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\begin{document}

\title{Component Groups of Quotients of $J_0(N)$}
\titlerunning{Component Groups}
\author{David Kohel\inst{1} \and William A. Stein\inst{2}}
\authorrunning{Kohel \and Stein}
\tocauthor{David Kohel (University of Sydney),
William A. Stein (University of California at Berkeley)}
%
\institute{University of Sydney\\
\email{kohel@maths.usyd.edu.au}\\
\texttt{http://www.maths.usyd.edu.au:8000/u/kohel/}
\and
University of California at Berkeley,\\
\email{was@math.berkeley.edu}\\
\texttt{http://shimura.math.berkeley.edu/\~{}was}
}

\maketitle
\begin{abstract}
Let~$f$ be a newform of weight~$2$ on $\Gamma_0(N)$, and let~$A_f$ be
the corresponding optimal abelian variety quotient of $J_0(N)$.  We
describe an algorithm to compute the order of the component group of
$A_f$ at primes~$p$ that exactly divide~$N$.  We give a table of
orders of component groups for all~$f$ of level $N\leq 127$ and four
examples in which the component group is very large, as predicted by
the Birch and Swinnerton-Dyer conjecture.
\end{abstract}

\section{Introduction}

Let $X_0(N)$ be the Riemann surface obtained by compactifying the
quotient of the upper half-plane by the action of $\Gamma_0(N)$.
Then $X_0(N)$ has a canonical structure of algebraic curve
over~$\Q$; denote its Jacobian by $J_0(N)$.  It is equipped with
an action of a commutative ring $\T=\Z[\ldots T_n \ldots]$ of
Hecke operators.  For more details on modular curves, Hecke operators,
and modular forms see, e.g.,~\cite{diamond-im}.

Now suppose that~$f=\sum_{n=1}^{\infty} a_n q^n$ is a modular newform
of weight~$2$ for the congruence subgroup $\Gamma_0(N)$.
The Hecke operators also act on~$f$ by $T_n(f) = a_n f$.
The eigenvalues $a_n$ generate an order $R_f = \Z[\ldots a_n \ldots]$
in a number field $K_f$.  Let $I_f$ be the kernel ideal of the map
$\T \rta R_f$ sending $T_n$ to~$a_n$.
Following Shimura~\cite{shimura:factors}, we associate to~$f$ the
quotient $A_f = J_0(N)/I_f J_0(N)$ of
$J_0(N)$.  Then $A_f$ is an abelian variety over~$\Q$ of dimension
$[K_f:\Q]$, with bad reduction exactly at the primes dividing~$N$.

One dimensional quotients of $J_0(N)$ have been intensely studied
in recent years, both computationally and theoretically.
The original conjectures of Birch and
Swinnerton-Dyer~\cite{birch-swd-I,birch-swd-II},
which concern elliptic curves,
were greatly influenced by computations.
The scale of these computations was extended and systematized by
Cremona in~\cite{cremona:algs}.

In another direction, Wiles~\cite{wiles:fermat} and
Taylor-Wiles~\cite{taylor-wiles} proved a special case of the
conjecture of Shimura-Taniyama, which asserts that every elliptic
curve over~$\Q$ is a quotient of some $J_0(N)$; this allowed them to establish
Fermat's Last Theorem.  The full Shimura-Taniyama
conjecture was later proved by Breuil, Conrad, Diamond,
This illustrates the central roll played by quotients of $J_0(N)$.

\section{Component Groups of $A_f$}

The N\'eron model $\cA/\Z$ of an abelian variety $A/\Q$ is by definition
a smooth commutative group scheme over~$\Z$ with generic fiber~$A$
such that for any smooth scheme~$S$ over~$\Z$, the restriction map
$$\Hom_\Z(S,\cA) \rta \Hom_\Q(S_\Q,A)$$ is a bijection.
For more details, including a proof of existence, see,
e.g.,~\cite{neronmodels}.

Suppose that $A_f$ is a quotient of $J_0(N)$ corresponding to a newform~$f$,
and let $\cA_f$ be the N\'eron model of~$A_f$.  For any prime divisor~$p$
of~$N$, the closed fiber~${\cA_f}_{/\Fp}$ is a group scheme
over~$\Fp$, which need not be connected.  Denote the connected
component of the identity by~${\cA^{\circ}_f}_{/\Fp}$.
There is an exact sequence
$$0 \rightarrow {\cA^{\circ}_f}_{/\Fp} \rightarrow {\cA_f}_{/\Fp}\rightarrow\Phi_{A_f,p} \rightarrow 0$$
with $\Phi_{A_f,p}$ a finite \'{e}tale group scheme over~$\Fp$
called the {\em component group} of $A_f$ at~$p$.

We note that a finite \'etale group scheme is equivalent to a
finite abelian group together with an action of $\Gal(\Fpbar/\Fp)$.
In this paper we describe an algorithm for computing the order of
this group.

\section{The Algorithm}

Let~$J = J_0(N)$.  As the Jacobian of a curve, $J$ is canonically
self-dual, so the projection $J \rta A_f$ induces a polarization
$A_f^{\vee} \rta A_f$, where $A_f^{\vee}$ denotes the abelian variety
dual of $A_f$.    The {\em modular degree} $\delta_{A_f}$
is the positive square root of the degree of this polarization.

Let $T_{J,p}$ be the toric part of the closed fiber of the
N\'eron model of $J$ at~$p$, and define $\cX_{J,p}$ to
be the group $\Hom_{\Fpbar}(T_{J,p},\G_m)$ of characters of $T_{J,p}$.
Then $\cX_{J,p}$ is a free abelian group equipped with an action
of both~$\T$ and $\Gal(\Fpbar/\Fp)$
(see, e.g., Ribet~\cite{ribet:modreps}); there is a bilinear
pairing
$$\langle\,,\,\rangle : \cX_{J,p} \times \cX_{J,p} \rightarrow \Z$$
called the {\em monodromy pairing} such that
$$\Phi_{J,p} \isom \coker(\cX_{J,p} \rightarrow \Hom(\cX_{J,p},\Z)).$$
Let $\cX_{J,p}[I_f]$ be the intersection of all kernels $\ker(t)$ for~$t$
in $I_f$, and let
$$\alpha_f: \cX_{J,p} \rightarrow \Hom(\cX_{J,p}[I_f],\Z)$$
be the map induced by the monodromy pairing.
The following theorem of Stein~\cite{stein:phd}, provides the
basis for the computation of orders of component groups.

\begin{theorem}\label{thm:main}
$$\#\Phi_{A_f,p} = \frac{\#\coker(\alpha_f) \cdot \delta_{A_f}} {\# (\alpha_f(\cX_{J,p})/\alpha_f(\cX_{J,p}[I_f]))}\,.$$
\end{theorem}

\subsection{Computing the modular degree $\delta_{A,f}$}

To compute the modular degree we require
$H_1(X_0(N),\Z;\mbox{\rm cusps})$ together with the action
of the Hecke ring $\T$.
Using modular symbols (see, e.g., Cremona~\cite{cremona:algs}), we
first compute $H_1(X_0(N),\Q;\mbox{\rm cusps})$.
Using lattice reduction, we compute the $\Z$-submodule generated
by all Manin symbols $(c,d)$; then $H_1(X_0(N),\Z)$ is the
{\em integer} kernel of the boundary map.  We have a natural restriction
map
$$\Hom(H_1(X_0(N),\Z),\Z)[I_f] \rightarrow \Hom(H_1(X_0(N),\Z)[I_f],\Z)$$
where $t \in \T$ acts on $\varphi \in \Hom(H_1(X_0(N),\Z),\Z)$ by
$(t.\varphi)(x) = \varphi(tx)$.
By considering $J_0(N)$, $A_f$, and $A_f^{\vee}$
as complex lattices, and applying the snake lemma,
we see that the order of the cokernel of this map is the square of the
modular degree $\delta_f$.

\subsection{Computing the character group $\cX_{J,p}$}

When $N=Mp$, with~$M$ small, the algorithm of Mestre and
Oesterl\'e~\cite{mestre:graphs} can be used.  This algorithm
constructs the graph of isogenies between $\Fpbar$-isomorphism
classes of pairs consisting of a supersingular elliptic curve
and an $M$-torsion subgroup.  In particular, the
method is elementary to apply when $X_0(M)$ is of genus~$0$.

In general, the above category of enhanced'' supersingular
elliptic curves can be replaced
by one of left (or right) ideals of a quaternion order~$\cO$ of
level~$M$ in the quaternion algebra over~$\Q$ ramified at~$p$.
This gives an equivalent category, in which the computation of
homomorphisms is efficient.  The character group $\cX_{J,p}$ is
known by Deligne-Rapoport~\cite{deligne-rapoport} to be canonically
equivalent to the degree zero subgroup $\cX(\cO)$ of the free
abelian divisor group'' on the isomorphism classes of enhanced
supersingular  elliptic curves and of quaternion ideals.  Moreover,
this isomorphism  is compatible with the operation of Hecke operators,
which are  effectively computable in $\cX(\cO)$ in terms of ideal
homomorphisms.

The inner product of two classes in this setting is defined
to be the number of isomorphisms between any two representatives.
The linear extension to $\cX(\cO)$ gives an inner product which
agrees, under the isomophism, with the monodromy pairing on
$\cX_{J,p}$.  This gives, in particular, an isomorphism $\Phi_{J,p} \isom \coker(\cX(\cO) \rightarrow \Hom(\cX(\cO),\Z))$, and an
effective means of computing $\#\coker(\alpha_f)$ and
$\#(\alpha_f(\cX_{J,p})/\alpha_f(\cX_{J,p}[I_f]))$.

The ideal arithmetic of quaternions has been implemented in
{\sc Magma} \cite{magma} by the first author.  Additional details and
the application to Shimura curves, generalizing $X_0(N)$, can
be found in Kohel~\cite{kohel}.

\subsection{The Galois action on $\Phi_{A_f,p}$}

To determine the Galois action on $\Phi_{A_f,p}$, we need only
know the action of the Frobenius automorphism $\Frob_p$.  However
$\Frob_p$ is known to equal $-W_p$ where $W_p$ is the $p$th
Atkin-Lehner involution, which can be computed using modular symbols.
Since~$f$ is an eigenform, the involution $W_p$ acts as either
$+1$ or $-1$ on $\Phi_{A_f,p}$.
Moreover, the operator $W_p$ is determined by an involution on
the set of quaternion ideals, so can be determined explicitly
on the character group.

\section{Tables}

\subsection{Component groups at low level}

Table~\ref{tbl:lowlevel} gives the component groups of the quotients
$A_f$ of $J_0(N)$ for $N\leq 127$; it was
computed using a {\sc Magma} implementation of an
algorithm based on Theorem~\ref{thm:main}.  The column~$d$ is
the dimension, and the column $\#\Phi_{A_f,p}$ contains a list
of orders of component groups, one for each divisor~$p$ of~$N$.
An order is starred if the $\Gal(\Fpbar/\Fp)$-action is nontrivial.
More data along these lines can be obtained from the second author.

\begin{table}
\begin{center}
\caption{Component groups at low level}
\end{center}
\label{tbl:lowlevel}
$$\begin{array}{lcl} N & \, d \, & \, \#\Phi_{A_f,p}\, \\ 11 & 1 & 5\\ 14 & 1 & 6^*,3\\ 15 & 1 & 4^*,4\\ 17 & 1 & 4\\ 19 & 1 & 3\\ 20 & 1 & ?,2^*\\ 21 & 1 & 4,2^*\\ 23 & 2 & 11\\ 24 & 1 & ?,2^*\\ 26 & 1 & 3^*,3\\ & 1 & 7,1^*\\ 27 & 1 & ?\\ 29 & 2 & 7\\ 30 & 1 & 4^*,3,1^*\\ 31 & 2 & 5\\ 32 & 1 & ?\\ 33 & 1 & 6^*,2\\ 34 & 1 & 6,1^*\\ 35 & 1 & 3^*,3\\ & 2 & 8,4^*\\ 36 & 1 & ?,?\\ 37 & 1 & 1^*\\ & 1 & 3\\ 38 & 1 & 9^*,3\\ & 1 & 5,1^*\\ 39 & 1 & 2^*,2\\ & 2 & 14,2^*\\ 40 & 1 & ?,2\\ 41 & 3 & 10\\ 42 & 1 & 8,2^*,1^*\\ 43 & 1 & 1^*\\ & 2 & 7\\ 44 & 1 & ?,1^*\\ 45 & 1 & ?,1^*\\ 46 & 1 & 10^*,1\\ 47 & 4 & 23\\ 48 & 1 & ?,2\\ 49 & 1 & ?\\ 50 & 1 & 1^*,?\\ & 1 & 5,?\\ 51 & 1 & 3,1^*\\ & 2 & 16^*,4\\ 52 & 1 & ?,2^*\\ 53 & 1 & 1^*\\ \end{array}\quad \begin{array}{lcl} N & \, d \, & \, \#\Phi_{A_f,p}\, \\ & 3 & 13\\ 54 & 1 & 3^*,?\\ & 1 & 3,?\\ 55 & 1 & 2,2^*\\ & 2 & 14^*,2\\ 56 & 1 & ?,1\\ & 1 & ?,1^*\\ 57 & 1 & 2^*,1^*\\ & 1 & 2,2^*\\ & 1 & 10,1^*\\ 58 & 1 & 2^*,1^*\\ & 1 & 10,1^*\\ 59 & 5 & 29\\ 61 & 1 & 1^*\\ & 3 & 5\\ 62 & 1 & 4,1^*\\ & 2 & 66^*,3\\ 63 & 1 & ?,1^*\\ & 2 & ?,3\\ 64 & 1 & ?\\ 65 & 1 & 1^*,1^*\\ & 2 & 3^*,3\\ & 2 & 7,1^*\\ 66 & 1 & 2^*,3,1^*\\ & 1 & 4,1^*,1^*\\ & 1 & 10,5,1\\ 67 & 1 & 1\\ & 2 & 1^*\\ & 2 & 11\\ 68 & 2 & ?,2^*\\ 69 & 1 & 2,1^*\\ & 2 & 22^*,2\\ 70 & 1 & 4,2^*,1^*\\ 71 & 3 & 5\\ & 3 & 7\\ 72 & 1 & ?,?\\ 73 & 1 & 2\\ & 2 & 1^*\\ & 2 & 3\\ 74 & 2 & 9^*,3\\ & 2 & 95,1^*\\ 75 & 1 & 1^*,?\\ & 1 & 1,?\\ & 1 & 5,?\\ \end{array}\quad \begin{array}{lcl} N & \, d \, & \, \#\Phi_{A_f,p}\, \\ 76 & 1 & ?,1^*\\ 77 & 1 & 2^*,1^*\\ & 1 & 3^*,2\\ & 1 & 6,3^*\\ & 2 & 2,2^*\\ 78 & 1 & 16^*,5^*,1\\ 79 & 1 & 1^*\\ & 5 & 13\\ 80 & 1 & ?,2\\ & 1 & ?,2^*\\ 81 & 2 & ?\\ 82 & 1 & 2^*,1^*\\ & 2 & 28,1^*\\ 83 & 1 & 1^*\\ & 6 & 41\\ 84 & 1 & ?,1^*,2^*\\ & 1 & ?,3,2\\ 85 & 1 & 2^*,1\\ & 2 & 2^*,1^*\\ & 2 & 6,1^*\\ 86 & 2 & 21^*,3\\ & 2 & 55,1^*\\ 87 & 2 & 5,1^*\\ & 3 & 92^*,4\\ 88 & 1 & ?,1^*\\ & 2 & ?,2^*\\ 89 & 1 & 1^*\\ & 1 & 2\\ & 5 & 11\\ 90 & 1 & 2^*,?,3\\ & 1 & 6,?,1^*\\ & 1 & 4,?,1\\ 91 & 1 & 1^*,1^*\\ & 1 & 1,1\\ & 2 & 7,1^*\\ & 3 & 4^*,8\\ 92 & 1 & ?,1^*\\ & 1 & ?,1\\ 93 & 2 & 4^*,1^*\\ & 3 & 64,2^*\\ 94 & 1 & 2,1^*\\ & 2 & 94^*,1\\ 95 & 3 & 10,2^*\\ & 4 & 54^*,6\\ \end{array}\quad \begin{array}{lcl} N & \, d \, & \, \#\Phi_{A_f,p}\, \\ 96 & 1 & ?,2\\ & 1 & ?,2^*\\ 97 & 3 & 1^*\\ & 4 & 8\\ 98 & 1 & 2^*,?\\ & 2 & 14,?\\ 99 & 1 & ?,1^*\\ & 1 & ?,1\\ & 1 & ?,1^*\\ & 1 & ?,1^*\\ 100 & 1 & ?,?\\ 101 & 1 & 1^*\\ & 7 & 25\\ 102 & 1 & 2^*,2^*,1^*\\ & 1 & 6^*,6,1^*\\ & 1 & 8,4,1\\ 103 & 2 & 1^*\\ & 6 & 17\\ 104 & 1 & ?,1^*\\ & 2 & ?,2\\ 105 & 1 & 1,1,1\\ & 2 & 10^*,2^*,2\\ 106 & 1 & 4^*,1^*\\ & 1 & 5^*,1\\ & 1 & 24,1^*\\ & 1 & 3,1^*\\ 107 & 2 & 1^*\\ & 7 & 53\\ 108 & 1 & ?,?\\ 109 & 1 & 1\\ & 3 & 1^*\\ & 4 & 9\\ 110 & 1 & 7^*,1^*,3\\ & 1 & 3,1^*,1^*\\ & 1 & 5,5,1\\ & 2 & 16^*,3,1^*\\ 111 & 3 & 10^*,2\\ & 4 & 266,2^*\\ 112 & 1 & ?,1^*\\ & 1 & ?,1\\ & 1 & ?,1^*\\ 113 & 1 & 2\\ & 2 & 2\\ & 3 & 1^*\\ \end{array}\quad \begin{array}{lcl} N & \, d \, & \, \#\Phi_{A_f,p}\, \\ & 3 & 7\\ 114 & 1 & 2^*,5^*,1\\ & 1 & 20,3^*,1^*\\ & 1 & 6,3,1\\ 115 & 1 & 5^*,1\\ & 2 & 4^*,1^*\\ & 4 & 32,4^*\\ 116 & 1 & ?,1^*\\ & 1 & ?,2^*\\ & 1 & ?,1^*\\ 117 & 1 & ?,1\\ & 2 & ?,3\\ & 2 & ?,1^*\\ 118 & 1 & 2^*,1^*\\ & 1 & 19^*,1\\ & 1 & 10,1^*\\ & 1 & 1,1^*\\ 119 & 4 & 9,3^*\\ & 5 & 48^*,8\\ 120 & 1 & ?,1,1^*\\ & 1 & ?,2,1\\ 121 & 1 & ?\\ & 1 & ?\\ & 1 & ?\\ & 1 & ?\\ 122 & 1 & 4^*,1^*\\ & 2 & 39^*,3\\ & 3 & 248,1^*\\ 123 & 1 & 1^*,1^*\\ & 1 & 5,1\\ & 2 & 7,1^*\\ & 3 & 184^*,4\\ 124 & 1 & ?,1^*\\ & 1 & ?,1\\ 125 & 2 & ?\\ & 2 & ?\\ & 4 & ?\\ 126 & 1 & 8^*,?,1^*\\ & 1 & 2,?,1\\ 127 & 3 & 1^*\\ & 7 & 21\\ &&\\ &&\\ &&\\ \end{array}$$
\end{table}

\subsection{Some large component groups}

Let $\Omega_{A_f}$ be the real period of $A_f$, as defined by
Tate~\cite{tate:bsd} in his extension of the Birch  and Swinnerton-Dyer
conjecture to dimension greater than one.   The second author computed
$L(A_f,1)/\Omega_{A_f}$ for every newform~$f$ of level $N\leq 1500$.
The four ratios with the largest prime divisors in the numerator
are given in Table~\ref{table:lratios}.
The Birch and Swinnerton-Dyer conjecture predicts that the large
prime divisors of the numerators of these special values must divide
either the order of some $\Phi_{A_f,p}$ or of the Shafarevich-Tate
group of~$A_f$.  In each instance this is the case.

\begin{table}
\label{table:lratios}
\begin{center}
\caption{Large $L(A_f,1)/\Omega_{A_f}$}
\end{center}
$$\begin{array}{ccll} \qquad N \qquad\quad & \quad \dim \quad & \quad\quad L(A_f,1)/\Omega_{A_f} \qquad\qquad & \qquad \#\Phi_{A_f,p} \\ 1154=2\tdot 577 & 20 & 2^?\tdot85495047371/17^2 & 2^?\tdot 17^2 \tdot 85495047371, 2^? \\ 1238=2\tdot 619 & 19 & 2^?\tdot 7553329019/5\tdot 31 & 2^?\tdot 5\tdot31\tdot7553329019 , 2^?\\ 1322=2\tdot 661 & 21 & 2^?\tdot 57851840099/331 & 2^?\tdot 331 \tdot 57851840099, 2^?\\ 1382=2\tdot 691 & 20 & 2^?\tdot 37 \tdot 1864449649 /173 & 2^?\tdot 37\tdot173\tdot 1864449649, 2^?\\ 1478=2\tdot 739 & 20 & 2^?\tdot 7\tdot 29\tdot 1183045463 / 5\tdot 37 & 2^?\tdot5\tdot 7\tdot29\tdot37\tdot1183045463, 2^?\\ \end{array}$$
\end{table}

\section{Further directions}

Further considerations are needed to compute the {\em group}
structure of $\Phi_{A_f,p}$.
Once the group structure is determined, however, since the
action of Frobenius is known, the structure as a group scheme
is immediate.

An equivalence with quaternion divisor groups is not known to
hold for the character group at~$p$ when $p^2$ divides~$N$.
Thus our methods say nothing about the component group at primes
whose {\em square} divides the level.  The free abelian group
on classes of nonmaximal
orders  of index~$p$ at a ramified prime give a well-defined
divisor group.  Do the resulting Hecke modules determine the
component groups for quotients of level $p^2M$?

Is it possible to define quantities as in Theorem~\ref{thm:main}
even when the weight of~$f$ is {\em greater than~$2$}?
If so, how are the resulting quantities related to the Bloch-Kato
Tamagawa  numbers (see~\cite{bloch-kato}) of the higher weight motive
attached to~$f$?

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\end{document}