CoCalc Public Fileswww / talks / ants4-compgroups / ants4-talk.tex
Author: William A. Stein
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8\newcommand{\cX}{\mathcal{X}}
9\newcommand{\cJ}{\mathcal{J}}
10\newcommand{\F}{\mathbf{F}}
11\newcommand{\T}{\mathbf{T}}
12\newcommand{\Fbar}{\overline{\F}}
13\newcommand{\Z}{\mathbf{Z}}
14\newcommand{\C}{\mathbf{C}}
15\newcommand{\coker}{\mbox{\rm coker}}
16\newcommand{\Hom}{\mbox{\rm Hom}}
17\newcommand{\Ann}{\mbox{\rm Ann}}
18\newcommand{\Norm}{\mbox{\rm Norm}}
19\newcommand{\deg}{\mbox{\rm deg}}
20\newcommand{\ra}{\rightarrow}
21\newcommand{\Gm}{\mathbf{G}_m}
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23
24\title{Component groups of quotients of $J_0(N)$\footnote{Joint work with David Kohel.}}
25\author{William A. Stein}
26\date{Friday, 7 July 2000}
27\newcommand{\head}[1]{\begin{center}\bf #1\end{center}}
28\begin{document}
29\maketitle
30
31\begin{slide}
32\head{Component groups of modular abelian varieties}
33Let $f=\sum a_n q^n\in S_2(\Gamma_0(N);\C)$ be a newform.\\
34\mbox{}$\qquad f \leadsto$ optimal quotient $A_f = J_0(N)/I J_0(N)$,
35where $I$ is annihilator of $f$ in
36$\T=\Z[\ldots T_n \ldots]$.
37$$\xymatrix{ 38 A_f^{\vee}\[email protected]{^(->}[r] & J_0(N)\[email protected]{->>}[d]\\ 39 & A_f 40}$$
41Let $\cA_f$ be the {\em N\'eron model} of $A_f$.  This is a
42smooth commutative group scheme over $\Z$.
43
44{\bf Component group:}
45$$\Phi_{\cA_f,p} := (\cA_{f,\F_p} / \cA_{f,\F_p}^\circ)(\overline{\F}_p).$$
46
47{\bf Goal:}
48Find $\#\Phi_{\cA_f,p}$ as function of $f$ and $p$.
49
50\end{slide}
51
52\begin{slide}
53\head{Computing component groups}
54Suppose $p\mid\mid N$.
55
56$T =$ maximal torus of $\cJ_0(N)_{\F_p}$\\
57$X = \Hom_{\Fbar_p}(T,\Gm)$ (character group)\\
58$\langle \, , \, \rangle : X \times X \ra \Z$ (natural pairing)\\
59$\alpha : X \ra \Hom(X[I],\Z)$ (induced map)\vspace{2ex}\\
60$H = H_1(X_0(N),\Z)$ (modular symbols)\\
61$S=S_2(\Gamma_0(N);\C)$ (cusp forms)\\
62$\beta : H \ra \Hom(S[I],\C)$ (integration pairing)
63
64{\bf Theorem.} (Stein)
65$$66\#\Phi_{\cA_f,p} 67 = \frac{\sqrt{\# \beta(H)/\beta(H[I])}} 68 {\# \alpha(X)/\alpha(X[I])}\,\cdot \#\coker(\alpha). 69$$
70
71Image of $\Phi_{\cJ_0(N),p}$ in $\Phi_{\cA_f,p}$ is
72isomorphic to $\coker(\alpha)$.
73
74{\bf Remark.} Every quantity in the formula can be
75computed, e.g., in M{\small AGMA}.
76\end{slide}
77
78\begin{slide}
79\head{Example}
80Biggest numerator of $L$-value:\\
81$N=1154 = 2\cdot 577$\\
82$\dim A_f = 20$\\
83$L(A_f,1)/\Omega_{A_f} = 2^?\tdot85495047371/17^2$\\
84BSD $\Longrightarrow$ Big component group!\\
85Our formula gives:
86$$\#\Phi_{\cA_f,2} = 2^?\tdot 17^2 \tdot 85495047371$$
87Abelian variety $B = \mathbf{577E}$ (dim $22$) has
88$a_2$ with $85495047371 \mid \Norm(a_2^2-(2+1)^2)$,
89so Ribet's level raising theorem implies
90that $A_f$ has component group of order divisible by
91$85495047371$.
92\end{slide}
93
94\begin{slide}
95\head{Conjectures and Questions}
96
97\begin{enumerate}
98\item Find group structure of $\Phi_{\cA_f,p}$.
99
100\item Find expression for
101$$\frac{\sqrt{\# \beta(H)/\beta(H[I])}} 102 {\# \alpha(X)/\alpha(X[I])}$$
103purely in terms of homology.\\
104E.g., when $N$ prime quotient appears to be $1$ (Emerton seems to
105have proved this).
106
107\item Formula in the additive reduction case.
108
109
110\end{enumerate}
111
112\end{slide}
113
114
115\end{document}
116