\\
&{\cT}\ar[d]\\
0\ar[r]&{\cC}\ar[r]\ar[d]&{\cA_k^0}\ar[r]&{\cB}\ar[r]&0\\
&{\cU}\ar[d]\\
&0}$$
with~$\cT$ a torus of dimension~$t$
and~$\cU$ a unipotent group of dimension~$u$.
The abelian variety~$A$ is said to have \defn{purely toric reduction}
\index{Purely toric reduction}
if $t=\dim A$, and have \defn{semistable reduction} if $u=0$.
\index{Semistable reduction}
\begin{definition}[Character group of torus]
The \defn{character group}\index{Character group of torus|textit}
$$X_A = \Hom_{\kbar}(\cT_{/{\kbar}},{\Gm}_{/{\kbar}})\label{defn:chargroup}$$
is a free abelian group of rank~$t$ contravariantly associated to~$A$.
\end{definition}
As discussed in, e.g., \cite{ribet:modreps}, if~$A$ is semistable
there is a \defn{monodromy pairing}
$X_A\cross X_{A^{\vee}}\ra \Z$ and an exact sequence
$$0 \ra X_{A^{\vee}} \ra \Hom(X_{A},\Z) \ra \Phi_A \ra 0.$$
\section{Rigid uniformization}
\index{Rigid uniformization|textit}
In this section we review the
rigid analytic uniformization
of a semistable\index{Semistable reduction!and uniformization}
abelian variety over a finite extension~$K$ of the
maximal unramified extension $\Qp^{\ur}$ of $\Qp$. We use this
uniformization to prove
that if~$A$ has purely toric reduction\index{Purely toric reduction},
and $\phi:A^{\vee}\ra A$ is a
symmetric isogeny\index{Symmetric isogeny} (as defined below), then
$$\deg(\phi) = (\# \coker(X_A\ra X_{A^{\vee}}))^2.$$
We also prove some lemmas about character groups\index{Character group of torus}.
It is possible to prove the assertions we will need without recourse
to rigid uniformization, as Ahmed Abbes has pointed out to the author.
\subsection{Raynaud's uniformization}
\label{subsec:raynaud}
\begin{theorem}[Raynaud\index{Raynaud}]\label{raynaud}
If~$A$ is a semistable abelian variety, its universal
covering (as defined in \cite{coleman:monodromy})
is isomorphic to an extension~$G$ of an abelian
variety~$B$ with good reduction by a torus~$T$. The
covering map from~$G$ to~$A$ is a homomorphism, and
its kernel is a twisted free abelian group~$\Gamma$ of finite rank.
\end{theorem}
This may be summarized by the diagram
$$\xymatrix{
&\Gamma\ar[d] \\
T\ar[r] & G\ar[r]\ar[d] & B\\
& A,
}$$
which we call the
\defn{uniformization cross}\index{Uniformization cross}
of~$A$.
\begin{remark}
The group~$\Gamma$ can be identified with the character
group $X_{A^{\vee}}$ of the previous and latter sections.
\end{remark}
The uniformization cross
of the dual abelian variety $A^{\vee}$ is
$$\xymatrix{
&\Gamma^{\vee}\ar[d] \\
T^{\vee}\ar[r] & G^{\vee}\ar[r]\ar[d] & B^{\vee}\\
& A^{\vee},
}$$
where $\Gamma^{\vee}=\Hom(T,\Gm)$, where $T^{\vee}=\Hom(\Gamma,\Gm)$,
and the morphisms $\Gamma^{\vee}\ra G^{\vee}$ and $T^{\vee}\ra G^{\vee}$
are the one-motif duals\index{One-motif dual}
of the morphisms $T\ra G$ and $\Gamma\ra G$, respectively.
For more details see, e.g., \cite{coleman:monodromy}.
To avoid confusion when considering the uniformization of more than
one abelian variety, we will often denote the objects $T$, $G$,
$\Gamma$, and~$B$ connected with $A$ by $T_A$, $G_A$, $\Gamma_A$, and
$B_A$, respectively.
\begin{example}[Tate curve]\index{Tate curve!uniformization of}
If $E/\Qp$ is an elliptic curve with split multiplicative reduction,
then the uniformization is $E=\Gm/q^\Z$ where $q=q(j)$ is
obtained by inverting the expression for~$j$ as a function of
$q(z)=e^{2\pi iz}$.
\end{example}
\subsection{Some lemmas}
Let $\pi:J\ra A$ be an optimal quotient
\index{Optimal quotient!and semistable reduction},
assume that~$J$ has semistable\index{Semistable reduction} reduction,
and that~$A$ has purely
toric\index{Purely toric reduction} reduction.
\begin{lemma}\label{lem:surj}
The map $\Gamma_J\ra \Gamma_A$ induced by~$\pi$ is surjective.
\end{lemma}
\begin{proof}
Since $G_J$ is simply connected,~$\pi$ induces a map
$G_J\ra T_A$ and a map $\Gamma_J\ra \Gamma_A$.
Because~$\pi$ is surjective and $T_A$ is a
torus, the map $G_J\ra T_A$ is surjective.
Upon applying the snake lemma\index{Snake lemma} to the
following diagram, we obtain a surjective map from $B=\ker(\pi)$ to
$M=\coker(\Gamma_J\ra\Gamma_A)$:
$$\xymatrix{
& \Gamma_J\ar[r]\ar[d] & \Gamma_A\ar[r]\ar[d] & M\ar[r]\ar[d]& 0 \\
& G_J\ar[r]\ar[d] & T_A\ar[d]\ar[r] & 0 \\
B\ar[r] & J\ar[r]^{\pi}& A.
}$$
Since~$\pi:J\ra A$ is an optimal quotient, the kernel~$B$ is connected.
Thus~$M$ must also be connected.
Since~$M$ is discrete it follows that $M=0$.
\end{proof}
\subsubsection{Abelian varieties with purely toric reduction}
Assume that~$A$ has purely toric reduction\index{Purely toric reduction}.
Then $B=0$, and the uniformization
cross\index{Uniformization cross} is simply
$$\xymatrix{\Gamma\ar[d]\\ T\ar[d] \\ A.}$$
\begin{definition}[Symmetric isogeny]\index{Symmetric isogeny|textit}
A \defn{symmetric isogeny} $\vphi:A^{\vee}\ra A$
is an isogeny such that the map
$\vphi^{\vee}:A^{\vee}\ra (A^{\vee})^{\vee}=A$
is equal to~$\vphi$.
\end{definition}
Let $\vphi:A^{\vee}\ra A$ be a symmetric isogeny.
Denote by $\vphi_t:T^{\vee}\ra T$ and $\vphi_a:\Gamma^{\vee}\ra\Gamma$ the
maps induced by~$\vphi$.
\begin{proposition}\label{prop:kerphi}
There is an exact sequence
$$0\ra\ker(\vphi_t) \ra \ker(\vphi) \ra \coker(\vphi_a)\ra 0,$$
and $\ker(\vphi_t)$ is the Cartier dual of $\coker(\vphi_a)$.
\end{proposition}
\begin{proof}
Since~$\vphi$ is an isogeny we obtain the following diagram:
$$\xymatrix{
0\ar[r]\ar[d] & \Gamma^{\vee}\ar[r]^{\vphi_a}\ar[d] & \Gamma\ar[r]\ar[d]
& \coker(\vphi_a)\ar[d]\\
\ker(\vphi_t)\ar[r]\ar[d]& T^{\vee}\ar[d]\ar[r]^{\vphi_t} & T\ar[r]\ar[d] & 0\\
\ker(\vphi)\ar[r] & A^{\vee}\ar[r]^{\vphi} & A.}$$
The snake lemma\index{Snake lemma} then gives the claimed exact sequence.
For the second assertion, observe that if we take one-motif
duals of every object in the diagram
$$\xymatrix{
& \Gamma^{\vee}\ar[r]^{\vphi_a}\ar[d] & \Gamma\ar[d]\ar[r] & \coker(\vphi_a) \\
\ker(\vphi_t)\ar[r] & T^{\vee}\ar[r]^{\vphi_t} & T}$$
we obtain the following diagram:
$$\xymatrix{
& T & T^{\vee}\ar[l]_{\vphi_a^{\vee}} & \coker(\vphi_a)^{\vee}\ar[l]\\
\ker(\vphi_t)^{\vee}& \Gamma\ar[l]\ar[u] &\Gamma^{\vee}.\ar[l]^{\vphi_t^{\vee}}\ar[u]
}$$
Since $\vphi$ is symmetric, $\vphi_a^{\vee}=\vphi_t$, so
$$\ker(\vphi_t) = \coker(\vphi_a)^{\vee}.$$
\end{proof}
\begin{lemma}\label{lem:isogcoker}
$\#\ker(\vphi)=\#\coker(\vphi_a)^2$
\end{lemma}
\begin{proof}
Use the exact sequence of Proposition~\ref{prop:kerphi} together
with the observation that the
order of a finite group scheme equals the order of its Cartier dual.
\end{proof}
\section{The main theorem}
\label{sec:compgrpmaintheorem}
Let $\pi:J\ra A$ be an optimal quotient\index{Optimal quotient},
with~$J$ a Jacobian\index{Jacobian!semistable}
having semistable reduction and~$A$
an abelian variety having purely toric reduction.
Let $X_A$, $X_{A^{\vee}}$, and $X_J$ denote the
character groups\index{Character group of torus} of the toric parts
of the closed fibers of the abelian varieties~$A$,
$A^{\vee}$, and~$J$, respectively.
\subsection{Description of the component group in terms of the monodromy pairing}
Recall that there is a pairing
$X_A\cross X_{A^{\vee}}\ra \Z$ called
the monodromy pairing\index{Monodromy pairing}.
We have an exact sequence
$$0 \ra X_{A^{\vee}} \ra \Hom(X_{A},\Z) \ra \Phi_A \ra 0.$$
If~$J$ is a Jacobian\index{Jacobian!is principally polarized}
then~$J$ is canonically self-dual via the
$\theta$-polarization, so
the monodromy pairing on~$J$
can be viewed as a pairing $X_J\cross X_J \ra \Z$, and
there is an exact sequence
$$0 \ra X_{J} \ra \Hom(X_J,\Z) \ra \Phi_J \ra 0.$$
\begin{example}[Tate curve]\index{Tate curve!and monodromy pairing}
\index{Tate curve!component group of}
Suppose $E=\Gm/q^{\Z}$ is a Tate curve over $\Qp^{\ur}$.
The monodromy pairing on $X_E=q^{\Z}$ is
$$\langle q, q\rangle = \ord_p(q)=-\ord_p(j).$$
Thus $\Phi_E$ is cyclic of order $-\ord_p(j)$.
\end{example}
\subsubsection{Proof of the main theorem}
We now prove the main theorem.
Let $\pi : J\ra A$ be an optimal quotient, and let
$\theta:A^{\vee}\ra A$ denote the induced polarization.
Let $\pi_*$, $\pi^*$, $\theta_*$, and $\theta^*$ be the
maps induced on character groups by the various functorialities,
as indicated in the following two key diagrams:
$$\[email protected]=3pc{A^{\vee} \[email protected]{^(->}[r]^{\pi^{\vee}}\ar[dr]_{\theta}
& J \[email protected]{->>}[d]^{\pi}\\
&A}
\qquad\qquad\qquad
\[email protected]=3pc{X_A \[email protected]{^(->}[r]^{\pi^*} \ar[dr]^{\theta^*}
& X_J \[email protected]{->>}[d]^{\pi_*} \\
& X_{A^{\vee}}.\[email protected]/^1.5pc/[ul]^{\theta_*}}
$$
The surjectivity of $\pi_*$ was proved in Lemma~\ref{lem:surj}.
The injectivity of $\pi^*$ follows because
$$\theta_*\pi_*\pi^*=\theta_*\theta^*=\deg(\theta)\neq 0,$$
and multiplication by a nonzero integer on a free abelian
group is injective.
Let
$$\alp:X_J\ra \Hom(\pi^* X_A,\Z)$$
be the map defined by the monodromy pairing\index{Monodromy pairing}
restricted to $X_J\cross \pi^* X_A$.
\begin{lemma}\label{lem:twokers}
$\ker(\pi_*) = \ker(\alp)$
\end{lemma}
\begin{proof}
Suppose $x\in \ker(\pi_*)$, and let $y=\pi^* z$ with
$z\in X_A$. Then
$$\langle x, y \rangle = \langle x, \pi^* z \rangle
= \langle \pi_* x, z \rangle = 0,$$
so $x\in\ker(\alp)$.
Next let $x\in\ker(\alp)$.
Then for all $z\in X_A$,
$$0 = \langle x, \pi^* z\rangle = \langle \pi_* x, z\rangle,$$
so $\pi_* x$ is in the kernel of the
monodromy map
$$X_{A^{\vee}} \ra \Hom(X_A,\Z).$$
Since $X_{A^{\vee}}$ and $\Hom(X_A,\Z)$ are free of the same finite
rank and the cokernel is torsion, the monodromy map is injective.
Thus $\pi_* x=0$ and $x\in\ker(\pi_*)$.
\end{proof}
\begin{lemma}\label{lem:compphi}
There is an exact sequence
$$X_J \ra \Hom(\pi^* X_A,\Z) \ra \Phi_A \ra 0.$$
\end{lemma}
\begin{proof}
Lemma~\ref{lem:twokers} gives the following
commutative diagram with exact rows
$$\xymatrix{0\ar[r]
& X_J/\ker(\alp)\ar[d]^{\isom} \ar[r]
& {\Hom(\pi^* X_A,\Z)}\ar[r] \ar[d]^{\isom} & \coker(\alp)\ar[r]\ar[d] & 0\\
0\ar[r] & X_{A^{\vee}}\ar[r] & {\Hom(X_A,\Z)}\ar[r] & {\Phi_A}\ar[r] & 0.}$$
By Lemma~\ref{lem:twokers}, the first vertical map is an isomorphism.
The second is an isomorphism because it is induced by the
isomorphism $\pi^*:X_A\ra \pi^* X_A$. It follows that
$\coker(\alp)\isom \Phi_A$, as claimed.
\end{proof}
Let $\cL$ be the \defn{saturation} of $\pi^* X_A$ in $X_J$;
thus $\pi^*X_A$ is a finite-index subgroup of~$\cL$
and the quotient $X_J/\cL$ is torsion free.
For~$L$ of finite index in $\cL$,
define the
\defn{modular degree}\index{Modular degree!and character group|textit}
of~$L$ to be
$$m_L = [\alp(X_J):\alp(L)],$$
and the
\defn{component group}\index{Component group!and character group|textit}
of~$L$ to be
$$\Phi_L = \coker( X_J \ra \Hom(L,\Z)).$$
When $L=\cL$ and $A$ is fixed, we often slightly abuse notation and
write $m_X=m_\cL$ and $\Phi_X=\Phi_\cL$.
We think of $m_X$ and $\Phi_X$ as the character group
``modular degree and component group''
of~$A$.
\begin{lemma}\label{lem:homog}
Choose a subgroup~$L$ of finite index in~$\cL$.
The rational number
$\ds \frac{\#\Phi_L}{m_L}$
is independent of the choice of~$L$.
\end{lemma}
\begin{proof}
Suppose $L'$ is another finite index subgroup of~$\cL$,
and let $n=[L:L']$. Here~$n$ is a rational number, the lattice
index of~$L'$ in~$L$.
Since~$\alp$ is injective when restricted to $\cL$, it follows that
$$m_{L'} = [\alp(X_J):\alp(L')]
= [\alp(X_J):\alp(L)]\cdot[\alp(L):\alp(L')] = m_L\cdot n.$$
Similarly, $\#\Phi_{L'} = \#\Phi_L\cdot n$.
\end{proof}
Recall that $m_A = \sqrt{\deg(\theta)}$ and
$$ \Phi_A \isom \coker(X_{A^{\vee}}\ra \Hom(X_A,\Z)),$$
where $m_A$ is the modular degree of~$A$ and $\Phi_A$ is the
component group of~$A$.
\begin{theorem}\label{formula}
For any subgroup~$L$ of finite index in $\cL$,
the following relation holds:
$$\frac{\#\Phi_A}{m_A} = \frac{\#\Phi_L}{m_L}.$$
\end{theorem}
\begin{proof}
By Lemma~\ref{lem:homog} we may assume that $L=\pi^*X_A$.
With this choice of $L$, Lemma~\ref{lem:compphi} asserts that
$\Phi_L \isom \Phi_A$.
By Lemma~\ref{lem:twokers}, properties of the index,
and Lemma~\ref{lem:isogcoker} we have
\begin{eqnarray*}
m_L&=&[\alp(X_J):\alp(L)] \\
&=& [\pi_*(X_J):\pi_*(L)]\\
&=& [X_{A^{\vee}}:\pi_*(\pi^*X_A)]\\
&=& [X_{A^{\vee}}:\theta^* X_A]\\
&=& \#\coker(\theta^*) \\
&=& \sqrt{\deg(\theta)} = m_A.
\end{eqnarray*}
\end{proof}
\begin{proposition}\label{prop:compim}
$$\image(\Phi_J\ra\Phi_A) \isom \Phi_\cL.$$
\end{proposition}
\begin{proof}
Since $\pi^*X_A\subset \cL \subset X_J$, an application
of Lemma~\ref{lem:compphi} gives the following commutative diagram
with exact rows:
$$\xymatrix{
X_J\ar[r]\[email protected]{=}[d]& \Hom(X_J,\Z)\ar[r]\ar[d]& \Phi_J \ar[r]\ar[d]& 0\\
X_J\ar[r]\[email protected]{=}[d]& \Hom(\cL,\Z)\ar[r]\ar[d]& \Phi_\cL \ar[r]\ar[d] & 0\\
X_J\ar[r]& \Hom(\pi^*X_A,\Z)\ar[r]& \Phi_A \ar[r]& 0.
}$$
The map $\Hom(\cL,\Z)\ra\Hom(\pi^*X_A,\Z)$ is an isomorphism,
so the map $\Phi_\cL\ra\Phi_A$ is injective. Thus
$$\image(\Phi_J\ra\Phi_A) \isom \image(\Phi_J\ra \Phi_\cL).$$
The cokernel of $\Hom(X_J,\Z)\ra\Hom(\cL,\Z)$
surjects onto the cokernel of $\Phi_J\ra \Phi_\cL$.
Using the exact sequence
$$0\ra \cL \ra X_J \ra X_J/\cL \ra 0,$$
we find that
$$\coker(\Hom(X_J,\Z)\ra\Hom(\cL,\Z)) \subset \Ext^1(X_J/\cL,\Z).$$
Because~$\cL$ is saturated, the quotient $X_J/\cL$ is torsion free,
so the indicated $\Ext^1$ group vanishes.
Thus the map $\Phi_J\ra\Phi_\cL$ is surjective,
from which the proposition follows.
\end{proof}
The following corollary
follows from Theorem~\ref{formula} and Proposition~\ref{prop:compim}.
\begin{corollary}\label{cor:div}
$$\#\coker(X_J\ra X_A) = \frac{m_A}{m_\cL}.$$
\end{corollary}
\begin{remark}
A non-obvious consequence of this corollary is that
$$m_\cL \mid m_A.$$
\end{remark}
\section{Optimal quotients of $J_0(N)$}\label{sec:optquoj0n}
\label{sec:compj0n}\index{Optimal quotient!of $J_0(N)$}
We now summarize some facts about $J_0(N)$ that will be used in
our numerical computations. Some of these facts were discussed in
greater generality in the previous chapters of this thesis.
\subsection{Modular curves and semistability}
Let $X_0(N)$ be the modular curve\index{Modular curve}
associated to the subgroup $\Gamma_0(N)$ of $\sltwoz$
that consists of those matrices which are upper
triangular modulo~$N$. Initially, $X_0(N)$ is constructed as
a Riemann surface as the quotient
$$\Gamma_0(N)\backslash (\{z : z \in \C,\,\Im(z)>0\}\union\P^1(\Q)).$$
With some work, we find that $X_0(N)$ has a canonical
structure of algebraic curve over~$\Q$.
Suppose that~$p$ is a prime divisor of~$N$ such that $N/p$ is
coprime to~$p$. We write $p\mid\mid N$. In this situation,
it is well-known that the Jacobian\index{Jacobian!of $X_0(N)$}
$J_0(N)$ of $X_0(N)$ has semistable
reduction at~$p$.
\subsection{Newforms and optimal quotients}
The Hecke algebra\index{Hecke algebra|textit}
$$\T=\Z[\ldots T_n\ldots]\subset\End(J_0(N))$$
is a commutative ring of endomorphisms of~$J_0(N)$ of $\Z$-rank
equal to the dimension $J_0(N)$.
The character group $X_{J_0(N)}$ of $J_0(N)$ at~$p$
is equipped with a functorial action of~$\T$.
The Hecke algebra~$\T$ also acts on the complex vector space
$S = S_2(\Gamma_0(N),\C)$
of cusp forms.
A newform~$f$ is an eigenform normalized so that the coefficient
of~$q$ in the Fourier expansion of~$f$ at the cusp~$\infty$ is~$1$, and
such that~$f$ is not a modular form of any level $N'\mid N$, with $N'$ a
proper divisor of~$N$.
Let~$f$ be a newform, and associate to~$f$ the ideal $I_f$ of
the Hecke algebra~$\T$ of elements which
annihilate~$f$. Then $\O_f=\T/I_f$ is an
order in the ring of integers of the totally real number field
$K_f$ obtained by adjoining the Fourier coefficients of~$f$ to~$\Q$.
The quotient
$$A_f = J_0(N)/ I_f J_0(N)$$
is an optimal quotient\index{Optimal quotient} of $J_0(N)$
of dimension equal to $[K_f:\Q]$. It is purely
toric\index{Purely toric reduction} at~$p$, since $p\mid \mid N$.
\subsection{Homology and the modular degree}
Let $H=H_1(X_0(N),\Z)$ be the integral homology of the
complex algebraic curve $X_0(N)$. Integration defines a
$\T$-equivariant nondegenerate\index{Integration pairing}
pairing
$S \cross H \ra \C$.
This pairing induces a map
$\alp: H \ra \Hom_\C(S,\C)$.
\begin{theorem}\label{Af}
We have the following commutative diagram of $\T$-modules:
$$\xymatrix{
H[I_f] \[email protected]{^(->}[r]\[email protected]{^(->}[d] & H \[email protected]{->>}[r]\[email protected]{^(->}[d]
& \alp(H)\ar[d]\[email protected]{^(->}[d]\\
\Hom_\C(S,\C)[I_f]\[email protected]{^(->}[r]\[email protected]{->>}[d] & \Hom_\C(S,\C)\[email protected]{->>}[r]\[email protected]{->>}[d]
&\Hom_\C(S[I_f],\C)\[email protected]{->>}[d]\\
A_f^{\vee}(\C)\[email protected]{^(->}[r]\[email protected]/_2pc/_{\theta_A}[rr] & J(\C) \[email protected]{->>}[r] & A_f(\C) \\
}$$
\end{theorem}
\begin{proof}
This can be deduced from \cite{shimura:factors}.
See also Section~\ref{sec:tori}.
\end{proof}
\begin{corollary}\label{moduluscomp}
$m_A^2 = [\alp(H) : \alp(H[I_f])]$.
\end{corollary}
\begin{proof}
Recall that $m_A$ is by definition equal to $\sqrt{\deg(\theta_A)}$.
The kernel of an isogeny between complex tori is
isomorphic to the cokernel of the induced map
on lattices. The corollary now follows from
the diagram of Theorem~\ref{Af},
which indicates that the index $[\alp(H):\alp(H[I_f])]$
is the cokernel of the map $H[I_f]\ra \alp(H).$
For more details, see Section~\ref{sec:moddeg}.
\end{proof}
\subsection{Rational points of the component group (Tamagawa numbers)}
Let $\Frob_p:X_J\ra X_J$ denote the map induced by the
Frobenius automorphism.
We have $\Frob_p=-W_p$, where $W_p$ is the map induced
by the Atkin-Lehner\index{Atkin-Lehner involution} involution on $J_0(p)$.
Let~$f$ be a newform, $A=A_f$ the corresponding optimal
quotient, and $w_p$ the sign of the eigenvalue of
$W_p$ on $f$.
\begin{proposition}
$$\Phi_A(\Fp)
= \begin{cases}
\Phi_A(\Fpbar) & \text{if $w_p=-1$},\\
\Phi_A(\Fpbar)[2] & \text{if $w_p=1$.}
\end{cases}$$
\end{proposition}
\begin{proof}
If $w_p=-1$, then $\Frob_p=1$ and the $\Gal(\Fpbar/\Fp)$-action
of $\Phi_A(\Fpbar)$ is trivial. In this case
$\Phi(\Fp)=\Phi(\Fpbar)$.
Next suppose $w_p=1$. Recall that we have an exact sequence
$$0\ra X_{A^{\vee}} \ra \Hom(X_A,\Z) \ra \Phi_A \ra 0.$$
Since $W_p$ acts as $+1$ on~$f$, it also acts as $+1$ on
each of the four modules~$A$,~$X_A$,~$\Hom(X_A,\Z)$, and~$\Phi_A$.
Thus $\Frob_p=-W_p$ acts as $-1$ on $\Phi_A$. Since the subgroup
of $2$-torsion elements of a finite abelian group equals the subgroup
of elements fixed under $-1$, it follows that
$\Phi_A(\Fp) = \Phi_A(\Fpbar)[2]$.
\end{proof}
{\bf WARNING:} When we extend this result to the whole
of $J_0(N)$, it is necessary to be exceedingly careful!
The action of $\Frob_p=T_p$ need
not be by $\pm 1$, even though it must be by an involution
of order~$2$. For example, the component group of
$J_0(65)$ at~$5$ is cyclic of order~$42$. The action
of $\Frob_5$ is by multiplication by $-13$. Note that
$(-13)^2 = 1 \pmod{42}$. The fixed points of
multiplication by~$-13$ is the order~$14$ subgroup
of $\Z/42\Z$.
\section{Computations}
Using the algorithms of Chapter~\ref{chap:computing},
we can enumerate the optimal
quotients $A_f$ of $J_0(N)$ and compute the modular degree $m_A$.
The method of graphs\index{Method of graphs}
(see \cite{mestre:graphs}) and
quaternion algebras (see \cite{kohel:hecke})\index{Quaternion algebras}
can be used to compute $X=X_{J_0(N)}$ with its $\T$-action
and the monodromy pairing\index{Monodromy pairing}. We can then
compute the following three modules:
the saturated submodule
$\cL=\bigcap_{t\in I_f} \ker(t)$
of~$X$,
the character group modular degree $m_X=m_\cL$,
and $\Phi_X=\Phi_\cL$.
By Theorem~\ref{formula} we obtain
$$\#\Phi_A = \#\Phi_X \cdot \frac{m_A}{m_X}.$$
Using this method, we have computed $\#\Phi_A$ in a number of cases.
We give tables that report on some of these computations in
Secton~\ref{sec:compgrptables}.
In the next section we discuss a conjecture and a question,
which were both suggested by our numerical computations.
\subsection{Conjectures and questions}
\label{sec:compgroupconjectures}
\index{Conjecture!about modular degree}
Suppose that $N=pM$ with $(p,M)=1$.
Let
$$H_{\new} =
\ker\,\Bigl( H_1(X_0(N),\Z)\lra
H_1(X_0(M),\Z)\oplus H_1(X_0(M),\Z)\Bigr),$$
where the map is induced by the two natural
degeneracy maps $X_0(N)\ra X_0(M)$.
The Hecke algebra\index{Hecke algebra}~$\T$ acts on $H_{\new}$,
and also on the submodule $H_{\new}[I_f]$ of those elements that
are annihilated by $I_f$.
Integration defines a map
$\alp: H_{\new}\ra \Hom(S[I_f],\C)$.
Define the $p$-new homology modular degree $m_H$ by
$$m_H^2 = [\alp(H_{\new}) : \alp(H_{\new}[I_f])].$$
We expect that there is a very close relationship
between $m_X$ and $m_H$.
\begin{question}
Is $m_X$ equal to $m_H$?
\end{question}
The following conjecture offers
a refinement of some of the results of \cite{mazur:eisenstein}.
\index{Conjecture!refined Eisenstein}
\begin{conjecture}[Refined Eisenstein conjecture]\label{conj:iso}
Let~$p$ be a prime and let $f_1,\ldots,f_n$ be a set of
representatives for the Galois-conjugacy classes of newforms
in $S_2(\Gamma_0(p))$. Let $A_1,\ldots, A_n$ be the optimal
quotients associated to $f_1,\ldots,f_n$, respectively.
Then for each~$i$, $i=1,\ldots,n$, we have
$$\#A_i(\Q)_{\tor}=\#\Phi_{A_i}(\Fpbar)=\#\Phi_{A_i}(\Fp).$$
Furthermore,
$$\#\Phi_{J_0(p)}(\Fpbar)= \prod_{i=1}^d \#\Phi_{A_i}(\Fpbar).$$
\end{conjecture}
We have verified Conjecture~\ref{conj:iso} for all $p\leq 757$,
and, up to a power of~$2$, for all $p< 2000$.
\begin{remark}
It is tempting to guess that, e.g., the natural map
$$\Phi_{J_0(113)}(\Fpbar)\ra \prod_{i=1}^4 \Phi_{A_i}(\Fpbar)$$
is an isomorphism. Two of the $\Phi_{A_i}(\Fpbar)$ have
order~$2$, so the product $\prod \Phi_{A_i}(\Fpbar)$ can not
be a cyclic group. However, the groups
$\Phi_{J_0(p)}(\Fpbar)$ are known to be
cyclic for all primes~$p$.
\end{remark}
\subsection{Tables}\label{sec:compgrptables}
We have computed component groups of many optimal quotients
$A_f$ of $J_0(N)$.
In this section we provide tables, which hint at the data
we have gathered. Our notation for optimal quotients
is described in Section~\ref{sec:optquo-notation}.
See also Table~\ref{table:shacompgps}.
\subsubsection{Table~\ref{tbl:lowlevel}: Component groups at low level}
Table~\ref{tbl:lowlevel} gives the component groups of the
quotients $A_f$ of $J_0(N)$ for $N\leq 106$.
The column labeled $d$ contains the
dimensions of the $A_f$,
and the column labeled $\#\Phi_{A,p}$ contains a list
of the orders of the component groups of $A_f$,
one for each divisor~$p$ of~$N$, ordered by increasing~$p$.
An entry of ``?'' indicates that $p^2\mid N$, so our algorithm
does not apply.
A component group order is starred if the $\Gal(\Fpbar/\Fp)$-action is
nontrivial.
\subsubsection{Table~\ref{table:big1}--\ref{table:big2}: Big component groups}
Using the algorithms described in Section~\ref{sec:ratpartformula},
we computed the rational numbers $L(A,1)/\Omega_A$
for every optimal quotient~$A$ that is attached to a newform
of level $\leq 1500$.
There are exactly~$5$ optimal quotients~$A$ such that the numerator of
$L(A,1)/\Omega_A$ is nonzero and divisible by a prime $>10^9$.
The Birch and Swinnerton-Dyer conjecture
\index{BSD conjecture!predicts large component groups}
predicts that these large
prime divisors must divide either $\#\Phi_A$ or
the Shafarevich-Tate\index{Shafarevich-Tate group}
group of~$A$. This is the case, as Table~\ref{table:big2} shows.
\subsubsection{Table~\ref{table:compj0n}: Quotients of $J_0(N)$}
Table~\ref{table:compj0n} contains all of the invariants involved
in the computation of component groups for
each of the newform optimal quotients of levels $65$, $66$, $68$, and $69$.
\subsubsection{Table~\ref{table:compprime}:
Quotients of $J_0(p)^-$}
We computed
the quantities $m_A$, $m_X$ and $\Phi_X$ for each abelian
variety $A=A_f$ associated to a newform of prime level~$p$ with
$p\leq 757$. The results are as follows:
\begin{enumerate}
\item In all cases $m_A=m_X$, so the map $\Phi_J\ra \Phi_A$
is surjective.
\item $\Phi_A=1$ whenever the sign of the Atkin-Lehner involution
$w_p$ on~$A$ is~$1$.
\item $\prod \#\Phi_A(\Fpbar) = \#\Phi_J(\Fpbar)$
\end{enumerate}
Table~\ref{table:compprime}
lists those~$A$ of level $\leq 631$ for which $w_p=-1$, along with
the order of the corresponding component group.
\begin{table}
\ssp
\begin{center}
\caption{Component groups at low level\label{tbl:lowlevel}}
\end{center}
\vspace{-.3in}
\index{Table of!component groups at low level}
\index{Component group!table of}
$$
\begin{array}{lcl}
A & \, d \, & \, \#\Phi_{A,p}\, \\
\vspace{-2ex}\\
{\bf 11A} & 1 & 5\\
{\bf 14A} & 1 & 6^*,3\\
{\bf 15A} & 1 & 4^*,4\\
{\bf 17A} & 1 & 4\\
\vspace{-2ex}\\
{\bf 19A} & 1 & 3\\
{\bf 20A} & 1 & ?,2^*\\
{\bf 21A} & 1 & 4,2^*\\
{\bf 23A} & 2 & 11\\
\vspace{-2ex}\\
{\bf 24A} & 1 & ?,2^*\\
{\bf 26A} & 1 & 3^*,3\\
{\bf 26B} & 1 & 7,1^*\\
{\bf 27A} & 1 & ?\\
\vspace{-2ex}\\
{\bf 29A} & 2 & 7\\
{\bf 30A} & 1 & 4^*,3,1^*\\
{\bf 31A} & 2 & 5\\
{\bf 32A} & 1 & ?\\
\vspace{-2ex}\\
{\bf 33A} & 1 & 6^*,2\\
{\bf 34A} & 1 & 6,1^*\\
{\bf 35A} & 1 & 3^*,3\\
{\bf 35B} & 2 & 8,4^*\\
\vspace{-2ex}\\
{\bf 36A} & 1 & ?,?\\
{\bf 37A} & 1 & 1^*\\
{\bf 37B} & 1 & 3\\
{\bf 38A} & 1 & 9^*,3\\
\vspace{-2ex}\\
{\bf 38B} & 1 & 5,1^*\\
{\bf 39A} & 1 & 2^*,2\\
{\bf 39B} & 2 & 14,2^*\\
{\bf 40A} & 1 & ?,2\\
\vspace{-2ex}\\
{\bf 41A} & 3 & 10\\
{\bf 42A} & 1 & 8,2^*,1^*\\
{\bf 43A} & 1 & 1^*\\
{\bf 43B} & 2 & 7\\
\vspace{-2ex}\\
{\bf 44A} & 1 & ?,1^*\\
{\bf 45A} & 1 & ?,1^*\\
{\bf 46A} & 1 & 10^*,1\\
{\bf 47A} & 4 & 23\\
\vspace{-2ex}\\
{\bf 48A} & 1 & ?,2\\
{\bf 49A} & 1 & ?\\
{\bf 50A} & 1 & 1^*,?\\
{\bf 50B} & 1 & 5,?\\
\end{array}\quad
\begin{array}{lcl}
A & \, d \, & \, \#\Phi_{A,p}\, \\
\vspace{-2ex}\\
{\bf 51A} & 1 & 3,1^*\\
{\bf 51B} & 2 & 16^*,4\\
{\bf 52A} & 1 & ?,2^*\\
{\bf 53A} & 1 & 1^*\\
\vspace{-2ex}\\
{\bf 53A} & 1 & 1^*\\
{\bf 53B} & 3 & 13\\
{\bf 54A} & 1 & 3^*,?\\
{\bf 54B} & 1 & 3,?\\
\vspace{-2ex}\\
{\bf 55A} & 1 & 2,2^*\\
{\bf 55B} & 2 & 14^*,2\\
{\bf 56A} & 1 & ?,1\\
{\bf 56B} & 1 & ?,1^*\\
\vspace{-2ex}\\
{\bf 57A} & 1 & 2^*,1^*\\
{\bf 57B} & 1 & 2,2^*\\
{\bf 57C} & 1 & 10,1^*\\
{\bf 58A} & 1 & 2^*,1^*\\
\vspace{-2ex}\\
{\bf 58B} & 1 & 10,1^*\\
{\bf 59A} & 5 & 29\\
{\bf 61A} & 1 & 1^*\\
{\bf 61B} & 3 & 5\\
\vspace{-2ex}\\
{\bf 62A} & 1 & 4,1^*\\
{\bf 62B} & 2 & 66^*,3\\
{\bf 63A} & 1 & ?,1^*\\
{\bf 63B} & 2 & ?,3\\
\vspace{-2ex}\\
{\bf 64A} & 1 & ?\\
{\bf 65A} & 1 & 1^*,1^*\\
{\bf 65B} & 2 & 3^*,3\\
{\bf 65C} & 2 & 7,1^*\\
\vspace{-2ex}\\
{\bf 66A} & 1 & 2^*,3,1^*\\
{\bf 66B} & 1 & 4,1^*,1^*\\
{\bf 66C} & 1 & 10,5,1\\
{\bf 67A} & 1 & 1\\
\vspace{-2ex}\\
{\bf 67B} & 2 & 1^*\\
{\bf 67C} & 2 & 11\\
{\bf 68A} & 2 & ?,2^*\\
{\bf 69A} & 1 & 2,1^*\\
\vspace{-2ex}\\
{\bf 69B} & 2 & 22^*,2\\
{\bf 70A} & 1 & 4,2^*,1^*\\
{\bf 71A} & 3 & 5\\
{\bf 71B} & 3 & 7\\
\end{array}\quad
\begin{array}{lcl}
A & \, d \, & \, \#\Phi_{A,p}\, \\
\vspace{-2ex}\\
{\bf 72A} & 1 & ?,?\\
{\bf 73A} & 1 & 2\\
{\bf 73B} & 2 & 1^*\\
{\bf 73C} & 2 & 3\\
\vspace{-2ex}\\
{\bf 74A} & 2 & 9^*,3\\
{\bf 74B} & 2 & 95,1^*\\
{\bf 75A} & 1 & 1^*,?\\
{\bf 75B} & 1 & 1,?\\
\vspace{-2ex}\\
{\bf 75C} & 1 & 5,?\\
{\bf 76A} & 1 & ?,1^*\\
{\bf 77A} & 1 & 2^*,1^*\\
{\bf 77B} & 1 & 3^*,2\\
\vspace{-2ex}\\
{\bf 77C} & 1 & 6,3^*\\
{\bf 77D} & 2 & 2,2^*\\
{\bf 78A} & 1 & 16^*,5^*,1\\
{\bf 79A} & 1 & 1^*\\
\vspace{-2ex}\\
{\bf 79B} & 5 & 13\\
{\bf 80A} & 1 & ?,2\\
{\bf 80B} & 1 & ?,2^*\\
{\bf 81A} & 2 & ?\\
\vspace{-2ex}\\
{\bf 82A} & 1 & 2^*,1^*\\
{\bf 82B} & 2 & 28,1^*\\
{\bf 83A} & 1 & 1^*\\
{\bf 83B} & 6 & 41\\
\vspace{-2ex}\\
{\bf 84A} & 1 & ?,1^*,2^*\\
{\bf 84B} & 1 & ?,3,2\\
{\bf 85A} & 1 & 2^*,1\\
{\bf 85B} & 2 & 2^*,1^*\\
\vspace{-2ex}\\
{\bf 85C} & 2 & 6,1^*\\
{\bf 86A} & 2 & 21^*,3\\
{\bf 86B} & 2 & 55,1^*\\
{\bf 87A} & 2 & 5,1^*\\
\vspace{-2ex}\\
{\bf 87B} & 3 & 92^*,4\\
{\bf 88A} & 1 & ?,1^*\\
{\bf 88B} & 2 & ?,2^*\\
{\bf 89A} & 1 & 1^*\\
\vspace{-2ex}\\
{\bf 89B} & 1 & 2\\
{\bf 89C} & 5 & 11\\
{\bf 90A} & 1 & 2^*,?,3\\
{\bf 90B} & 1 & 6,?,1^*\\
\end{array}\quad
\begin{array}{lcl}
A & \, d \, & \, \#\Phi_{A,p}\, \\
\vspace{-2ex}\\
{\bf 90C} & 1 & 4,?,1\\
{\bf 91A} & 1 & 1^*,1^*\\
{\bf 91B} & 1 & 1,1\\
{\bf 91C} & 2 & 7,1^*\\
\vspace{-2ex}\\
{\bf 91D} & 3 & 4^*,8\\
{\bf 92A} & 1 & ?,1^*\\
{\bf 92B} & 1 & ?,1\\
{\bf 93A} & 2 & 4^*,1^*\\
\vspace{-2ex}\\
{\bf 93B} & 3 & 64,2^*\\
{\bf 94A} & 1 & 2,1^*\\
{\bf 94B} & 2 & 94^*,1\\
{\bf 95A} & 3 & 10,2^*\\
\vspace{-2ex}\\
{\bf 95B} & 4 & 54^*,6\\
{\bf 96A} & 1 & ?,2\\
{\bf 96B} & 1 & ?,2^*\\
{\bf 97A} & 3 & 1^*\\
\vspace{-2ex}\\
{\bf 97B} & 4 & 8\\
{\bf 98A} & 1 & 2^*,?\\
{\bf 98B} & 2 & 14,?\\
{\bf 99A} & 1 & ?,1^*\\
\vspace{-2ex}\\
{\bf 99B} & 1 & ?,1\\
{\bf 99C} & 1 & ?,1^*\\
{\bf 99D} & 1 & ?,1^*\\
{\bf 100A} & 1 & ?,?\\
\vspace{-2ex}\\
{\bf 101A} & 1 & 1^*\\
{\bf 101B} & 7 & 25\\
{\bf 102A} & 1 & 2^*,2^*,1^*\\
{\bf 102B} & 1 & 6^*,6,1^*\\
\vspace{-2ex}\\
{\bf 102C} & 1 & 8,4,1\\
{\bf 103A} & 2 & 1^*\\
{\bf 103B} & 6 & 17\\
{\bf 104A} & 1 & ?,1^*\\
\vspace{-2ex}\\
{\bf 104B} & 2 & ?,2\\
{\bf 105A} & 1 & 1,1,1\\
{\bf 105B} & 2 & 10^*,2^*,2\\
{\bf 106A} & 1 & 4^*,1^*\\
\vspace{-2ex}\\
{\bf 106B} & 1 & 5^*,1\\
{\bf 106C} & 1 & 24,1^*\\
{\bf 106D} & 1 & 3,1^*\\
&&\\
\end{array}$$
\end{table}
\begin{table}
\ssp
\caption{Big $L(A,1)/\Omega_A$\label{table:big1}}
\index{Table of!big $L(A,1)/\Omega_A$}
\index{Component group!table of}
$$\begin{array}{lccc}
A & \text{dim} &
N & \text{\qquad $L(A,1)/\Omega_A\cdot \text{Manin constant}$\qquad }\\
\vspace{-2ex} && & \\
\text{\bf 1154E} & 20&2\cdot 577 & 2^?\cdot 85495047371/17^2\\
\text{\bf 1238G} & 19& 2\cdot 619 & 2^?\cdot 7553329019/5\cdot 31\\
\text{\bf 1322E} & 21& 2\cdot 661 & 2^?\cdot 57851840099/331\\
\text{\bf 1382D} & 20& 2\cdot 691 & 2^?\cdot 37 \cdot 1864449649 /173\\
\text{\bf 1478J} & 20
& 2\cdot 739 & 2^?\cdot 7\cdot 29\cdot 1183045463 / 5\cdot 37\\
\end{array}$$
\end{table}
\begin{table}
\ssp
\caption{Big component groups\label{table:big2}}
\index{Table of!big component groups}
\index{Component group!table of}
$$\begin{array}{lcccccc}
A &p & w & \#\Phi_X & m_X & \#\Phi_A(\Fpbar) \\
\vspace{-1ex} && & & & & \\
\text{\bf 1154E} &2 & - & 17^2 & 2^{24}
& 2^?\cdot 17^2 \cdot 85495047371 \\
&577& + & 1 & 2^{26}\cdot85495047371
& 2^? \\
\vspace{-1ex}&&&&&\\
\text{\bf 1238G} & 2 &- & 5\cdot31 & 2^{26} &2^?\cdot 5\cdot31\cdot7553329019 \\
& 619 &+ & 1 & 2^{28}\cdot 7553329019 & 2^? \\
\vspace{-1ex}&&&&&\\
\text{\bf 1322E} & 2 & - & 331 & 2^{28} & 2^?\cdot 331 \cdot 57851840099\\
& 661& + & 1 & 2^{32}\cdot 57851840099 & 2^?\\
\vspace{-1ex}&&&&&\\
\text{\bf 1382D} & 2 & - & 173 & 2^{29} & 2^?\cdot 37\cdot173\cdot 1864449649 \\
& 691 & + & 1 & 2^{31}\cdot 37\cdot 1864449649 & 2^?\\
\vspace{-1ex}&&&&&\\
\text{\bf 1478J} & 2 & - & 5\cdot37 &2^{31}
& 2^?\cdot5\cdot 7\cdot29\cdot37\cdot1183045463 \\
& 739 & + & 1 &2^{33}\cdot 7\cdot 29\cdot 1183045463
& 2^? \\
\end{array}$$
\end{table}
\begin{table}
\ssp
\caption{Component groups of quotients of $J_0(N)$\label{table:compj0n}}
\index{Table of!component groups of quotients}
\index{Component group!table of}
$$\begin{array}{lccccccc}
A & \text{dim} & p & w_p & \#\Phi_X & m_X & m_A & \#\Phi_A \\
\vspace{-1ex} && & & & & & \\
\text{\bf 65A} & 1& 5 & +& 1 &2 & 2 & 1\\
&& 13 &+& 1 & 2 & & 1\\
\text{\bf 65B} &2& 5 &+& 3 & 2^2 & 2^2 & 3\\
&& 13 &- & 3 & 2^2 & & 3\\
\text{\bf 65C}&2 & 5 &-& 7 & 2^2& 2^2 &7 \\
&& 13 &+ & 1 & 2^2& & 1\\
\vspace{-1ex} && & & & & & \\
\text{\bf 66A}&1& 2 &+ & 1 &2 & 2^2&2 \\
&& 3 &- & 3 &2^2 & & 3\\
&& 11 &+ & 1 &2^2 & &1 \\
\text{\bf 66B}&1& 2 &- & 2 &2 & 2^2& 2^2\\
&& 3 &+ & 1 &2^2& & 1\\
&& 11 &+ & 1 &2^2 & & 1\\
\text{\bf 66C}&1& 2 & -& 1 & 2& 2^2\cdot 5& 2\cdot5\\
&& 3 &- & 1 & 2^2 & & 5\\
&& 11 &- & 1 & 2^2\cdot5 & &1 \\
\vspace{-1ex} && & & & & & \\
\text{\bf 68A}&2&17&+&2 &2\cdot3 &2\cdot3 & 2 \\
\vspace{-1ex} && & & & & & \\
\text{\bf 69A} &1&3 &-&2 &2 & 2& 2\\
&&23 &+& 1&2 & & 1\\
\text{\bf 69B} &2&3 &+&2 &2 &2\cdot11& 2\cdot11 \\
&&23 &-&2 &2\cdot11 && 2 \\
\end{array}$$
\end{table}
\begin{table}
\ssp
\caption{Component groups of quotients of $J_0(p)^{-}$
\index{Table of!component groups at prime level}
\index{Component group!table of}
\label{table:compprime}}
\vspace{-.3in}
$$
\begin{array}{lcc}
\vspace{-2ex}\\
A & d & \#\Phi_A \\
\vspace{-2ex}\\
{\bf 11A}&1&5\\
{\bf 17A}&1&2^2\\
{\bf 19A}&1&3\\
{\bf 23A}&2&11\\
\vspace{-2ex}& &\\
{\bf 29A}&2&7\\
{\bf 31A}&2&5\\
{\bf 37B}&1&3\\
{\bf 41A}&3&2\cdot5\\
\vspace{-2ex}& &\\
{\bf 43B}&2&7\\
{\bf 47A}&4&23\\
{\bf 53B}&3&13\\
{\bf 59A}&5&29\\
\vspace{-2ex}& &\\
{\bf 61B}&3&5\\
{\bf 67A}&1&1\\
{\bf 67C}&2&11\\
{\bf 71A}&3&5\\
\vspace{-2ex}& &\\
{\bf 71B}&3&7\\
{\bf 73A}&1&2\\
{\bf 73C}&2&3\\
{\bf 79B}&5&13\\
\vspace{-2ex}& &\\
{\bf 83B}&6&41\\
{\bf 89B}&1&2\\
{\bf 89C}&5&11\\
{\bf 97B}&4&2^3\\
\vspace{-2ex}& &\\
{\bf 101B}&7&5^2\\
{\bf 103B}&6&17\\
{\bf 107B}&7&53\\
{\bf 109A}&1&1\\
\vspace{-2ex}& &\\
{\bf 109C}&4&3^2\\
{\bf 113A}&1&2\\
{\bf 113B}&2&2\\
{\bf 113D}&3&7\\
\vspace{-2ex}& &\\
{\bf 127B}&7&3\cdot7\\
{\bf 131B}&10&5\cdot13\\
{\bf 137B}&7&2\cdot17\\
{\bf 139A}&1&1\\
\vspace{-2ex}& &\\
{\bf 139C}&7&23\\
{\bf 149B}&9&37\\
{\bf 151B}&3&1\\
{\bf 151C}&6&5^2\\
\end{array}\,\,
\begin{array}{lcc}
\vspace{-2ex}\\
A & d& \#\Phi_A \\
\vspace{-2ex}\\
{\bf 157B}&7&13\\
{\bf 163C}&7&3^3\\
{\bf 167B}&12&83\\
{\bf 173B}&10&43\\
\vspace{-2ex} &\\
{\bf 179A}&1&1\\
{\bf 179C}&11&89\\
{\bf 181B}&9&3\cdot5\\
{\bf 191B}&14&5\cdot19\\
\vspace{-2ex} &\\
{\bf 193C}&8&2^4\\
{\bf 197C}&10&7^2\\
{\bf 199A}&2&1\\
{\bf 199C}&10&3\cdot11\\
\vspace{-2ex} &\\
{\bf 211A}&2&5\\
{\bf 211D}&9&7\\
{\bf 223C}&12&37\\
{\bf 227B}&2&1\\
\vspace{-2ex} &\\
{\bf 227C}&2&1\\
{\bf 227E}&10&113\\
{\bf 229C}&11&19\\
{\bf 233A}&1&2\\
\vspace{-2ex} &\\
{\bf 233C}&11&29\\
{\bf 239B}&17&7\cdot17\\
{\bf 241B}&12&2^2\cdot5\\
{\bf 251B}&17&5^3\\
\vspace{-2ex} &\\
{\bf 257B}&14&2^6\\
{\bf 263B}&17&131\\
{\bf 269C}&16&67\\
{\bf 271B}&16&3^2\cdot5\\
\vspace{-2ex} &\\
{\bf 277B}&3&1\\
{\bf 277D}&9&23\\
{\bf 281B}&16&2\cdot5\cdot7\\
{\bf 283B}&14&47\\
\vspace{-2ex} &\\
{\bf 293B}&16&73\\
{\bf 307A}&1&1\\
{\bf 307B}&1&1\\
{\bf 307C}&1&1\\
\vspace{-2ex} &\\
{\bf 307D}&1&1\\
{\bf 307E}&2&3\\
{\bf 307F}&9&17\\
{\bf 311B}&22&5\cdot31\\
\end{array}\,\,
\begin{array}{lcc}
\vspace{-2ex}\\
A & d &\#\Phi_A \\
\vspace{-2ex}\\
{\bf 313A}&2&1\\
{\bf 313C}&12&2\cdot13\\
{\bf 317B}&15&79\\
{\bf 331D}&16&5\cdot11\\
\vspace{-2ex} &\\
{\bf 337B}&15&2^2\cdot7\\
{\bf 347D}&19&173\\
{\bf 349B}&17&29\\
{\bf 353A}&1&2\\
\vspace{-2ex} &\\
{\bf 353B}&3&2\\
{\bf 353D}&14&2\cdot11\\
{\bf 359D}&24&179\\
{\bf 367B}&19&61\\
\vspace{-2ex} &\\
{\bf 373C}&17&31\\
{\bf 379B}&18&3^2\cdot7\\
{\bf 383C}&24&191\\
{\bf 389A}&1&1\\
\vspace{-2ex} &\\
{\bf 389E}&20&97\\
{\bf 397B}&2&1\\
{\bf 397C}&5&11\\
{\bf 397D}&10&3\\
\vspace{-2ex} &\\
{\bf 401B}&21&2^2\cdot5^2\\
{\bf 409B}&20&2\cdot17\\
{\bf 419B}&26&11\cdot19\\
{\bf 421B}&19&5\cdot7\\
\vspace{-2ex} &\\
{\bf 431B}&1&1\\
{\bf 431D}&3&1\\
{\bf 431F}&24&5\cdot43\\
{\bf 433A}&1&1\\
\vspace{-2ex} &\\
{\bf 433B}&3&1\\
{\bf 433D}&16&2^2\cdot3^2\\
{\bf 439C}&25&73\\
{\bf 443C}&1&1\\
\vspace{-2ex} &\\
{\bf 443E}&22&13\cdot17\\
{\bf 449B}&23&2^4\cdot7\\
{\bf 457C}&20&2\cdot19\\
{\bf 461D}&26&5\cdot23\\
\vspace{-2ex} &\\
{\bf 463B}&22&7\cdot11\\
{\bf 467C}&26&233\\
{\bf 479B}&32&239\\
{\bf 487A}&2&1\\
\end{array}\,\,
\begin{array}{lcc}
\vspace{-2ex}&\\
A & d & \#\Phi_A \\
\vspace{-2ex}&\\
{\bf 487B}&2&3\\
{\bf 487C}&3&1\\
{\bf 487D}&16&3^3\\
{\bf 491C}&29&5\cdot7^2\\
\vspace{-2ex} &\\
{\bf 499C}&23&83\\
{\bf 503B}&1&1\\
{\bf 503C}&1&1\\
{\bf 503D}&3&1\\
\vspace{-2ex} &\\
{\bf 503F}&26&251\\
{\bf 509B}&28&127\\
{\bf 521B}&29&2\cdot5\cdot13\\
{\bf 523C}&26&3\cdot29\\
\vspace{-2ex} &\\
{\bf 541B}&24&3^2\cdot5\\
{\bf 547C}&25&7\cdot13\\
{\bf 557B}&1&1\\
{\bf 557D}&26&139\\
\vspace{-2ex} &\\
{\bf 563A}&1&1\\
{\bf 563E}&31&281\\
{\bf 569B}&31&2\cdot71\\
{\bf 571A}&1&1\\
\vspace{-2ex} &\\
{\bf 571B}&1&1\\
{\bf 571C}&2&1\\
{\bf 571D}&2&1\\
{\bf 571F}&4&1\\
\vspace{-2ex} &\\
{\bf 571I}&18&5\cdot19\\
{\bf 577A}&2&3\\
{\bf 577B}&2&1\\
{\bf 577C}&3&1\\
\vspace{-2ex} &\\
{\bf 577D}&18&2^4\\
{\bf 587C}&31&293\\
{\bf 593B}&1&2\\
{\bf 593C}&2&1\\
\vspace{-2ex} &\\
{\bf 593E}&27&2\cdot37\\
{\bf 599C}&37&13\cdot23\\
{\bf 601B}&29&2\cdot5^2\\
{\bf 607D}&31&101\\
\vspace{-2ex} &\\
{\bf 613C}&27&3\cdot17\\
{\bf 617B}&28&2\cdot7\cdot11\\
{\bf 619B}&30&103\\
{\bf 631B}&32&3\cdot5\cdot7\\
\end{array}$$
\end{table}
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