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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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Let $r_o,\ldots,r_p$ be coset representatives for $\Gamma_0(p)$
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in $\SL_2(\Z)$. So
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$$\SL_2(\Z) = \Gamma_0(p)r_o \, \union\, \Gamma_0(p)r_1 \, \union\,
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\cdots \, \union\, \Gamma_0(p)r_p.$$
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E.g.,
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$$\mtwo{1}{0}{0}{1},
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\mtwo{1}{0}{1}{1},
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\mtwo{1}{0}{2}{1},
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\mtwo{1}{0}{3}{1},
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\ldots,
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\mtwo{0}{-1}{1}{0}.$$
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\end{slide}
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\begin{slide}
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Let $V$ be the $p+1$-dimensional vector space with
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basis
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$$x_0,\,x_1,\,\ldots,\,x_p.$$
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{\bf Theorem} (Manin).
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There is an isomorphism
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$$V / R = \text{ModSym}(\Gamma_0(p)),$$
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where $x_i$
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maps to $r_i\{0,\infty\} = \{r_i(0),r_i(\infty)\}$,
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and the relations~$R$ are described explicitly below.
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\end{slide}
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\begin{slide}
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The subspace~$R$ of {\color{red} relations} is the subspace generated by
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$$x_i+ x_iS$$
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$$x_i + x_iT + x_iT^2,$$
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where
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$$S = \mtwo{0}{-1}{1}{\hfill 0}\text{ and }T = \mtwo{1}{-1}{1}{\hfill 0},$$
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and $i=0,\ldots,p$.
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\end{slide}
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\begin{slide}
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Generating modular symbols:
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$$\{0,\infty\}, \,\, \{0,1\},\,\, \{0, 1/2\}, \ldots,
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\{0,1/10\},\,\, \{\infty, 0\}$$
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Doing the linear algebra, we find
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that $\{0,\infty\}$, $\{0,1/5\}$, $\{0,1/7\}$
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are a basis. And, e.g.,
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\begin{eqnarray*}
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\{0,1/2\} &=& -\{0,1/5\}\\
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\{0,1/3\} &=& -\{0,1/7\}\\
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\{0,1/4\} &=& \,\,\,\,\, \{0,1/5\} - \{0,1/7\}
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\end{eqnarray*}
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\end{slide}
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\end{document}
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