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\chapter{Modular symbols}
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\label{chap:modsym}
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Modular symbols are central to most of the algorithms and computations
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of this thesis. This chapter closely follows Merel~\cite{merel:1585}
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and Cremona~\cite{cremona:algs}. However, we work with modular forms
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with character throughout; this makes the theory trickier, but is
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extremely import in computational applications.
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\section{The definition of modular symbols}
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\label{sec:defnofmodsyms}
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Fix a triple $(N,k,\eps)$ consisting of a level~$N$, a
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weight~$k$, and a mod~$N$ character
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$\eps:(\Z/N\Z)^*\ra\C^*$
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such that $\eps(-1)=(-1)^k$.
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Let $\sM$ be the torsion-free abelian group generated by symbols
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$\{\alp,\beta\}$ with $\alp, \beta\in\P^1(\Q)=\Q\union\{\infty\}$
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subject to the following relations:
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$$\{\alp,\beta\}+\{\beta,\gamma\}+\{\gamma,\alp\} = 0,
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\qquad \text{ for all }\alp,\beta,\gamma\in\P^1(\Q).$$
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Because $\sM$ is torsion free, $\{\alp,\alp\}=0$ and
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$\{\alp,\beta\} = -\{\beta,\alp\}$.
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Let $V_{k-2}$\label{defn:vk} be the $\Z$-submodule of $\Z[X,Y]$ made up of
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homogeneous polynomials of degree $k-2$, and set
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$\sM_k := V_{k-2}\tensor\sM.$
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\label{pg:higherweightmodsym}
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For $g=\abcd{a}{b}{c}{d}\in\GL_2(\Q)$ and $P\in V_{k-2}$, let
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\begin{align*}
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gP(X,Y) &= P\left(\det(g)g^{-1}\vtwo{X}{Y}\right)
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= P\left(\mtwo{\hfill d}{-b}{-c}{\hfill a}\vtwo{X}{Y}\right)\\
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&= P(dX-bY,-cX+aY).
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\end{align*}
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This defines a left action of $\GL_2(\Q)$ on $V_{k-2}$;
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it is a left action because
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\begin{align*}
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(gh)P(v) &= P(\det(gh)(gh)^{-1}v)
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= P(\det(h)h^{-1}\det(g)g^{-1}v)\\
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&= gP(\det(h)h^{-1}v) = g(hP(v)).
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\end{align*}
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Combining this action with the action of $\GL_2(\Q)$ on $\P^1(\Q)$
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by linear fractional transformations gives
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a left action of $\GL_2(\Q)$ on $\sM_k$:
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$$g (P \tensor \{\alp,\beta\}) = g(P)\tensor\{g(\alp),g(\beta)\}.$$
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Finally, for $g=\abcd{a}{b}{c}{d}\in\Gamma_0(N)$, let
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$\eps(g) := \eps(\overline{a})$,
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where~$a$ reduces to $\overline{a}\in\Z/N\Z$.
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\begin{definition}[Modular symbols]\label{defn:modsym}
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The space $\sM_k(N,\eps)$ of \defn{modular symbols} is
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the torsion-free quotient of $\sM_k\tensor\Z[\eps]$ by the
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relations $gx=\eps(g)x$ for all $x\in\sM_k$
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and all $g\in\Gamma_0(N)$.
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\end{definition}
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Denote by $P\{\alp,\beta\}$ the image
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of $P\tensor\{\alp,\beta\}$ in $\sM_k(N,\eps)$.
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For any $\Z[\eps]$-algebra~$R$, let
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$\sM_k(N,\eps;R) := \sM_k(N,\eps)\tensor R$.
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\section{Cuspidal modular symbols}\label{cuspidalsymbols}
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Let~$\sB$ be the free abelian group generated by the symbols
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$\{\alp\}$ for all $\alp\in\P^1(\Q)$.
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There is a left action of~$\GL_2(\Q)$ on~$\sB$ given by
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$g\{\alp\}=\{g(\alp)\}$.
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Let $\sB_k := V_{k-2}\tensor \sB$, and let $\GL_2(\Q)$ act
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on $\sB_k$ by $g(P\{\alp\}) = (gP)\{g(\alp)\}$.
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\begin{definition}[Boundary modular symbols]\label{def:boundarysymbols}
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The space of \defn{boundary modular symbols} is the
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torsion-free quotient
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of $\sB_k\tensor\Z[\eps]$ by the relations
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$gx = \eps(g) x$ for all
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$g\in \Gamma_0(N)$ and $x\in \sB_k$.
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\end{definition}
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Denote by $P\{\alp\}$ the image of $P\tensor\{\alp\}$.
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The \defn{boundary map} is
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$$\delta: \sM_k(N,\eps) \ra \sB_k(N,\eps),
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\qquad \delta(P\{\alp,\beta\}) =
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P\{\beta\}-P\{\alp\}.$$
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\begin{definition}[Cuspidal modular symbols]\label{defn:cuspidalmodularsymbols}
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The space $\sS_k(N,\eps)$ of \defn{cuspidal modular symbols} is
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the kernel of~$\delta$:
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$$0\ra \sS_k(N,\eps) \ra\sM_k(N,\eps)\xrightarrow{\,\delta\,}
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\sB_k(N,\eps).$$
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\end{definition}
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\section{Duality between modular symbols and modular forms}
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For any integer~$k$, any $\C$-valued function~$f$ on
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the complex upper half plane $\h:=\{z \in \C : \im(z) > 0\}$,
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and any $\gamma\in\GL_2(\Q)$, define a function
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$f|[\gamma]$ on~$\h$ by
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$$(f|[\gamma]_k)(z) = \det(\gamma)^{k-1}\frac{f(\gamma z)}{(cz+d)^{k}}.$$
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\begin{definition}[Cusp forms]
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Let $S_k(N,\eps)$ be the complex vector space of holomorphic
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functions $f(z)$ on~$\h$ that satisfy
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the equation
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$$f|[\gamma]_k = \eps(\gamma)f$$
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for all $\gamma\in\Gamma_0(N)$, and such that~$f$
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is holomorphic at all cusps in the sense of
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\cite[pg.~42]{diamond-im}.
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\end{definition}
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\begin{definition}[Antiholomorphic cusp forms]
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Let $\Sbar_k(N,\eps)$ be the space of \defn{antiholomorphic
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cusp forms}; the definition is as above, except
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$$\frac{f(\gamma z)}{(c\overline{z}+d)^k} = \overline{\eps}(\gamma) f(z)$$
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for all $\gamma\in\Gamma_0(N)$.
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\end{definition}
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There is a canonical isomorphism of real vector spaces
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between $S_k(N,\eps)$ and $\Sbar_k(N,\eps)$ that associates
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to~$f$ the anitholomorphic cusp form $z\mapsto \overline{f(z)}$.
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\begin{theorem}[Duality]\label{thm:perfectpairing}
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There is a pairing
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\begin{equation*}
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\langle\,\, , \, \, \rangle:
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(S_k(N,\eps)\oplus \Sbar_k(N,\eps)) \cross \sM_k(N,\eps;\C)
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\ra \C
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\end{equation*}
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given by
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$$\langle f\oplus g, P\{\alp,\beta\}\rangle =
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\int_{\alp}^{\beta} f(z)P(z,1) dz
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+ \int_{\alp}^{\beta} g(z)P(\zbar,1) d\zbar,$$
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where the path from~$\alp$ to~$\beta$ is,
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except for the endpoints, contained in~$\h$.
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The pairing is perfect when restricted to $\sS_k(N,\eps;\C)$.
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\end{theorem}
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\begin{proof}
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Take the~$\eps$ part of each side of~\cite[Thm.~3]{merel:1585}.
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\end{proof}
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\section{Linear operators}
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\label{sec:heckeops}
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\subsection{Hecke operators}\label{heckeops:modsym}
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The Hecke operators~$T_n$, for $n\geq 1$ act on
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modular symbols and modular forms. The Hecke algebra~$\T$
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is the subring of the ring of endomorphism of modular
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symbols or modular forms generated by all~$T_n$.
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For the definition of Hecke operators on modular symbols,
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see~\cite[\S2]{merel:1585}. For example, when $n=p$ is prime,
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$$T_p(x) = \left[ \mtwo{p}{0}{0}{1} + \sum_{r \md p}
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\mtwo{1}{r}{0}{p}\right] x,$$
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where the first matrix is omitted if $p\mid N$.
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\begin{proposition}\label{prop:modsympairing}
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The pairing of Theorem~\ref{thm:perfectpairing} respects the
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action of the Hecke operators, in the sense that
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$\langle f T, x \rangle = \langle f , T x \rangle$
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for all $T\in \T$, $x\in\sM_k(\Gamma)$,
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and $f\in S_k(\Gamma)\oplus \Sbar_k(\Gamma)$.
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\end{proposition}
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\subsection{The $*$-involution}\label{sec:starinvolution}
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The matrix $j=\abcd{-1}{0}{\hfill0}{1}$ defines
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an involution~$*$ of $\sM_k(N,\eps)$ given by
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$x\mapsto x^*=j(x)$. Explicitly,
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\begin{equation*}
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(P(X,Y)\{\alp,\beta\})^* = P(X,-Y)\{-\alp,-\beta\}.
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\end{equation*}
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We are extremely fortunate to have the following proposition,
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because modular symbols correspond to both holomorphic and
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non-holomorphic cusp forms.
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\begin{proposition}
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The $*$-involution is well defined.
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\end{proposition}
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\begin{proof}
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$\sM_k(N,\eps)$ is the torsion-free quotient of the
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free $\Z[\eps]$-module generated by symbols
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$x=P\{\alp,\beta\}$ by the submodule generated by
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relations $\gamma x - \eps(\gamma)x$ for
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all $\gamma\in \Gamma_0(N)$.
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In order to check that~$*$ is well defined, it
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suffices to check that $*(\gamma x - \eps(\gamma)x)$ is of
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the form $\gamma' y - \eps(\gamma') y$.
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Note that if $\gamma=\abcd{a}{b}{c}{d}\in \Gamma_0(N)$, then
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$j\gamma j^{-1} = \abcd{\hfill a}{-b}{-c}{\hfill d}$ is also in $\Gamma_0(N)$
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and $\eps(j\gamma j^{-1}) = \eps(\gamma)$. We have
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\begin{align*}
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j(\gamma x - \eps(\gamma) x) &=
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j \gamma x - j \eps(\gamma) x \\
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&= j \gamma j^{-1} j x - \eps(\gamma) j x\\
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&= (j\gamma j^{-1}) (j x) - \eps(j \gamma j^{-1}) (jx).
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\end{align*}
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\end{proof}
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If~$f$ is a modular form, let $f^*$ be the holomorphic
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function $\overline{f(-\overline{z})}$, where the bar
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denotes complex conjugation.
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The Fourier coefficients
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of $f^*$ are the complex conjugates of those of~$f$; though $f^*$
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is again a holomorphic modular form, it's character
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is $\overline{\eps}$ instead of~$\eps$.
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The pairing of Theorem~\ref{thm:perfectpairing}
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is the restriction of a pairing on the full spaces without
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character, and we have the following proposition.
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\begin{proposition}\label{prop:starpairing}
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We have
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\begin{equation*}
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\langle f^*, x^* \rangle = \overline{\langle f, x\rangle}.
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\end{equation*}
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\end{proposition}
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\begin{definition}[Plus one quotient]
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The \defn{plus one quotient} $\sM_k(N,\eps)_+$ is the torsion-free
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quotient of $\sM_k(N,\eps)$ by the relations
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$x^*-x=0$ for all $x\in \sM_k(N,\eps)$.
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The \defn{minus one quotient} is defined similarly.
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\end{definition}
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\begin{warning} The choice of~$*$ above
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agrees with~\cite[\S2.1.3]{cremona:algs},
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but not with~\cite[\S1.6]{merel:1585}.
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\end{warning}
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\subsection{The Atkin-Lehner involutions}\label{sec:atkin-lehner}
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In this section we assume
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that~$k$ is even and $\eps^2=1$.
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To each divisor~$d$ of $N$ such that $(d,N/d)=1$
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there is an \defn{Atkin-Lehner involution}~$W_d$ on $\sM_k(N,\eps)$,
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which is defined as follows.
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Using the Euclidean algorithm,
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choose integers $x,y,z,w$ such that $dxw - (N/d)yz = 1$;
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let $g=\abcd{dx}{y}{Nz}{dw}$ and define
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$W_d(x) \define g(x) / d^{\frac{k-2}{2}}.$
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For example, when $d=N$ we have $g=\abcd{0}{-1}{N}{\hfill 0}$.
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There is also an Atkin-Lehner involution, also denoted $W_d$,
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that acts on modular forms by $W_d(f) = f|[W_d]_k$.
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These involutions are compatible with the integration pairing:
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$\langle W_d(f), x\rangle = \langle f, W_d(x)\rangle$.
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\section{Degeneracy maps}
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\label{sec:degeneracymaps}
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\label{pg:degeneracymaps}
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In this section, we describe natural maps between spaces of
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modular symbols of different levels. These are useful in
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investigating level lowering and raising questions,
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and in computing kernels of certain natural maps between
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Jacobians of modular curves.
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Fix a positive integer~$N$ and a Dirichlet character
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$\eps : (\Z/N\Z)^*\ra \C^*$.
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Let~$M$ be a positive divisor of~$N$ that
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is divisible by the conductor of~$\eps$, in the sense that~$\eps$
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factors through $(\Z/M\Z)^*$ via the natural
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map $(\Z/N\Z)^*\ra (\Z/M\Z)^*$ composed with
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some uniquely defined character $\eps':(\Z/M\Z)^*\ra\C^*$.
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For any positive divisor~$t$ of $N/M$, let
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$T=\abcd{1}{0}{0}{t}$ and
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fix a choice
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$D_t=\{T\gamma_i : i=1,\ldots, n\}$
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of coset representatives for $\Gamma_0(N)\backslash T\Gamma_0(M)$.
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{\bf Warning:} There is a mistake in \cite[\S2.6]{merel:1585}:
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The quotient ``$\Gamma_1(N)\backslash\Gamma_1(M)T$'' should be replaced
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by ``$\Gamma_1(N)\backslash T\Gamma_1(M)$''.
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\begin{proposition}
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For each divisor $t$ of $N/M$ there are well-defined linear maps
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\begin{align*}
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\alp_t : \sM_k(N,\eps) \ra \sM_k(M,\eps'),&\qquad
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\alp_t(x) = tT^{-1}x = \mtwo{t}{0}{0}{1} x\\
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\beta_t : \sM_k(M,\eps') \ra \sM_k(N,\eps),&\qquad
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\beta_t(x) = \sum_{T\gam_i\in D_t} \eps'(\gam_i)^{-1}T\gam_i{} x.
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\end{align*}
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Furthermore
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$\alp_t\circ \beta_t$ is multiplication
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by $t\cdot [\Gamma_0(M) : \Gamma_0(N)].$
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\end{proposition}
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\begin{proof}
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To show that $\alp_t$ is well defined, we must show that for
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$x\in\sM_k(N,\eps)$ and $\gam=\abcdmat\in\Gamma_0(N)$, that
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$\alp_t(\gamma x -\eps(\gamma) x)=0\in\sM_k(M,\eps')$.
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We have
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$$\alp_t(\gam x) = \mtwo{t}{0}{0}{1}\gam x
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= \mtwo{a}{tb}{c/t}{d}\mtwo{t}{0}{0}{1} x
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= \eps'(a)\mtwo{t}{0}{0}{1} x,$$
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so
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$\alp_t(\gamma x -\eps(\gamma) x)
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= \eps'(a)\alp_t(x) - \eps(\gamma)\alp_t(x) = 0$.
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We next verify that~$\beta_t$ is well defined.
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Suppose that $x\in\sM_k(M,\eps')$ and $\gamma\in\Gamma_0(M)$;
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then $\eps'(\gam)^{-1}\gam x = x$, so
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\begin{align*}
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\beta_t(x)
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&= \sum_{T\gam_i\in D_t}
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\eps'(\gam_i)^{-1}T\gam_i{}\eps'(\gam)^{-1}\gam{} x\\
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&= \sum_{T\gam_i\gam\in D_t}
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\eps'(\gam_i\gam)^{-1}T\gam_i{}\gam{} x.
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\end{align*}
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This computation shows that the definition of~$\beta_t$
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does not depend on the choice~$D_t$ of coset representatives.
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To show that~$\beta_t$ is well defined
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we must show that, for $\gam\in\Gamma_0(M)$, we have
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$\beta_t(\gam x) = \eps'(\gam)\beta_t(x)$ so that $\beta_t$
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respects the relations that define $\sM_k(M,\eps)$.
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Using that~$\beta_t$ does not depend on the choice of
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coset representative, we find that for $\gamma\in\Gamma_0(M)$,
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\begin{align*}
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\beta_t(\gam x)
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&= \sum_{T\gam_i\in D_t} \eps'(\gam_i)^{-1}T\gam_i{} \gam{} x\\
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&= \sum_{T\gam_i\gam^{-1}\in D_t}
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\eps'(\gam_i\gam^{-1})^{-1}T\gam_i{}\gam{}^{-1} \gam{} x\\
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&= \eps'(\gam)\beta_t(x).\\
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\end{align*}
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To compute $\alp_t\circ\beta_t$, we use
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that $\#D_t = [\Gamma_0(N) : \Gamma_0(M)]$:
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\begin{align*}
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\alp_t(\beta_t(x)) &=
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\alp_t \left(\sum_{T\gamma_i}
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\eps'(\gam_i)^{-1}T\gam_i x\right)\\
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&= \sum_{T\gamma_i}
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\eps'(\gam_i)^{-1}(tT^{-1})T\gam_i x\\
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&= t\sum_{T\gamma_i}
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\eps'(\gam_i)^{-1}\gam_i x\\
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&= t\sum_{T\gamma_i} x \\
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&= t \cdot [\Gamma_0(N) : \Gamma_0(M)] \cdot x.
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\end{align*}
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\end{proof}
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\begin{definition}[New and old modular symbols]\label{def:newandoldsymbols}
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The subspace $\sM_k(N,\eps)^{\new}$
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of \defn{new modular symbols} is the
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intersection of the kernels of the $\alp_t$ as~$t$
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runs through all positive divisors of $N/M$ and~$M$
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runs through positive divisors of~$M$ strictly less than~$N$
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and divisible by the conductor of~$\eps$.
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The subspace $\sM_k(N,\eps)^{\old}$
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of \defn{old modular symbols} is the
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subspace generated by the images of the $\beta_t$
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where~$t$ runs through all positive divisors of $N/M$ and~$M$
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runs through positive divisors of~$M$ strictly less than~$N$.
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\end{definition}
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\begin{remark}
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The map $\beta_t\circ\alp_t$ can not be multiplication by
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a scalar since $\sM_k(M,\eps')$
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usually has smaller dimension than $\sM_k(N,\eps)$.
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\end{remark}
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\begin{remark}
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The proposition implies that $\beta_t$ is injective.
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For each prime~$p$ there is also a map
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$\beta_{t,p}:\sM_k(M,\eps';\F_p[\eps]) \ra
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\sM_k(N,\eps;\F_p[\eps])$.
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When~$p$ does not divide $t\cdot [\Gamma_0(M) : \Gamma_0(N)]$,
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the proposition shows that $\beta_{t,p}$ is injective.
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However, $\beta_t\tensor\F_p$ need not be injective for all~$p$.
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For example, suppose $M=14$, $N=28$, and $\eps=1$. Then there
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are basis with respect to which the matrix of $\beta_1$ is
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$$\left(
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\begin{matrix}
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1&0&0&1&0&0&0&0&0\\
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0&1&0&0&1&0&0&0&0\\
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0&0&1&0&0&1&0&0&0\\
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0&0&0&0&0&0&2&1&-1\\
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0&0&0&0&0&0&0&1&1
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\end{matrix}
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\right).$$
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The row vector $(0,0,0,1,1)$ is in the kernel of the mod~$2$
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reduction of this matrix.
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\end{remark}
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\begin{algorithm}\label{alg:degenreps}
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Let~$M$ be a positive divisor of~$N$ and~$t$
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a divisor of~$N/M$. This algorithm
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computes a set~$D_t$ of representatives for
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the orbit space
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$\Gamma_0(M)\backslash T\Gamma_0(N).$
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Let $\Gamma_0(N/t,t)$ denote the subgroup of $\SL_2(\Z)$
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consisting of matrices that are upper triangular modulo $N/t$ and lower
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triangular modulo~$t$. Observe that two right cosets
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of $\Gamma_0(N/t,t)$ in $\SL_2(\Z)$, represented by
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$\abcd{a}{b}{c}{d}$ and $\abcd{a'}{b'}{c'}{d'}$,
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are equivalent if and only if
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$(a,b)=(a',b')$ as points of $\P^1(t)$, i.e.,
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$ab'\con ba'\pmod{t}$,
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and $(c,d)=(c',d')$ as points of $\P^1(N/t)$.
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Using the following algorithm, we compute right coset
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representatives for $\Gamma_0(N/t,t)$
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inside~$\Gamma_0(M)$.
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\begin{enumerate}
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\item Compute the number $[\Gamma_0(M):\Gamma_0(N)]$ of cosets.
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\item Compute a random element $x \in \Gamma_0(M)$.
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\item If~$x$ is not equivalent to anything generated so
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far, add it to the list.
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\item Repeat steps (2) and (3) until the list is as long
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as the bound of step (1).
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\end{enumerate}
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There is a natural bijection between
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$\Gamma_0(N)\backslash T \Gamma_0(M)$
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and $\Gamma_0(N/t,t)\backslash \Gamma_0(M)$,
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under which~$T\gamma$ corresponds to~$\gamma$.
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Thus we obtain coset representatives for
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$\Gamma_0(N)\backslash T\Gamma_0(M)$
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by left multiplying each
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coset representative of $\Gamma_0(N/t,t)\backslash\Gamma_0(M)$ by~$T$.
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\end{algorithm}
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\subsection{Compatibility}
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Suppose that the characteristic of the base field is zero.
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The degeneracy maps defined above
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are compatible with the corresponding degeneracy maps
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$\tilde{\alp}_t$ and $\tilde{\beta}_t$
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on modular forms. This is because the degeneracy
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maps on modular forms are defined by summing over the
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same coset representatives $D_t$.
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Thus we have the following compatibilities.
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\begin{align*}
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\langle \tilde{\alp}_t(f), x \rangle &= \langle f, \alp_t(x)\rangle,\\
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\langle \tilde{\beta}_t(f), x\rangle &= \langle f, \beta_t(x) \rangle .
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\end{align*}
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If~$p$ is prime to~$N$, then $T_p\alp_t = \alp_t T_p$
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and $T_p\beta_t = \beta_t T_p$.
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415
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\section{Manin symbols}
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\label{sec:maninsymbols}
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From the definition given in
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Section~\ref{sec:defnofmodsyms}, it is not even clear
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that $\sM_k(N,\eps)$ is of finite rank. The Manin
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symbols provide a finite presentation of~$\sM_k(N,\eps)$
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that is much more useful from a computational point of view.
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\begin{definition}[Manin symbols]\label{defn:maninsymbols}
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The \defn{Manin symbols} are the set of pairs
426
$$[P(X,Y),(u,v)]$$
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where $P(X,Y)\in V_{k-2}$ and
428
$0\leq u,v < N$ with $\gcd(u,v,N)=1$.
429
\end{definition}
430
Define a {\em right} action of $\GL_2(\Q)$ on
431
the free $\Z[\eps]$-module~$M$ generated by the Manin
432
symbols as follows. The element $g=\abcd{a}{b}{c}{d}$ acts by
433
\begin{equation*}
434
[P,(u,v)]g=[g^{-1}P(X,Y),(u,v) g]
435
= [P(aX+bY,cX+dY),(au+cv,bu+dv)].
436
\end{equation*}
437
Let $\sigma=\abcd{0}{-1}{1}{\hfill 0}$ and $\tau=\abcd{0}{-1}{1}{-1}$\label{defn:sigmatau}.
438
Let $\sM_k(N,\eps)'$ be the torsion-free quotient
439
of~$M$ by
440
\begin{align*}
441
\mbox{}x + x\sigma &= 0,\\
442
\mbox{}x + x\tau+ x\tau^2 &= 0,\\
443
\eps(\lambda) [P,(u,v)]- [P,(\lambda u, \lambda v)] &=0.
444
\end{align*}
445
446
\begin{theorem}\label{thm:maninsymbols}
447
There is a natural isomorphism
448
$\vphi:\sM_k(N,\eps)'\lra\sM_k(N,\eps)$ given by
449
$$[X^iY^{2-k-i},(u,v)] \mapsto g(X^iY^{k-2-i}\{ 0,\infty\})$$
450
where $g=\abcd{a}{b}{c}{d}\in\SL_2(\Z)$ is some matrix
451
such that $(u,v)\con (c,d) \pmod{N}$.
452
\end{theorem}
453
\begin{proof}
454
In~\cite[\S1.2, \S1.7]{merel:1585} it is proved that $\vphi\tensor\C$ is
455
an isomorphism, so~$\vphi$ is injective and well defined.
456
The discussion in Section~\ref{sec:modmanconv} below shows that
457
every element in $\sM_k(N,\eps)$ is a $\Z$-linear combination
458
of elements in the image, so~$\vphi$ is surjective as well.
459
\end{proof}
460
461
\subsection{Conversion between modular and Manin symbols}
462
\label{sec:modmanconv}
463
For some purposes it is better to work with modular symbols,
464
and for others it is better to work with Manin symbols.
465
For example, there are descriptions of the Atkin-Lehner involution
466
in terms of both Manin and modular symbols; in practice it is
467
more efficient to compute this involution using modular symbols.
468
It is thus essential to convert between these two representations.
469
The conversion from Manin to modular symbols
470
is straightforward, and follows immediately from
471
Theorem~\ref{thm:maninsymbols}.
472
The conversion back is accomplished using the
473
method used to prove Theorem~\ref{thm:maninsymbols};
474
it is known as ``Manin's trick'',
475
and involves continued fractions.
476
477
Given a Manin symbol $[X^iY^{k-2-i},(u,v)]$,
478
we write down a corresponding modular symbol as follows.
479
Choose $\abcd{a}{b}{c}{d}\in\SL_2(\Z)$ such that
480
$(c,d)\con (u,v)\pmod{N}$. This is possible
481
by Lemma~1.38 of~\cite[pg.~20]{shimura:intro}; in practice
482
it is not completely trivial, but can be accomplished
483
using the extended Euclidean algorithm.
484
Then
485
\begin{eqnarray*}
486
[X^iY^{k-2-i},(u,v)] &\corrto&
487
\abcd{a}{b}{c}{d}(X^iY^{k-2-i}\{ 0,\infty\})\\
488
&&= (dX-bY)^i(-cX+aY)^{2-k-i}
489
\left\{\frac{b}{d},\,\frac{a}{c}\right\}.\\
490
\end{eqnarray*}
491
492
In the other direction, suppose that we are given a modular
493
symbol $P(X,Y)\{\alp,\beta\}$ and wish to represent it as a
494
sum of Manin symbols.
495
Because
496
$P\{a/b,c/d\} = P\{a/b,0\}+P\{0,c/d\}$
497
it suffices to write $P\{0,a/b\}$ in
498
terms of Manin symbols.
499
Let
500
$$0=\frac{p_{-2}}{q_{-2}} = \frac{0}{1},\,\,
501
\frac{p_{-1}}{q_{-1}}=\frac{1}{0},\,\,
502
\frac{p_0}{1}=\frac{p_0}{q_0},\,\,
503
\frac{p_1}{q_1},\,\,
504
\frac{p_2}{q_2},\,\ldots,\,\frac{p_r}{q_r}=\frac{a}{b}$$
505
denote the continued fraction convergents of the
506
rational number $a/b$.
507
Then
508
$$p_j q_{j-1}
509
- p_{j-1} q_j = (-1)^{j-1}\qquad \text{for }-1\leq j\leq r.$$
510
If we let
511
$g_j = \mtwo{(-1)^{j-1}p_j}{p_{j-1}}{(-1)^{j-1}q_j}{q_{j-1}}$,
512
then $g_j\in\sltwoz$ and
513
\begin{align*}
514
P(X,Y)\{0,a/b\}
515
&=P(X,Y)\sum_{j=-1}^{r}\left\{\frac{p_{j-1}}{q_{j-1}},\frac{p_j}{q_j}\right\}\\
516
&=\sum_{j=-1}^{r} g_j((g_j^{-1}P(X,Y))\{0,\infty\})\\
517
&=\sum_{j=-1}^{r} [g_j^{-1}P(X,Y),((-1)^{j-1}q_j,q_{j-1})].
518
\end{align*}
519
Note that in the $j$th summand, $g_j^{-1}P(X,Y)$, replaces $P(X,Y)$.
520
Since $g_j\in\sltwoz$ and $P(X,Y)$ has integer coefficients,
521
the polynomial $g_j^{-1}P(X,Y)$ also has integer coefficients,
522
so no denominators are introduced.
523
524
The continued fraction expansion $[c_1,c_2,\ldots,c_n]$
525
of the rational number $a/b$ can be computed
526
using the Euclidean algorithm.
527
The first term, $c_1$, is the ``quotient'': $a = bc_1+r$,
528
with $0\leq r < b$.
529
Let $a'=b$, $b'=r$ and compute $c_2$ as
530
$a'=b'c_2+r'$, etc., terminating when the
531
remainder is $0$. For example, the expansion
532
of $5/13$ is $[0,2,1,1,2]$.
533
The numbers $$d_i=c_1+\frac{1}{c_2+\frac{1}{c_3+\cdots}}$$
534
will then be the (finite) convergents.
535
For example if $a/b=5/13$, then the convergents are
536
$$0/1,\,\, 1/0,\,\, d_1=0,\,\, d_2=\frac{1}{2},\,\, d_3=\frac{1}{3},\,\,
537
d_4=\frac{2}{5},\,\, d_5=\frac{5}{13}.$$
538
539
540
541
\subsection{Hecke operators on Manin symbols}
542
\label{subsec:heckeonmanin}
543
Thoerem~2 of \cite{merel:1585} gives a description of
544
the Hecke operators~$T_n$ directly on the space of Manin symbols.
545
This avoids the expense of first converting a Manin
546
symbol to a modular symbol, computing~$T_n$ on the modular symbol,
547
and then converting back. For the reader's convenience, we very
548
briefly recall Merel's theorem here, along with an enhancement
549
of Cremona.
550
551
As in~\cite[\S2.4]{cremona:algs}, define~$R_p$ as follows.
552
When $p=2$,
553
$$R_2 := \left\{\mtwo{1}{0}{0}{2},
554
\mtwo{2}{0}{0}{1}, \mtwo{2}{1}{0}{1},
555
\mtwo{1}{0}{1}{2}\right\}.$$
556
When~$p$ is odd,~$R_p$ is the set of $2\times 2$ integer
557
matrices $\abcd{a}{b}{c}{d}$ with determinant~$p$, and either
558
\begin{enumerate}
559
\item $a>|b|>0$, $d>|c|>0$, and $bc<0$; or
560
\item $b=0$, and $|c|<d/2$; or
561
\item $c=0$, and $|b|<a/2$.
562
\end{enumerate}
563
\begin{proposition}
564
For $[P(X,Y),(u,v)]\in\sM_k(N,\eps)$ and~$p$ a prime, we have
565
\begin{align*}T_p([P(X,Y),(u,v)])
566
&= \sum_{g\in R_p} [P(X,Y),(u,v)].g \\
567
&= \sum_{\abcd{a}{b}{c}{d}\in R_p} [P(aX+bY,cX+dY),(au+cv,bu+dv)]
568
\end{align*}
569
where the sum is restricted to matrices $\abcd{a}{b}{c}{d}$
570
such that $\gcd(au+cv,bu+dv,N)=1$.
571
\end{proposition}
572
\begin{proof}
573
For the case $k=2$ and an algorithm to compute $R_p$,
574
see \cite[\S2.4]{cremona:algs}.
575
The general case follows from~\cite[Theorem 2]{merel:1585} applied
576
to the set~$\sS$ of~\cite[\S3]{merel:1585} by observing that
577
when~$p$ is an odd {\em prime} $\sS_p'$ is empty.
578
\end{proof}
579
580
\subsection{The cuspidal and boundary spaces in terms of Manin symbols}
581
This section is a review of Merel's explicit description
582
of the boundary map in terms of Manin symbols for $\Gamma=\Gamma_1(N)$
583
(see~\cite[\S1.4]{merel:1585}). In the next section, we
584
describe a very efficient way to compute the boundary map.
585
586
Let~$\cR$ be the equivalence relation
587
on $\Gamma\backslash\Q^2$ which identifies
588
the element
589
$[\Gamma\smallvtwo{\lambda u}{\lambda v}]$
590
with $\sign(\lambda)^k[\Gamma\smallvtwo{u}{v}]$,
591
for any $\lambda\in\Q^*$. Denote by $B_k(\Gamma)$
592
the finite dimensional $\Q$-vector space
593
with basis the equivalence classes
594
$(\Gamma\backslash\Q^2)/\cR$.
595
The dimension of this space is $\#(\Gamma\backslash\P^1(\Q))$.
596
\begin{proposition}
597
The map
598
$$\mu:\sB_k(\Gamma)\ra B_k(\Gamma),
599
\qquad P\left\{\frac{u}{v}\right\}\mapsto
600
P(u,v)\left[\Gamma\vtwo{u}{v}\right]$$
601
is well defined and injective.
602
Here $u$ and $v$ are assumed coprime.
603
\end{proposition}
604
Thus the kernel of $\delta:\sS_k(\Gamma)\ra \sB_k(\Gamma)$
605
is the same as the kernel of $\mu\circ \delta$.
606
\begin{proposition}\label{prop:boundary}
607
Let $P\in V_{k-2}$ and $g=\abcd{a}{b}{c}{d}\in\sltwoz$. We have
608
$$\mu\circ\delta([P,(c,d)])
609
= P(1,0)[\Gamma\smallvtwo{a}{c}]
610
-P(0,1)[\Gamma\smallvtwo{b}{d}].$$
611
\end{proposition}
612
613
614
\subsection{Computing the boundary map}
615
\label{sec:computeboundary}
616
In this section we describe how to efficiently compute the
617
map $\sM_k(N,\eps)\ra B_k(N,\eps)$
618
given in the previous section. Henceforth, we will denote
619
this map by~$\delta$.
620
Our algorithm generalizes~\cite[\S2.2]{cremona:algs}.
621
To compute the image of $[P,(c,d)]$, with
622
$g=\abcd{a}{b}{c}{d}\in\sltwoz$,
623
we must compute the class of $[\smallvtwo{a}{c}]$ and of
624
$[\smallvtwo{b}{d}]$.
625
Instead of finding a canonical form for cusps, we
626
use a quick test for equivalence modulo scalars.
627
In the following algorithm, by the $i$th standard cusp we mean
628
the $i$th basis vector for a basis of $B_k(N,\eps)$. The
629
basis is constructed as the algorithm is called successively.
630
We present the algorithm first, and then prove the results
631
used by the algorithm in testing equivalence.
632
633
\begin{algorithm}\label{alg:cusplist}
634
Given a cusp $[\smallvtwo{u}{v}]$ this algorithm computes
635
an integer~$i$ and a scalar~$\alp$ such that
636
$[\smallvtwo{u}{v}]$ is equivalent to~$\alp$ times
637
the $i$th standard cusp.
638
First, using Proposition~\ref{prop:cusp1} and
639
Algorithm~\ref{alg:cusp1}, check
640
whether or not $[\smallvtwo{u}{v}]$ is equivalent,
641
modulo scalars, to any cusp found so far. If so,
642
return the index of the representative and the scalar.
643
If not, record $\smallvtwo{u}{v}$ in the representative list.
644
Then, using Proposition~\ref{prop:cuspdies},
645
check whether or not $\smallvtwo{u}{v}$
646
is forced to equal zero by the relations.
647
If it does not equal zero, return its position
648
in the list and the scalar~$1$.
649
If it equals zero, return the scalar~$0$ and the position~$1$;
650
keep $\smallvtwo{u}{v}$ in the list, and record that it is zero.
651
\end{algorithm}
652
653
In the case considered in \cite{cremona:algs}, the relations
654
between cusps involve only the trivial character, so they
655
do not force any cusp classes to vanish. He gives the
656
following two criteria for equivalence.
657
\begin{proposition}[Cremona]\label{prop:cusp1}
658
Let $\smallvtwo{u_i}{v_i}$, $i=1,2$ be written so that
659
$\gcd(u_i,v_i)=1$.
660
\begin{enumerate}
661
\item There exists $g\in\Gamma_0(N)$ such that
662
$g\smallvtwo{u_1}{v_1}=\smallvtwo{u_2}{v_2}$ if and only if
663
$$s_1 v_2 \con s_2 v_1 \pmod{\gcd(v_1 v_2,N)},\,
664
\text{ where $s_j$ satisfies $u_j s_j\con 1\pmod{v_j}$}.$$
665
\item There exists $g\in\Gamma_1(N)$ such that
666
$g\smallvtwo{u_1}{v_1}=\smallvtwo{u_2}{v_2}$ if and only if
667
$$v_2 \con v_1 \pmod{N}\text{ and } u_2 \con u_1 \pmod{\gcd(v_1,N)}.$$
668
\end{enumerate}
669
\end{proposition}
670
\begin{proof}
671
The first is Proposition 2.2.3 of \cite{cremona:algs}, and
672
the second is Lemma 3.2 of \cite{cremona:gammaone}.
673
\end{proof}
674
675
\begin{algorithm}\label{alg:cusp1}
676
Suppose $\smallvtwo{u_1}{v_1}$ and
677
$\smallvtwo{u_2}{v_2}$
678
are equivalent modulo $\Gamma_0(N)$.
679
This algorithm computes a matrix $g\in\Gamma_0(N)$ such
680
that $g\smallvtwo{u_1}{v_1}=\smallvtwo{u_2}{v_2}$.
681
Let $s_1, s_2, r_1, r_2$ be solutions to
682
$s_1 u_1 -r_1 v_1 =1$ and
683
$s_2 u_2 -r_2 v_2 =1$.
684
Find integers $x_0$ and $y_0$ such
685
that $x_0v_1v_2+y_0N=1$.
686
Let $x=-x_0(s_1v_2-s_2v_1)/(v_1v_2,N)$
687
and $s_1' = s_1 + xv_1$.
688
Then $g=\mtwo{u_2}{r_2}{v_2}{s_2}
689
\cdot \mtwo{u_1}{r_1}{v_1}{s_1'}^{-1}$
690
sends $\smallvtwo{u_1}{v_1}$ to $\smallvtwo{u_2}{v_2}$.
691
\end{algorithm}
692
\begin{proof}
693
This follows from the proof of Proposition~\ref{prop:cusp1} in
694
\cite{cremona:algs}.
695
\end{proof}
696
697
698
To see how the~$\eps$ relations, for nontrivial~$\eps$,
699
make the situation more complicated, observe that it is
700
possible that $\eps(\alp)\neq \eps(\beta)$ but
701
$$\eps(\alp)\left[\vtwo{u}{v}\right] =\left[\gamma_\alp \vtwo{u}{v}\right]=
702
\left[\gamma_\beta \vtwo{u}{v}\right]=\eps(\beta)\left[\vtwo{u}{v}\right];$$
703
One way out of this difficulty is to construct
704
the cusp classes for $\Gamma_1(N)$, then quotient
705
out by the additional~$\eps$ relations using
706
Gauss elimination. This is far too
707
inefficient to be useful in practice because the number of
708
$\Gamma_1(N)$ cusp classes is huge. Instead,
709
we give a quick test to determine whether or not
710
a cusp vanishes modulo the $\eps$-relations.
711
712
\begin{lemma}\label{lem:canlift}
713
Suppose $\alp$ and $\alp'$ are integers
714
such that $\gcd(\alp,\alp',N)=1$.
715
Then there exist integers $\beta$ and $\beta'$,
716
congruent to $\alp$ and $\alp'$ modulo $N$, respectively,
717
such that $\gcd(\beta,\beta')=1$.
718
\end{lemma}
719
\begin{proof}
720
By \cite[1.38]{shimura:intro} the map
721
$\SL_2(\Z)\ra\SL_2(\Z/N\Z)$ is surjective.
722
By the Euclidean algorithm, there exists
723
integers $x$, $y$ and $z$ such that
724
$x\alp + y\alp' + zN = 1$.
725
Consider the matrix
726
$\abcd{y}{-x}{\alp}{\hfill\alp'}\in \SL_2(\Z/N\Z)$
727
and take $\beta$, $\beta'$ to be the bottom
728
row of a lift of this matrix to $\SL_2(\Z)$.
729
\end{proof}
730
731
\begin{proposition}\label{prop:cuspdies}
732
Let $N$ be a positive integer and $\eps$ a Dirichlet
733
character modulo $N$.
734
Suppose $\smallvtwo{u}{v}$ is a cusp with $u$ and $v$ coprime.
735
Then $\smallvtwo{u}{v}$ vanishes modulo the relations
736
$$\left[\gamma\smallvtwo{u}{v}\right]=
737
\eps(\gamma)\left[\smallvtwo{u}{v}\right],\qquad
738
\text{all $\gamma\in\Gamma_0(N)$}$$
739
if and only if there exists $\alp\in(\Z/N\Z)^*$,
740
with $\eps(\alp)\neq 1$, such that
741
\begin{align*}
742
v &\con \alp v \pmod{N},\\
743
u &\con \alp u \pmod{\gcd(v,N)}.
744
\end{align*}
745
\end{proposition}
746
\begin{proof}
747
First suppose such an $\alp$ exists.
748
By Lemma~\ref{lem:canlift}
749
there exists $\beta, \beta'\in\Z$ lifting
750
$\alp,\alp^{-1}$ such that $\gcd(\beta,\beta')=1$.
751
The cusp $\smallvtwo{\beta u}{\beta' v}$
752
has coprime coordinates so,
753
by Proposition~\ref{prop:cusp1} and our
754
congruence conditions on $\alp$, the cusps
755
$\smallvtwo{\beta{}u}{\beta'{}v}$
756
and $\smallvtwo{u}{v}$ are equivalent by
757
an element of $\Gamma_1(N)$.
758
This implies that $\left[\smallvtwo{\beta{}u}{\beta'{}v}\right]
759
=\left[\smallvtwo{u}{v}\right]$.
760
Since $\left[\smallvtwo{\beta{}u}{\beta'{}v}\right]
761
= \eps(\alp)\left[\smallvtwo{u}{v}\right]$,
762
our assumption that $\eps(\alp)\neq 1$
763
forces $\left[\smallvtwo{u}{v}\right]=0$.
764
765
Conversely, suppose $\left[\smallvtwo{u}{v}\right]=0$.
766
Because all relations are two-term relations, and the
767
$\Gamma_1(N)$-relations identify $\Gamma_1(N)$-orbits,
768
there must exists $\alp$ and $\beta$ with
769
$$\left[\gamma_\alp \vtwo{u}{v}\right]
770
=\left[\gamma_\beta \vtwo{u}{v}\right]
771
\qquad\text{ and }\eps(\alp)\ne \eps(\beta).$$
772
Indeed, if this did not occur,
773
then we could mod out by the $\eps$ relations by writing
774
each $\left[\gamma_\alp \smallvtwo{u}{v} \right]$
775
in terms of $\left[\smallvtwo{u}{v}\right]$, and there would
776
be no further relations left to kill
777
$\left[\smallvtwo{u}{v}\right]$.
778
Next observe that
779
$$
780
\left[\gamma_{\beta^{-1}\alp}
781
\vtwo{u}{v}\right]
782
= \left[\gamma_{\beta^{-1}}\gamma_\alp
783
\vtwo{u}{v}\right]
784
= \eps(\beta^{-1})\left[\gamma_\alp
785
\vtwo{u}{v}\right]
786
= \eps(\beta^{-1})\left[\gamma_\beta
787
\vtwo{u}{v}\right]
788
= \left[\vtwo{u}{v}\right].$$
789
Applying Proposition~\ref{prop:cusp1} and
790
noting that $\eps(\beta^{-1}\alp)\neq 1$ shows
791
that $\beta^{-1}\alp$ satisfies the properties
792
of the ``$\alp$'' in the statement of the
793
proposition we are proving.
794
\end{proof}
795
796
The possible $\alp$ in Proposition~\ref{prop:cuspdies}
797
can be enumerated as follows. Let $g=(v,N)$ and list the
798
$\alp=v\cdot\frac{N}{g}\cdot{}a+1$, for $a=0,\ldots,g-1$,
799
such that $\eps(\alp)\neq 0$.
800
801
802
803
{\vspace{3ex}\em\par\noindent Working in the plus or minus quotient. }
804
Let~$s$ be a sign, either~$+1$ or~$-1$.
805
To compute $\sS_k(N,\eps)_s$ it is necessary
806
to replace $B_k(N,\eps)$ by its quotient modulo the
807
additional relations
808
$\left[ \smallvtwo{-u}{\hfill v}\right]
809
= s \left[\smallvtwo{u}{v}\right]$
810
for all cusps $\smallvtwo{u}{v}$.
811
Algorithm~\ref{alg:cusplist} can be modified to deal
812
with this situation as follows.
813
Given a cusp $x=\smallvtwo{u}{v}$, proceed as
814
in Algorithm~\ref{alg:cusplist} and check if
815
either $\smallvtwo{u}{v}$ or $\smallvtwo{-u}{\hfill v}$
816
is equivalent (modulo scalars) to any cusp seen so far. If not,
817
use the following trick to determine whether
818
the $\eps$ and $s$-relations
819
kill the class of $\smallvtwo{u}{v}$:
820
use the unmodified Algorithm~\ref{alg:cusplist}
821
to compute the scalars $\alp_1, \alp_2$ and
822
standard indices $i_1$, $i_2$ associated to
823
$\smallvtwo{u}{v}$ and $\smallvtwo{-u}{\hfill v}$, respectively.
824
The $s$-relation kills the class of $\smallvtwo{u}{v}$
825
if and only if $i_1=i_2$ but $\alp_1\neq s\alp_2$.
826
827
828
\section{The complex torus attached to a modular form}
829
\label{sec:tori}
830
Fix integers $N\geq 1$, $k\geq 2$, and let~$\eps$ be a mod~$N$
831
Dirichilet character. {\bf For the rest of this section assume that
832
$\eps^2=1$.} We construct a Hecke and complex
833
conjugation invariant lattice in~$S$, hence a complex torus
834
$J_k(N,\eps)$ equipped with an action of Hecke operators and an
835
$\R$-structure. The reader may wish to compare our construction to
836
the one given by Shimura in~\cite{shimura:surles}, where in addition
837
he observes that the Petterson pairing gives his torus the structure
838
of an abelian variety. When $k=2$, the torus comes from an abelian
839
variety defined over~$\Q$; when $k>2$, the study of these complex tori
840
is of interest in trying to understand the conjectures of Bloch and
841
Kato (see \cite{bloch-kato}) on motifs attached to modular forms.
842
843
Let $\sS=\sS_k(N,\eps)$ (resp., $S=S_k(N,\eps)$)
844
be the associated space of cuspidal modular symbols (resp., forms).
845
The Hecke algebra~$\T$ acts in a way compatible with the
846
integration pairing
847
$\langle\quad,\quad\rangle
848
: S \cross \sS \ra \C,$
849
which we view as a $\T$-module homomorphism $\Phi:\sS\ra S^*=\Hom(S,\C)$
850
called the \defn{period mapping}.
851
Because $\eps^2=1$, the $*$-involution preserves~$S$.
852
\begin{proposition}
853
The period mapping~$\Phi$ is injective and the
854
image of~$\Phi$ is a lattice in $S^*$.
855
\end{proposition}
856
\begin{proof}
857
By Theorem~\ref{thm:perfectpairing},
858
$\sS\tensor_{\R}\C\isom
859
\Hom_\C(S\oplus \Sbar,\C).$
860
Because $\eps^2=1$, we have $S = S_k(N,\eps;\R)\tensor_{\R}\C$.
861
Set $S_\R := S_k(N,\eps;\R)$ and likewise defined $\Sbar_\R$.
862
We have
863
$$\Hom_\C(S\oplus \Sbar,\C) =
864
\Hom_\R(S_\R \oplus \Sbar_\R,\R)\tensor_\R \C.$$
865
Let $\sS_{\R} = \sS_k(N,\eps;\R)$ and $\sS_{\R}^+$ be the
866
subspace fixed under~$*$. By Proposition~\ref{prop:starpairing}
867
we have maps
868
$$\sS_{\R}^+ \ra \Hom_{\R}(S_{\R}\oplus\Sbar_\R,\R)
869
\ra \Hom_{\R}(S_{\R},\R)$$
870
and
871
$$\sS_{\R}^- \ra \Hom_{\R}(S_{\R}\oplus\Sbar_\R,i\R)
872
\ra \Hom_{\R}(S_{\R},i\R).$$
873
The map $\sS_{\R}^+\ra \Hom_{\R}(S_{\R},\R)$ is
874
an isomorphism: the point is that if
875
$\langle \bullet, x\rangle$, for $x\in \sS_{\R}^+$,
876
vanishes on $S_\R$ then it vanishes on the
877
whole of $S\oplus \Sbar$. Likewise, the map
878
$\sS_{\R}^-\ra \Hom_{\R}(S_{\R},i\R)$
879
is an isomorphism. Thus
880
$$\sS_{\R} \isom \Hom_{\R}(S_{\R},\R)
881
\oplus \Hom_{\R}(S_{\R},i\R)
882
\isom \Hom_{\C}(S,\C).$$
883
Finally, observing that $\sS$ is by definition
884
torsion free completes the proof.
885
\end{proof}
886
887
Thus we have a torus
888
$J_k(N,\eps)$
889
that fits into an exact sequence
890
$$0\lra \sS \xrightarrow{\quad\Phi\quad}
891
\Hom(S,\C) \lra J_k(N,\eps) \lra 0.$$
892
Let $f\in S$ be a newform and $S_f$ the complex vector
893
space spanned by the Galois conjugates of~$f$.
894
The period map $\Phi_f$ associated to~$f$ is the map $\sS\ra \Hom(S_f,\C)$
895
obtaind by composing~$\Phi$ with restriction to $S_f$.
896
Set
897
$$A_f := \Hom(S_f,\C) / \Phi_f(\sS).$$
898
899
Associate\label{pg:dual} an abelian subvariety of~$J$ to~$f$ as follows.
900
Let $I_f = \Ann_{\T}(f)$ be the annihilator of $f$ in the Hecke algebra,
901
and set
902
$$\Adual_f := \Hom(S,\C)[I_f]/\Phi(\sS[I_f])$$
903
where $\Hom(S,\C)[I_f] = \intersect_{t \in I_f} \ker(t)$.
904
905
The following diagram summarizes the tori just defined;
906
its columns are exact
907
but its rows are {\em not}.
908
\begin{equation}\label{dgm:uniformization}
909
\[email protected]=.9pc{
910
0\ar[d] & 0\ar[d] & 0\ar[d] \\
911
\sS[I_f]\ar[r]\ar[dd] & \sS\ar[r]\ar[dd]&\Phi_f(\sS)\ar[dd] \\
912
& & \\
913
\Hom(S,\C)[I_f]\ar[r]\ar[dd] &\Hom(S,\C)\ar[r]\ar[dd] &\Hom(S[I_f],\C)\ar[dd]\\
914
& & \\
915
{\Adual_f}\ar[r]\ar[d]
916
& J_k(N,\eps) \ar[r]\ar[d]& A_f \ar[d]\\
917
0 & 0 & 0 \\
918
}\end{equation}
919
920
921
\subsection{Weight~$2$}
922
When $k=2$ and $\eps=1$ the above is just Shimura's classical
923
association of an abelian variety to a modular form;
924
see~\cite[Thm.~7.14]{shimura:intro} and~\cite{shimura:factors}.
925
In this case $A_f$ and $\Adual_f$ are abelian varieties
926
that are defined over~$\Q$ and are dual to each other.
927
Furthermore $A_f$ is an \defn{optimal} quotient of $J$, in the
928
sense that the kernel of $J\ra A_f$ is connected.
929
930
931
932
933
934
935
936
937
938
939
940
941