CoCalc Public Fileswww / talks / ants4-compgroups / ants4-talk.texOpen with one click!
Author: William A. Stein
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\documentclass{slides}
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\usepackage[all]{xy}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\newcommand{\tdot}{\!\cdot\!}
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\newcommand{\cA}{\mathcal{A}}
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\newcommand{\cX}{\mathcal{X}}
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\newcommand{\cJ}{\mathcal{J}}
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\newcommand{\F}{\mathbf{F}}
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\newcommand{\T}{\mathbf{T}}
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\newcommand{\Fbar}{\overline{\F}}
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\newcommand{\Z}{\mathbf{Z}}
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\newcommand{\C}{\mathbf{C}}
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\newcommand{\coker}{\mbox{\rm coker}}
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\newcommand{\Hom}{\mbox{\rm Hom}}
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\newcommand{\Ann}{\mbox{\rm Ann}}
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\newcommand{\Norm}{\mbox{\rm Norm}}
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\newcommand{\deg}{\mbox{\rm deg}}
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\newcommand{\ra}{\rightarrow}
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\newcommand{\Gm}{\mathbf{G}_m}
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\title{Component groups of quotients of $J_0(N)$\footnote{Joint work with David Kohel.}}
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\author{William A. Stein}
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\date{Friday, 7 July 2000}
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\newcommand{\head}[1]{\begin{center}\bf #1\end{center}}
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\begin{document}
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\maketitle
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\begin{slide}
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\head{Component groups of modular abelian varieties}
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Let $f=\sum a_n q^n\in S_2(\Gamma_0(N);\C)$ be a newform.\\
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\mbox{}$\qquad f \leadsto $ optimal quotient $A_f = J_0(N)/I J_0(N)$,
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where $I$ is annihilator of $f$ in
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$\T=\Z[\ldots T_n \ldots]$.
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$$\xymatrix{
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A_f^{\vee}\[email protected]{^(->}[r] & J_0(N)\[email protected]{->>}[d]\\
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& A_f
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}$$
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Let $\cA_f$ be the {\em N\'eron model} of $A_f$. This is a
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smooth commutative group scheme over $\Z$.
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{\bf Component group:}
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$$\Phi_{\cA_f,p} := (\cA_{f,\F_p} / \cA_{f,\F_p}^\circ)(\overline{\F}_p).$$
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{\bf Goal:}
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Find $\#\Phi_{\cA_f,p}$ as function of $f$ and $p$.
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\end{slide}
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\begin{slide}
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\head{Computing component groups}
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Suppose $p\mid\mid N$.
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$T =$ maximal torus of $\cJ_0(N)_{\F_p}$\\
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$X = \Hom_{\Fbar_p}(T,\Gm)$ (character group)\\
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$\langle \, , \, \rangle : X \times X \ra \Z$ (natural pairing)\\
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$\alpha : X \ra \Hom(X[I],\Z)$ (induced map)\vspace{2ex}\\
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$H = H_1(X_0(N),\Z)$ (modular symbols)\\
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$S=S_2(\Gamma_0(N);\C)$ (cusp forms)\\
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$\beta : H \ra \Hom(S[I],\C)$ (integration pairing)
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{\bf Theorem.} (Stein)
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$$
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\#\Phi_{\cA_f,p}
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= \frac{\sqrt{\# \beta(H)/\beta(H[I])}}
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{\# \alpha(X)/\alpha(X[I])}\,\cdot \#\coker(\alpha).
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$$
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Image of $\Phi_{\cJ_0(N),p}$ in $\Phi_{\cA_f,p}$ is
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isomorphic to $\coker(\alpha)$.
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{\bf Remark.} Every quantity in the formula can be
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computed, e.g., in M{\small AGMA}.
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\end{slide}
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\begin{slide}
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\head{Example}
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Biggest numerator of $L$-value:\\
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$N=1154 = 2\cdot 577$\\
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$\dim A_f = 20$\\
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$L(A_f,1)/\Omega_{A_f} = 2^?\tdot85495047371/17^2$\\
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BSD $\Longrightarrow$ Big component group!\\
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Our formula gives:
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$$\#\Phi_{\cA_f,2} = 2^?\tdot 17^2 \tdot 85495047371$$
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Abelian variety $B = \mathbf{577E}$ (dim $22$) has
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$a_2$ with $85495047371 \mid \Norm(a_2^2-(2+1)^2)$,
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so Ribet's level raising theorem implies
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that $A_f$ has component group of order divisible by
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$85495047371$.
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\end{slide}
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\begin{slide}
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\head{Conjectures and Questions}
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\begin{enumerate}
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\item Find group structure of $\Phi_{\cA_f,p}$.
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\item Find expression for
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$$\frac{\sqrt{\# \beta(H)/\beta(H[I])}}
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{\# \alpha(X)/\alpha(X[I])}$$
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purely in terms of homology.\\
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E.g., when $N$ prime quotient appears to be $1$ (Emerton seems to
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have proved this).
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\item Formula in the additive reduction case.
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\end{enumerate}
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\end{slide}
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\end{document}
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