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Author: William A. Stein
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% modsymbols.tex
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\documentclass[11pt]{article}
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\title{Manin symbols and modular forms\\{\large (first draft)}}
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\author{W. Stein\footnote{UC Berkeley, Department of Mathematics, Berkeley, CA 94720, USA.}}
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\include{macros}
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\begin{document}
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\maketitle
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%\begin{abstract}
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%\end{abstract}
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\tableofcontents
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\section{Introduction}
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\begin{quote}
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``The object of numerical computation is theoretical advance.''
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-- Atkin
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\end{quote}
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The definition of the
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spaces $S_k(\Gamma)$ of modular forms as functions on the upper half
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plane $\h$ satisfying a certain equation is very abstract.
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The definition of the Hecke operators even more so.
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We are fortunate that we now have methods available which allow us
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to transform the vector space of cusp forms of given weight $\geq 2$
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and level $N$ into a concrete object, which can be
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explicitely computed. We have the work
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of Atkin-Lehner, Birch-Swinnerton-Dyer,
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Cremona, Manin, Mazur, Merel, and many others
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to thank for this.
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The Eichler-Selberg trace formulas, as developed in \cite{hijikata} and \cite{wada},
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can be used to compute characteristic polynomials of Hecke operators and hence
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gain some information about spaces of modular forms.
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It is also sometimes possible to write down explicit basis in terms of
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$\Theta$-series and to compute the action of Hecke operators on their $q$-expansions.
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Other methods include computing the Hecke operators and $q$-expansions
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using Brandt matrices and quaternion algebras as in \cite{pizer}
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or \cite{kohel}, or the module of supersingular points in
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``characteristic $N$'' as exploited by Mestre and Oesterle in \cite{mestre}.
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Though the above methods are each beautiful and well suited to certain applications,
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we will not discuss them further here. Instead we focus
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on the modular symbols method, as it also has many advantages. We will only
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discusss the theory in this summary paper, leaving an explicit description
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of the objects involved for later. Nonetheless there is a definite
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gap between the {\em theory} on the one hand, and an efficient running
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machine implimentation on the other. To impliment the algorithms
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hinted at below requires making absolutely everything completely explnicit
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and then finding intelligent and efficient ways of performing the
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necessary manipulations. This is a nontrivial and tedious task, with
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room for error at every step.
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Our exposition follows very closely that of \cite{merel}.
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\subsection{Notation}
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Let $\Gamma$ be a finite index subgroup of $\sltwoz$ and
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$k\geq 2$ an integer. If $k$ is odd, assume $-1\not\in\Gamma$,
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so that the modular forms theory is nonempty.
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Let $\P^1(\Q) = \Q\union\{\infty\}$.
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\section{Modular symbols and modular forms}
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\subsection{Modular symbols}
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Let $\M$ be the $\Z$-module generated by formal symbols
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$\{\alpha,\beta\}$, $\alpha, \beta\in\P^1(\Q)$, subject
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to the relations
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$$\{\alpha,\beta\}+\{\beta,\gamma\}+\{\gamma,\alpha\}=0.$$
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Thus $\{\alpha,\beta\}=-\{\beta,\alpha\}$
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and $\{\alpha,\alpha\}=0$. There is a left action
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of $g\in\gltwoq$ given by
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$g.\{\alpha,\beta\}=\{g\alpha,g\beta\}$.
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\comment{% This map has no good properties, so far as I can see!
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There is a natural homomorphism
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$$\cD_0\into\M:\quad \alp\mapsto \{0,\alp\}$$
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By Manin's trick Lemma~\ref{maninstrick} this map is surjective,
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[and I think the relations defining $\M$ exactly make
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it injective].
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}
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Let $$V_k=\Sym^{k-2}_\Z(\Z\cross\Z)=\Z_{k-2}[X,Y]$$ be the
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free $\Z$-module of homogeneous polynomimals in two variables
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of degree $k-2$.
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There is a {\em left} action of $g=\abcd{a}{b}{c}{d}\in\mtwoz$
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given by $$g.P(X,Y) = P(\det(g)g^{-1}(X,Y)) = P(dX-bY,-cX+aY).$$
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%Note that $g$ induces an automorphism of the algebra
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%$\Z[X,Y]$ which restricts to a linear endomorphism
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%of each homogeneous piece, and that
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%$$g.(h.P(X,Y))=g.P(\det(h)h^{-1}(X,Y))=P(((X,Y)g)h)=(gh).P(X,Y).$$
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The space
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$$\M_k := V_k\tensor_\Z \M.$$
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is equipped with a left action of $\mtwoz$ given by
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$$g.(P\tensor x)
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=g.P\tensor g.x.$$
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%Note that $\M_k$ is {\em not} $V_k\tensor_{\Z[\mtwoz]}\M_k$
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%with its induced action.
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Let $$\M_k(\Gamma):=H_0(\Gamma,\M_k)$$ be the zeroth homology group.
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Thus $\M_k(\Gamma)$ is the quotient of $\M_k$ by the relations
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$g.x=x$ for all $x\in\M_k$ and $g\in\Gamma$.
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The elements of $\M_k(\Gamma)$ are called
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{\em modular symbols of weight $k$ for $\Gamma$}.
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As we will see later, using Shapiro's lemma and an explicit computation,
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$$\M_k(\Gamma)\tensor\C\isom H^1(\Gamma,V_k\tensor\C).$$
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The theory of Eichler and Shimura embeds modular forms
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in $H^1(\Gamma,V_k\tensor\C)$.
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\subsection{Manin symbols}
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Let $e=\{0,\infty\}\in\M$.
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\comment{Note furthermore that $\psltwoz=\sltwoz/\{\pm 1\}$
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is the free product of the
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cyclic group of order $2$ generated by $S$ and the cyclic group of order
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$3$ generated by $\tau$.}
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\begin{proposition}[Manin's trick]~\label{maninstrick}
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The elements $g.e$ for $g\in\sltwoz$ generate $\M$.
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\end{proposition}
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\begin{proof}
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(From \cite{cremona1}.) Writing $\{\alpha,\beta\}=\{0,\beta\}-\{0,\alp\}$,
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it suffices to show that every symbol of the form
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$\{0,\alp\}$ is in the group generated by
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the $g.e$. Let
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$$\frac{p_{-2}}{q_{-2}} = \frac{0}{1},\,
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\frac{p_{-1}}{q_{-1}}=\frac{1}{0},\,
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\frac{p_0}{1}=\frac{p_0}{q_0},\,
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\frac{p_1}{q_1},\,
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\frac{p_2}{q_2},\ldots,\frac{p_r}{q_r}=\alp$$
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denote the continued fraction convergents of the rational number $\alp$.
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Then
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$$p_j q_{j-1} - p_{j-1} q_j = (-1)^{j-1}\qquad \text{for }-1\leq j\leq r.$$
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Hence
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$$\{0,\alp\}
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=\sum_{j=-1}^{r}\left\{\frac{p_{j-1}}{q_{j-1}},\frac{p_j}{q_j}\right\}
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= \sum_{j=-1}^{r}g_j.e$$
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where
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$g_j = \mtwo{(-1)^{j-1}p_j}{p_{j-1}}{(-1)^{j-1}q_j}{q_{j-1}}$.
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\end{proof}
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To a pair $g\in\sltwoz$ and $P\in V_k$ define
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the {\em Manin symbol}
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$$[P,g]=g.(P\tensor e)\in\M_k(\Gamma).$$
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The matrices
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$$\sigma=\mtwo{0}{-1}{1}{0}\quad \text{and} \quad \tau=\mtwo{0}{-1}{1}{-1}$$
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satisfy
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$$\sigma^4=1,\,\,\tau^3=1$$
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and generate $\sltwoz$.
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\begin{proposition}~\label{maninsymbols}
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Let $g\in\sltwoz$ and $P\in V_k$. The symbol
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$[P,g]$ depends only on $P$ and the class $\Gamma g$. When
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$g$ runs through $\sltwoz$ and $P$ runs through $V_k$,
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the Manin symbols generate $\M_k(\Gamma)$ ({\em maybe}, at
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least this should be true after tensoring with $\Q$).
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Furthermore, they satisfy
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\begin{eqnarray*}
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\mbox{}[P,g]+[\sigma^{-1}P,g\sigma]&=&0,\\
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\mbox{}[P,g]+[\tau^{-1}P,g\tau]+[\tau^{-2}P,g\tau^2]&=&0, \\
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\end{eqnarray*}
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\end{proposition}
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\begin{proof}
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The first assertion follows from the construction of $\M_k(\Gamma)$.
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The correct version of the second assertion
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(should somehow) follow from Lemma~\ref{maninstrick}.
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For the third assertion note the following relations:
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\begin{eqnarray*}
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e + \sigma(e) &=& \{0,\infty\}
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+ \{\sigma(0),\sigma(\infty)\}\\
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&=&\{0,\infty\}+\{\infty,0\}=0,\\
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\\
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e+\tau(e)+\tau^2(e) &=&
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\{0,\infty\} + \{\tau(0),\tau(\infty)\}
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+ \{\tau^2(0),\tau^2(\infty)\}\\
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&=& \{0,\infty\} + \{1,0\} + \{\infty,1\} = 0.
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\end{eqnarray*}
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Thus
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\begin{eqnarray*}
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\mbox{}[P,g]+[\sigma^{-1}P,g\sigma] &=&
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g(P\tensor e) + g\sigma(\sigma^{-1}P\tensor e)\\
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&=& gP\tensor e + gP\tensor g\sigma e\\
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&=& gP\tensor g(e+\sigma(e)) = 0,\\
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\\
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\mbox{}[P,g]+[\tau^{-1}P,g\tau]+[\tau^{-2}P,g\tau^2]&=&
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g(P\tensor e) + g\tau(\tau^{-1}P\tensor e) + g\tau^2(\tau^{-2}P\tensor e) \\
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&=& gP\tensor g(e+\tau(e)+\tau^2(e)) = 0.
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\end{eqnarray*}
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\end{proof}
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\begin{theorem}
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The above relations generate all relations satisfied by the Manin symbols.
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(Maybe one must tensor with $\Q$.)
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\end{theorem}
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\begin{remark}
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Not only do not know whether or not this is true before tensoring with $\Q$,
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I do not know how to prove this.
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\end{remark}
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%Note that the relation $[P,g]=[j^{-1}P,gj]$ is imposed
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%by repeated application of the $\sigma$ relation because $\sigma^2=j$.
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The Manin symbol $[P,g]$ can be written as $\Z$-linear combinations of
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Manin symbols $$[X^q Y^{k-2-q},g],\quad \text{with } 0 \leq q \leq k-2.$$ Since
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$[\sltwoz:\Gamma]$ is finite, $\M_k(\Gamma)$ is a finitely
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generated abelian group. In particular, we can write down an
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explicit basis which is then readily amenable to machine computation.
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We write $[X^q Y^{k-2-q},g]=[q,g]$ to simplify notation.
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Let $g_1,\ldots,g_n$ be a set of coset representatives for
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$\Gamma$ in $\sltwoz$. Then $\M_k(\Gamma)$ is generated by
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$$\{[q,a_i] : 0\leq q\leq k-2, \, 1\leq i\leq n\}$$
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subject to the relations given by Proposition~\ref{maninsymbols}.
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Note that $\M_k(\Gamma)$ may contain nontrivial torsion, which is
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not well understood (by me!).
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% what is the torsion?
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\subsection{Cuspidal modular symbols}
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\comment{
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%About my paper "Universal...", there are a few mistakes there
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%(one reason other than my own fault, is the fact that that I never
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%saw the proofs). Some of them are insignificant. One you have to
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%be careful about is the fact that the parameterization of the cusps
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%of X_1(N) is wrong. It is easy to fix it anyway. One should replace
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%the set I called P_N by the disjoint union of the (Z/gcd(d,N)Z)^* where
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%d runs through Z/NZ. Then everything works mutatis mutandis.
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}
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In this section we assume $k$ is even. There is a similar definition
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when $k$ is odd.
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Let $\cD_0=\Div^0(\P^1(\Q))$ be the group of divisors
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of degree zero supported on $\P^1(\Q)$ and note that
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$\gltwoq$ acts on $\cD_0$ on the left by linear
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fractional transformations.
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Let $\cC(\Gamma)=H_0(\Gamma)=H_0(\Gamma,\cD_0)$ be the zeroth
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homology of $\Gamma\subset\sltwoz$ acting on $\cD_0=\Div^0(\P^1(\Q))$,
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so $H_0(\Gamma)$ is the free abelian group generated
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by the set of orbits $\Gamma\backslash P^1(\Q)$. It is the free abelian group on the
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cusps of the modular curve $X({\Gamma})$. There is a map
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$\M_k(\Gamma)\into\cC(\Gamma)$ which on Manin symbols is
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$$[P,g]\mapsto P(1,0)[g(\infty)]-P(0,1)[g(0)].$$
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Let $\cS_k(\Gamma)$ be the kernel of this map.
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\subsection{Duality between modular symbols and modular forms}
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\subsubsection{Weight 2}
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The first homology group $H_1(X_{\Gamma},\Z)$ of the modular curve
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$X_{\Gamma}$, viewed as a real $2$-manifold, is a free abelian group
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of rank $2g$, where $g$ is the genus of $X_{\Gamma}$.
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The global differentials $\Omega(X) = H^0(X,\Omega)$
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on $X_{\Gamma}$, now viewed as a Riemann surface, form a
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$g$ dimensional complex vector space. It is equal to the
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complex vector space $S_2(\Gamma)$ of cusp forms.
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There is a nondegenerate pairing
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\begin{eqnarray*}
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H_1(X_{\Gamma},\Z)\tensor \Omega(X)&\ra& \C\\
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\langle \gamma, \omega \rangle &\mapsto& \int_{\gamma} \omega.
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\end{eqnarray*}
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Taking coefficients in $\R$ we have
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$$H_1(X_{\Gamma},\R)=H_1(X_{\Gamma},\Z)\tensor\R.$$
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Extending the above pairing gives
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a natural injection of $H_1(X_{\Gamma},\R)$
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into the dual space of $\Omega(X)$. Since the
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two spaces have the same real dimension, this injection
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must be an isomorphism.
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Suppose now that $k=2$. We can identify modular symbols
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$\{\alp,\beta\}$ for $\Gamma$ as elements of
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$H_1(X_{\Gamma},\R)$, and we have the formula,
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$$\langle \{\alp,\beta\}, \omega\rangle
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= \int_{\alp}^{\beta} \omega.$$
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In fact, modular symbols were first introduced in this way
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by Birch in \cite{birch} in his work
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with Swinnerton-Dyer on the special value at $s=1$ of
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the $L$-function associated to a (modular) elliptic curve.
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\subsubsection{Higher weight}
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The duality generalizes to higher weight.
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Let $f:\h\into\C$ be a map. For $g=\abcd{a}{b}{c}{d}\in\gltwoq$ and $z\in\h$,
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define
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\begin{eqnarray*}
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f|[g]_k(z)&=&(cz+d)^{-k} f(gz)(\det g)^{k-1}\\
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f|[\overline{g}]_k(z)&=&(c\overline{z}+d)^{-k} f(gz)(\det g)^{k-1}
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\end{eqnarray*}
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We denote by $S_k(\Gamma)$ (resp. $\overline{S_k(\Gamma)}$) the complex
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vector space of holomorphic (resp. antiholomorphic) cusp forms of
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weight $k$ for $\Gamma$. There is a canonical isomorphism of
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real vector spaces between $S_k(\Gamma)$ and $\overline{S_k(\Gamma)}$
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which associates to $f$ the antiholomorphic modular form
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$z\mapsto \overline{f(z)}$.
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There is a pairing
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$$(S_k(\Gamma)\oplus \overline{S_k(\Gamma)})\cross \M_k(\Gamma)\into\C$$
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given by the rule
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$$\langle f_1+f_2,P\tensor\{\alpha,\beta\}\rangle
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= \int_{\alp}^{\beta} f_1(z)P(z,1)dz
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+ \int_{\alp}^{\beta} f_2(z)P(\overline{z},1)dz $$
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where $f_1\in S_k(\Gamma)$ and $f_2\in \overline{S_k(\Gamma)}$.
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\begin{theorem}~\label{pairing}
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The following pairing, obtained from the above one,
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is nondegenerate:
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$$(S_k(\Gamma)\oplus\overline{S_k(\Gamma)})\cross \cS_k(\Gamma)
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\into \C.$$
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\end{theorem}
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\subsection{Complex conjugation}
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Let $\eta=\abcd{-1}{0}{0}{1}$ and $\tilde{\eta}=\abcd{1}{0}{0}{-1}$.
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Assume in this section that $\eta^{-1}\Gamma\eta=\Gamma$.
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\begin{proposition} The map $\iota$ which associates to
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$f\in S_k(\Gamma)\oplus\overline{ S_k(\Gamma)}$ the
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function $z\mapsto f(-\overline{z})$ is a complex
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linear involution of $S_k(\Gamma)\oplus\overline{ S_k(\Gamma)}$
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which exchanges $S_k(\Gamma)$ and $\overline{ S_k(\Gamma)}$.
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\end{proposition}
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Define an involution $\iota^{\star}$ on $\M_k(\Gamma)$ by
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$$\iota^{\star}(P\tensor x)= -\tilde{\eta}P\tensor\eta x.$$
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This involution is adjoint to $\iota$ with respect to
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the pairing of Theorem~\ref{pairing}. Moreover $\iota^{\star}$
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acts as follows on Manin symbols
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$$\iota^{\star}([P,g]) = -[\tilde{\eta}P,\eta g \eta^{-1}].$$
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Let $\cS_k(\Gamma)^{+}$ denote the subspace of elements
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of $\cS_k(\Gamma)$ fixed by $\iota^{\star}$.
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\begin{proposition}
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The bilinear pairing induced by the pairing $\langle . , . \rangle$
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$$S_k(\Gamma)\cross \cS_k(\Gamma)^{+}\into\C$$
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is nondegenerate.
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\end{proposition}
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\subsection{Eichler-Shimura}
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Eichler and Shimura found a way to embed
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modular forms into a cohomology group.
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There is also a way to embed modular symbols into the same cohomology
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group.
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The complex
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vector space $V_k\tensor_{\Gamma} \sltwoz$ is endowed
344
with a {\em right} action of $\sltwoz$ given by the formula
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$$(P\tensor g).\gamma = (\gamma^{-1}P)\tensor (g\gamma).$$
346
\begin{proposition}
347
We have an isomorphism of complex vector spaces
348
$$H^1(\sltwoz,V_k\tensor_{\Gamma} \sltwoz\tensor\C)\isom\M_k(\Gamma)\tensor\C.$$
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\end{proposition}
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\begin{proof}
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This is Proposition 9 of \cite{merel}.
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The proof involves explicit computations with cocycles using the fact that
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$\sltwoz$ is generated by $\sigma$ and $\tau$.
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\end{proof}
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\begin{remark}
357
It might be possible to replace tensoring with $\C$ by something
358
less severe.
359
\end{remark}
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\begin{lemma}[Shapiro]
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Let $H$ be a subgroup of a group $G$ and let $A$ be a $\Z[H]$-module.
363
Then
364
$$H^q(G,\Hom_H(\Z[G],A))=H^q(H,A)\qquad\text{for all $q\geq 0$}.$$
365
\end{lemma}
366
367
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\begin{corollary}
369
There is an isomorphism
370
$$\M_k(\Gamma)\tensor\C \isom H^1(\Gamma, V_k\tensor\C).$$
371
\end{corollary}
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\begin{proof}
373
Since $\Gamma$ has finite index in $\sltwoz$ there is an isomorphism
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$$\Hom_{\Gamma}(\Z[\sltwoz],V_k\tensor\C)
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\isom V_k\tensor_{\Gamma} \sltwoz\tensor\C.$$
376
Now apply Shapiro's lemma.
377
\end{proof}
378
379
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Define the {\em parabolic} cohomology group $H^1_P$ by the exactness
381
of the following sequence
382
$$0\into H^1_P(\Gamma,V_k\tensor \C) \into
383
H^1(\Gamma,V_k\tensor\C) \into
384
\bigoplus_{\text{cusps $\alp$}} H^1(\Gamma_{\alp},V_k\tensor\C)$$
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where $\Gamma_{\alp}$ is the stabilizer in $\Gamma$ of the cusp $\alp$
386
of $X_{\Gamma}$.
387
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For $f\in M_k(\Gamma)$ define a class in $H^1(\Gamma,V_k\tensor\C)$
389
by the cocycle
390
$$\gamma\mapsto \int_{z_0}^{\gamma(z_0)} f(z)
391
\binom{z}{1}^{k-2}dz.$$
392
Here $z_0$ is a basepoint, $v^{k-2}$ denotes the image
393
of $v\tensor\cdots\tensor v$ in $\Sym^{k-2}(\C\cross\C)$
394
and the integral is that of a vector-valued differential.
395
There is a similiar construction for holomorphic
396
differentials.
397
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\begin{theorem}[Eichler-Shimura]
399
The map above gives rise to isomorphisms
400
\begin{eqnarray*}
401
M_k(\Gamma)\oplus\overline{S_k(\Gamma)}&\into &H^1(\Gamma,V_k\tensor \C)\\
402
S_k(\Gamma)\oplus\overline{S_k(\Gamma)}&\into &H^1_P(\Gamma,V_k\tensor \C).
403
\end{eqnarray*}
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\end{theorem}
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406
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%\subsection{Characters}
408
% put Hijikata trace formula for the case $N=1$ here.
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\section{Linear maps}
410
411
\subsection{Linear operators}
412
Let $\Delta\subset\mtwoz$ such that
413
$\Gamma\Delta\Gamma=\Delta$ and such that
414
$\Gamma\backslash\Delta$ is finite. Note
415
that $\Delta$ is a union of double cosets
416
of $\Gamma\backslash\mtwoz/\Gamma$. Let
417
$R$ be a set of representatives of $\Gamma\backslash\Delta$.
418
419
\comment{
420
Two examples are $\Gamma=\Gamma_1(N)$ with
421
$$\Delta = \Delta_1(N) = \{\abcd{a}{b}{c}{d}\in M_2(\Z)
422
: \det > 0, \,\,c\con a-1\con 0\pmod {N}\},$$
423
and $\Gamma=\Gamma_0(N)$ with
424
$$\Delta = \Delta_0(N) = \{\abcd{a}{b}{c}{d}\in M_2(\Z)
425
: \det > 0,\,\, c\con 0\pmod {N}, \,(a,N)=1\}.$$
426
427
Define the {\em Hecke ring} $R(\Gamma,\Delta)$ as follows.
428
It is the free $\Z$-module generated by the double cosets
429
$\Gamma\alp\Gamma$, $\alp\in\Delta$. Define multiplication
430
between two double cosets $u=\Gamma\alp\Gamma$ and
431
$v=\Gamma\beta\Gamma$ as follows. Consider their coset decompositions
432
$\Gamma\alp\Gamma = \coprod_{i}\Gamma\alpha_i$
433
and $\Gamma\beta\Gamma = \coprod_{i}\Gamma\beta_i$.
434
Then $\Gamma\alp\Gamma\beta\Gamma=\union_{i,j}\Gamma\alp_i\beta_j$
435
(not necessarily disjoint), and so $\Gamma\alp\Gamma\beta\Gamma$
436
is a finite union of double cosets of the form $\Gamma\gamma\Gamma$.
437
Define
438
$$u\cdot v = \sum_{w} m(u,v;w)w$$
439
where the sum is extended over all double cosets
440
$w=\Gamma\gamma\Gamma\subset\Gamma\alp\Gamma\beta\Gamma$, and
441
$$m(u,v;w)=\#\{(i,j) : \Gamma\alp_i\beta_j = \Gamma\gamma\}$$
442
for $w=\Gamma\gamma\Gamma$. Thus equipped,
443
$R(\Gamma,\Delta)$ becomes an associate, and in fact commutative,
444
ring with $\Gamma=\Gamma\cdot 1\cdot\Gamma$ as the unit element.
445
}
446
\subsubsection{Action on modular forms}
447
Let $M_k(\Gamma)$ be the space of modular forms of weight $k$ for $\Gamma$.
448
For $f\in M_k(\Gamma)$, define an operator $T_{\Delta}$ by
449
$$T_\Delta(f) = \sum_{\alp\in R} f|[\alp]_k$$
450
This is a well-defined linear action
451
on $M_k(\Gamma)$ which preserves the subspace $S_k(\Gamma)$.
452
453
\subsubsection{Action on modular symbols}
454
Similiarly, define an operator $T_{\Delta}$ on the
455
space $\M_k(\Gamma)$ of modular symbols by
456
$$T_\Delta(x) = \sum_{\alp\in R} \alp.x.$$
457
458
\subsubsection{Hecke operators}
459
Suppose now that $\Gamma=\Gamma_1(N)$. Let $n\geq 1$ be an
460
integer and set
461
$$\Delta_n=\{\abcd{a}{b}{c}{d}\in M_2(\Z) :
462
\det = n,\,N|c,\, N|(a-1)\}.$$
463
Then the $n$th {\em Hecke operator} is $T_{\Delta_n}$.
464
If $\Gamma=\Gamma_0(N)$ the condition that $N|(a-1)$ is relaxed to
465
$(N,a)=1$.
466
467
\subsection{Action on Manin symbols}
468
We now describe how to explicitely compute the action of the
469
Hecke operators on $\M_k(\Gamma)$.
470
Recall, we have an explicit ``generators and relations''
471
description of $\M_k(\Gamma)$ in terms of Manin symbols.
472
The action of the Hecke operators (and other linear operators)
473
described in the previous section is given in terms of
474
modular symbols. We {\em could} describe the action of
475
an operator on a Manin symbol by taking the Manin symbol,
476
finding the corresponding modular symbol, acting by the
477
operator, and then converting back to a sum of Manin symbols.
478
This process is painfully inefficient as it involves repeated
479
application of Proposition~\ref{maninstrick}. This was
480
how computations were originally done until Mazur and Merel
481
described the action of the Hecke operators directly in terms
482
of Manin symbols.
483
484
Let $n>0$ be an integer. We denote by $\mtwoz_n$ the set of
485
matrices of $\mtwoz$ of determinant $n$.
486
487
\begin{definition}[Condition (M)]
488
We say that an element $\sum_{g} a_g g\in\C[\mtwoz_n]$
489
satisfies condition (M) if for all cosets
490
$\cC\in \mtwoz_n/\sltwoz$, we have in $\C[\P^1(\Q)]$,
491
$$\sum_{g\in \cC} a_g ([g(\infty)] - [g(0)]) = [\infty]-[0].$$
492
\end{definition}
493
Note that the condition (M) depends neither on the level
494
or the weight.
495
496
Suppose now that $\Gamma=\Gamma_1(N)$ (or $\Gamma_0(N)$).
497
There is a bijection between
498
cosets $\Gamma\backslash\sltwoz$ and pairs of integers
499
$(u,v)$ satisfying a certain equivalence. The bijection associates
500
to a $2\times 2$ matrix its bottom two entries. We may thus view
501
the Manin symbols as pairs $[P,(u,v)]$.
502
503
\begin{theorem}[Merel]
504
Let $[P,(u,v)]$ be a Manin symbol. Suppose $\sum_{g} a_g g\in\C[\mtwoz_n]$
505
satisfies condition (M). Then we have
506
$$T_n([P(X,Y),(u,v)]) = \sum_{g=\abcd{a}{b}{c}{d}\in\mtwoz_n}
507
a_g [P(aX+bY,cX+dY),(au+cv,bu+dv)]$$
508
where the sum is restricted to the matrices $g$ such that
509
$\gcd(au+cv,bu+dv)=1$ (if $(n,N)=1$ this restriction is
510
unnecessary).
511
\end{theorem}
512
\begin{proof}
513
See section 2 of \cite{merel}.
514
\end{proof}
515
516
The element $\sum_{g \in\mtwoz_n} g \in \C[\mtwoz_n]$
517
satisfies condition (M). In Merel's paper one can find other families of simpler
518
(more sparse) elements satisfying condition (M).
519
520
\comment{
521
For one of the families $S_n$ there is an
522
asymptotic formula for the number $|S_n|$ of nonzero summands:
523
$$|S_n| \sim \frac{12\log 2}{\pi^2} \sigma(n)\log n, \quad\text{as $n\ra\infty$}$$
524
where $\sigma(n)$ is the sum of the positive divisors of $n$.
525
Let $s(n)$ be the right hand side, rounded down to the nearest integer. Then
526
$$s(10)=34,\quad s(100)=842,\quad s(250)=2177,\quad
527
s(500)=5719,\quad s(1000)=13622.$$
528
Finding a family minimizing these numbers is extremely important in
529
computing many Hecke eigenvalues.
530
531
\begin{remark}
532
Cremona has improved on this slightly to give even simpler
533
elements $\mathcal{X}_n\in\C[\mtwoz_n]$ which can be used to compute
534
the Hecke action, but which I don't think satisfies condition (M).
535
He proves this in his book for weight $2$, but I think his elements
536
work for any positive weight.
537
\end{remark}
538
}
539
%\subsection{Atkin-Lehner operators}
540
%\subsection{Newforms}
541
\comment{
542
\subsection{Characters}
543
Let $N$ be a positive integer and let
544
$\chi:(\Z/N\Z)^{\star}\into\C^{\star}$ be
545
a character. Let $\Z[\chi]=\Z[\chi(\Z/N\Z)^{\star}]$.
546
Define $\M_k(N,\chi)$ to be the quotient of
547
$\M_k(\Gamma_1(N))\tensor\Z[\chi]$ by the equivalence relation
548
which identifies the Manin symbol
549
$[P,(\lambda u,\lambda v)]$ with $\chi(\lambda)[P,(u,v)]$.
550
Define $\S_k(N,\chi)\subset \M_k(N,\chi)$.
551
\begin{proposition}
552
The pairing
553
$$(S_k(N,\chi)\sum\overline{ S_k(N,\chi)} \cross
554
\cS_k(N,\chi)\into \C$$
555
is nondegenerate.
556
\end{proposition}
557
Thus the space $\S_k(N,\chi)\tensor\C$ obtained from modular symbols
558
is related to the cusp forms $S_k(N,\chi)$.
559
}
560
561
\comment{
562
\section{Computation}
563
\subsection{Coset representatives~\label{cosetrep}}
564
\begin{proposition}
565
For $j=1,2$, let $g_j=\abcd{a_j}{b_j}{c_j}{d_j}\in\sltwoz$.
566
The following are equivalent.
567
\begin{enumerate}
568
\item The right cosets $\Gamma_0(N)g_1$ and
569
$\Gamma_0(N) g_2$ are equal,
570
\item $c_1d_2\con c_2 d_1\pmod{N}$,
571
\item There exists $u\in(\Z/N\Z)^{\star}$ such that
572
$c_1\con u c_2$ and $d_1\con u d_2$ $\pmod{N}$.
573
\end{enumerate}
574
\end{proposition}
575
\begin{proof}
576
Proposition 2.2.1 of \cite{cremona1}.
577
\end{proof}
578
579
\begin{proposition}
580
For $j=1,2$, let $\alp_j=p_j/q_j$ be cusps written in lowest terms.
581
The following are equivalent:
582
\begin{enumerate}
583
\item $\alp_2=g\alp_1$ for some $g\in\Gamma_0(N)$,
584
\item $q_2\con u q_1\pmod{N}$ and $up_2\con p_1\pmod{\gcd(q_1,N)}$,
585
with $\gcd(u,N)=1$,
586
\item $s_1 q_2\con s_2 q_1\pmod{gcd(q_1 q_2,N)}$, where
587
$s_j$ satisfies $p_j s_j \con 1\pmod {q_j}$.
588
\end{enumerate}
589
\end{proposition}
590
\begin{proof}
591
Proposition 2.2.3 of \cite{cremona1}.
592
\end{proof}
593
}
594
595
%\subsection{Cusp equivalence}
596
%\subsection{Newforms}
597
%\subsection{Computing Hecke eigenvalues}
598
599
%\subsection{Computing $L^{(r)}(f,1)$}
600
%\section{Computing in characteristic $p$}
601
%\section{Equations for modular curves}
602
%\section{Congruences between newforms}
603
%\section{$\Spec(\T)$}
604
%\section{Implimentation}
605
%\subsection{Coset representatives}
606
%\subsection{Linear algebra}
607
%\subsection{Complexity}
608
609
%\section{Examples}
610
%\subsection{Level $37$}
611
%\subsection{$X_0(389)$}
612
613
614
\section{Generating $H_1(\Gamma,\Z)$}
615
How can we generate $H_1(\Gamma,\Z)$ using modular symbols?
616
\begin{theorem}
617
Choose any $\alp\in\Q\union\{\infty\}$, it doesn't
618
matter which. Then the map
619
$$\Gamma\ra H_1(\Gamma,\Z): \quad \gamma\mapsto\{\alp,\gamma(\alp)\}$$
620
is a surjective group homomorphism.
621
\end{theorem}
622
So, knowing generators for $\Gamma$ would be enough.
623
624
Now specialize to the case $\Gamma=\Gamma_0(N)$.
625
Here is one guess for what {\em might} be true.
626
\begin{question}
627
Do the Manin symbols $(c,d)$ with $(c,N)=(d,N)=1$
628
generate $H_1(X_0(N),\Z)$?
629
\end{question}
630
When $N$ is prime those Manin symbols lie in
631
$H_1(X_0(N),\Z)$ because they correspond
632
to paths from the non-$\infty$ cusp to itself.
633
Let $(c,d)$ be such a Manin symbol and choose
634
$a,b$ so that $M=\abcd{a}{b}{c}{d}\in\sltwoz$.
635
Then the Modular symbol corresponding to $(c,d)$
636
is $M.\e=M.\{0,\infty\}=\{M(0),M(\infty)\}$.
637
Since $c$ and $d$ are both coprime to $N$, the
638
cusps $[\frac{a}{c}]$ and $[\frac{b}{d}]$ are
639
the same (remember, we are assuming $N$ is prime
640
so that there are only two cusps). The only
641
other way to force the cusps to be the same would
642
be to force $b$ and $d$ to both be divisible by $N$,
643
but then $(c,d)$ would not be a Manin symbol.
644
645
I do not know if these Manin symbols are enough
646
to generate all integral modular symbols. But,
647
we can set up some computer computations to get an
648
idea of whether or not we should expect this.
649
650
{\bf Computation 1.} Let $H=H_1(X_0(N),\Z)$.
651
Let $V$ be the submodule of $H$ generated by
652
the modular symbols $\{\infty,\gamma(\infty)\}$
653
where $\gamma=\abcd{a}{b}{N}{d}$ and $0<a<N$.
654
Let $W$ be the submodule of $H_1(X_0(N),\Q)$ generated
655
by the Manin symbols $(c,d)$ for which both
656
$c$ and $d$ are coprime to $N$.
657
If things were as easy as imaginable then
658
both $V$ and $W$ would be equal to $H$ (and
659
to each other). If $W$ is properly contained
660
in $V$ then we learn that the answer to the question
661
is {\em NO}. In the prime case, if $V$ is properly
662
contained in $W$ we learn that $V$ does not
663
generate $H$, which is also interesting.
664
665
We compute $V$ and $W$ for $11\leq N\leq 100$, and the
666
module index $[V:W]$.
667
Whenever there is a - for the index, this means that
668
that {\em neither} $V$ nor $W$ span $H_1(X_0(N),\Q)$
669
over $\Q$. In these cases we didn't compute the actual index.
670
671
\begin{center}
672
\begin{tabular}{|c|c|}\hline
673
$N$&$[W:V]$\\ \hline
674
11 & 1\\
675
14 & 1\\
676
15 & 1\\
677
17 & 1\\
678
19 & 1\\
679
20 & 1\\
680
21 & 1\\
681
22 & 1\\
682
23 & 1\\
683
24 & 1\\
684
26 & 1\\
685
27 & 1\\
686
28 & 1\\
687
29 & 1\\
688
30 & -\\
689
31 & 1\\
690
32 & 1\\
691
33 & 1\\
692
34 & 1\\
693
35 & 1\\
694
37 & 1\\
695
38 & 1\\
696
39 & 1\\
697
41 & 1\\
698
42 & -\\
699
43 & 1\\
700
44 & 1\\
701
45 & 1\\
702
46 & 1\\
703
47 & 1\\
704
48 & -\\
705
49 & 1\\
706
50 & 1\\
707
51 & 1\\
708
52 & 1\\
709
53 & 1\\
710
54 & -\\
711
55 & 1\\
712
56 & -\\
713
57 & 1\\
714
58 & 1\\
715
59 & 1\\
716
60 & -\\
717
61 & 1\\
718
62 & 1\\
719
\hline
720
\end{tabular}
721
\begin{tabular}{|c|c|}\hline
722
$N$&$[W:V]$\\ \hline
723
63 & 1\\
724
64 & 1\\
725
65 & 1\\
726
66 & -\\
727
67 & 1\\
728
68 & 1\\
729
69 & 1\\
730
70 & -\\
731
71 & 1\\
732
72 & -\\
733
73 & 1\\
734
74 & 1\\
735
75 & 1\\
736
76 & 1\\
737
77 & 1\\
738
78 & -\\
739
79 & 1\\
740
80 & -\\
741
81 & 1\\
742
82 & 1\\
743
83 & 1\\
744
84 & -\\
745
85 & 1\\
746
86 & 1\\
747
87 & 1\\
748
88 & -\\
749
89 & 1\\
750
90 & -\\
751
91 & 1\\
752
92 & 1\\
753
93 & 1\\
754
94 & 1\\
755
95 & 1\\
756
96 & -\\
757
97 & 1\\
758
98 & 1\\
759
99 & 1\\
760
100 & -\\
761
101 & 1\\
762
102 & -\\
763
103 & 1\\
764
104 & -\\
765
105 & -\\
766
106 & 1\\
767
107 & 1\\
768
\hline
769
\end{tabular}
770
\begin{tabular}{|c|c|}\hline
771
$N$&$[W:V]$\\ \hline
772
108 & -\\
773
109 & 1\\
774
110 & -\\
775
111 & 1\\
776
112 & -\\
777
113 & 1\\
778
114 & -\\
779
115 & 1\\
780
116 & 1\\
781
117 & 1\\
782
118 & 1\\
783
119 & 1\\
784
120 & -\\
785
121 & 1\\
786
122 & 1\\
787
123 & 1\\
788
124 & 1\\
789
125 & 1\\
790
126 & -\\
791
127 & 1\\
792
128 & 1\\
793
129 & 1\\
794
130 & -\\
795
131 & 1\\
796
132 & -\\
797
133 & 1\\
798
134 & 1\\
799
135 & -\\
800
136 & -\\
801
137 & 1\\
802
138 & -\\
803
139 & 1\\
804
140 & -\\
805
141 & 1\\
806
142 & 1\\
807
143 & 1\\
808
144 & -\\
809
145 & 1\\
810
146 & 1\\
811
147 & 1\\
812
148 & 1\\
813
149 & 1\\
814
150 & -\\
815
151 & 1\\
816
152 & -\\
817
\hline
818
\end{tabular}
819
\begin{tabular}{|c|c|}\hline
820
$N$&$[W:V]$\\ \hline
821
153 & 1\\
822
154 & -\\
823
155 & 1\\
824
156 & -\\
825
157 & 1\\
826
158 & 1\\
827
159 & 1\\
828
160 & -\\
829
161 & 1\\
830
162 & -\\
831
163 & 1\\
832
164 & 1\\
833
165 & -\\
834
166 & 1\\
835
167 & 1\\
836
168 & -\\
837
169 & 1\\
838
170 & -\\
839
171 & 1\\
840
172 & 1\\
841
173 & 1\\
842
174 & -\\
843
175 & 1\\
844
176 & -\\
845
177 & 1\\
846
178 & 1\\
847
179 & 1\\
848
180 & -\\
849
181 & 1\\
850
182 & -\\
851
183 & 1\\
852
184 & -\\
853
185 & 1\\
854
186 & -\\
855
187 & 1\\
856
188 & 1\\
857
189 & -\\
858
190 & -\\
859
191 & 1\\
860
192 & -\\
861
193 & 1\\
862
194 & 1\\
863
195 & -\\
864
196 & -\\
865
197 & 1\\
866
867
\hline
868
\end{tabular}
869
\end{center}
870
871
{\bf Conclusion:} Neither obvious set of elements
872
of $H_1(X_0(N),\Z)$ will, in general, generate.
873
It might be necessary to look at
874
\begin{verbatim}
875
R. S. Kulkarni, An arithmetic-geometry method of the study
876
of the subgroups of the modular group, American Journal of mathematics
877
113, 1991, 1053-1133
878
\end{verbatim}
879
and find explicit generators.
880
881
882
\begin{thebibliography}{HHHHHHH}
883
\bibitem[B]{birch} B. Birch, {\em Elliptic curves, a progress report,}
884
AMS conference on number theory, Stonybrook (1969), 396--400.
885
886
\bibitem[CR1]{cremona1} J.E. Cremona,
887
{\em Algorithms for modular elliptic curves, 2nd edition},
888
Cambridge University Press, (1997).
889
890
\bibitem[CR2]{cremona2} J.E. Cremona,
891
{\em Modular symbols for $\Gamma_1(N)$ and elliptic curves with
892
everywhere good reduction}, Math. Proc. Camb. Phil. Soc.
893
{\bf 111} (1992), 199--218.
894
895
\bibitem[HJ]{hijikata} H. Hijikata,
896
{\em Explicit formula of the traces of Hecke operators for
897
$\Gamma_0(N)$}, J. Math. Soc. Japan. {\bf 26} (1974), 56--82.
898
899
\bibitem[K]{kohel} D. Kohel, {\em Hecke module structure of quaternions},
900
(1998), preprint.
901
902
\bibitem[M]{merel} L. Merel,
903
{\em Universal fourier expansions of modular forms},
904
Springer L.N.M. 1585, (1994).
905
906
\bibitem[M1]{mestre} J.F. Mestre,
907
{\em La m\'{e}thode des graphs. Exemples et applications},
908
Taniguchi Symp., Proceedings of the international
909
conference on class numbers and fundamental units of
910
algebraic number fields (Katata, 1986), 217--242, Nagoya Univ., Nagoya, 1986.
911
912
\bibitem[P]{pizer} A. Pizer, {\em An algorithm for computing modular forms
913
on $\Gamma_0(N)$}, Journal of Algebra {\bf 64} (1980), 340--390.
914
915
\bibitem[S]{stein} W. Stein,
916
{\em Modular forms database},
917
{\tt http://boole.berkeley.edu/\~{\mbox{}}was/Tables}.
918
919
\bibitem[ST1]{steven1}
920
G. Stevens, {\em Rigid analytic modular symbols},
921
(preprint).
922
923
\bibitem[WA]{wada}
924
H. Wada, {\em A table of Hecke operators. II},
925
Proc. Japan Acad., {\bf 49} (1973), 380--384.
926
927
\bibitem[W1]{wang1}
928
X. Wang, {\em The hecke algebra on the cohomology of $\Gamma_0(p_0)$},
929
Nagoya Math. J., {\bf 121} (1991), 97--125.
930
931
\bibitem[W1]{wang2}
932
X. Wang, {\em The hecke operators on $S_k(\Gamma_1(N))$},
933
J. Symbolic Computation, {\bf 18} (1994), 187--198.
934
935
\end{thebibliography}
936
937
\end{document}
938