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Author: William A. Stein
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\documentclass{slides}
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\usepackage{graphicx}
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\usepackage{psfrag}
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\usepackage{amsmath}
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\newcommand{\C}{\mathbf{C}}
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\newcommand{\F}{\mathbf{F}}
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\newcommand{\Q}{\mathbf{Q}}
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\newcommand{\Qbar}{\bar \Q}
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\newcommand{\Z}{\mathbf{Z}}
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\newcommand{\tors}{\mbox{\rm\small tors}}
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\font\cyr=wncyr10 scaled \magstep 4
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\newcommand{\Sha}{\mbox{\cyr X}}
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\newcommand{\Gal}{\mbox{\rm Gal}}
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\newcommand{\mtwo}[4]{\left(
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\begin{matrix}#1&#2\\#3&#4
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\end{matrix}\right)}
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\newcommand{\SL}{\mbox{\rm SL}}
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\newcommand{\union}{\cup}
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\usepackage{color}
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\begin{document}
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\begin{slide}
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There is an amazing sequence $T_1, T_2, T_3, \ldots$ of commuting
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linear maps on modular symbols. The corresponding
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systems of eigenvalues
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$$\{a_1,a_2,a_3,\ldots\}$$
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are
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the coefficients $a_n$ of the $q$-expansions of modular forms.
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When $n=p$ is prime to~$N$, we have
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$$T_p(\{\alpha,\beta\}) = \left[ \mtwo{p}{0}{0}{1}
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+ \sum_{r \hspace{-.6em}\mod p}
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\mtwo{1}{r}{0}{p}\right] \{\alpha,\beta\}.$$
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\end{slide}
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\begin{slide}
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Example: $N=11$
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\begin{eqnarray*}
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T_2(\{0,1/5\}) &=& \{0,2/5\} + \{0,1/10\} + \{1/2,3/5\}\\
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&=& -2\{0,1/5\} \\
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&=& \,\, a_2 \{0,1/5\}
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\end{eqnarray*}
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Consequently, the modular form of level $11$ is
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$$f=q + a_2 q^2 + a_3 q^3 + \cdots,$$
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where $T_\ell(\{0,1/5\}) = a_\ell\{0,1/5\}.$
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There is a deep connection with elliptic curves (due to Shimura):
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$$a_\ell = \ell + 1 - \# E(\F_\ell),$$
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where $E$ is $y^2 + y = x^3 - x^2 -10x -20$.
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\end{slide}
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\end{document}
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