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\title{Component groups of quotients of $J_0(N)$\footnote{Joint work with David Kohel.}}
\author{William A. Stein}
\date{Friday, 7 July 2000}
\newcommand{\head}[1]{\begin{center}\bf #1\end{center}}
\begin{document}
\maketitle

\begin{slide}
\head{Component groups of modular abelian varieties}
Let $f=\sum a_n q^n\in S_2(\Gamma_0(N);\C)$ be a newform.\\
\mbox{}$\qquad f \leadsto $ optimal quotient $A_f = J_0(N)/I J_0(N)$,
where $I$ is annihilator of $f$ in 
$\T=\Z[\ldots T_n \ldots]$.
$$\xymatrix{
 A_f^{\vee}\ar@{^(->}[r] & J_0(N)\ar@{->>}[d]\\
            & A_f
}$$
Let $\cA_f$ be the {\em N\'eron model} of $A_f$.  This is a
smooth commutative group scheme over $\Z$.

{\bf Component group:}
$$\Phi_{\cA_f,p} := (\cA_{f,\F_p} / \cA_{f,\F_p}^\circ)(\overline{\F}_p).$$

{\bf Goal:}
Find $\#\Phi_{\cA_f,p}$ as function of $f$ and $p$.

\end{slide}

\begin{slide}
\head{Computing component groups}
Suppose $p\mid\mid N$. 

$T =$ maximal torus of $\cJ_0(N)_{\F_p}$\\
$X = \Hom_{\Fbar_p}(T,\Gm)$ (character group)\\
$\langle \, , \, \rangle : X \times X \ra \Z$ (natural pairing)\\
$\alpha : X \ra \Hom(X[I],\Z)$ (induced map)\vspace{2ex}\\
$H = H_1(X_0(N),\Z)$ (modular symbols)\\
$S=S_2(\Gamma_0(N);\C)$ (cusp forms)\\
$\beta : H \ra \Hom(S[I],\C)$ (integration pairing)

{\bf Theorem.} (Stein)
$$
\#\Phi_{\cA_f,p}
   = \frac{\sqrt{\# \beta(H)/\beta(H[I])}}
          {\# \alpha(X)/\alpha(X[I])}\,\cdot \#\coker(\alpha).
$$

Image of $\Phi_{\cJ_0(N),p}$ in $\Phi_{\cA_f,p}$ is
isomorphic to $\coker(\alpha)$.  

{\bf Remark.} Every quantity in the formula can be 
computed, e.g., in M{\small AGMA}.
\end{slide}

\begin{slide}
\head{Example}
Biggest numerator of $L$-value:\\
$N=1154 = 2\cdot 577$\\
$\dim A_f = 20$\\
$L(A_f,1)/\Omega_{A_f} =  2^?\tdot85495047371/17^2$\\
BSD $\Longrightarrow$ Big component group!\\
Our formula gives:
$$\#\Phi_{\cA_f,2} = 2^?\tdot 17^2 \tdot 85495047371$$
Abelian variety $B = \mathbf{577E}$ (dim $22$) has
$a_2$ with $85495047371 \mid \Norm(a_2^2-(2+1)^2)$,
so Ribet's level raising theorem implies
that $A_f$ has component group of order divisible by
$85495047371$. 
\end{slide}

\begin{slide}
\head{Conjectures and Questions}

\begin{enumerate}
\item Find group structure of $\Phi_{\cA_f,p}$.

\item Find expression for 
$$\frac{\sqrt{\# \beta(H)/\beta(H[I])}}
          {\# \alpha(X)/\alpha(X[I])}$$
purely in terms of homology.\\
E.g., when $N$ prime quotient appears to be $1$ (Emerton seems to
have proved this).

\item Formula in the additive reduction case.


\end{enumerate}

\end{slide}


\end{document}