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\chapter{Modular symbols}%
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\label{chap:modsym}\index{Modular symbols}%
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Modular symbols permeate this thesis. In their simplest incarnation,
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modular symbols provide a finite presentation for the homology group
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$H_1(X_0(N),\Z)$ of the Riemann surface $X_0(N)$. This presentation
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is equipped with such a rich structure that from it we can deduce the
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action of the Hecke operators; this is already sufficient information for
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us to compute a basis for the space $S_2(\Gamma_0(N),\C)$ of cusp
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forms.
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We recall the definition of spaces of modular symbols in
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Sections~\ref{sec:defnofmodsyms}--\ref{cuspidalsymbols}. Then in
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Section~\ref{sec:duality}, we review the
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duality between modular symbols and modular forms.
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In Section~\ref{sec:heckeops}, we see that
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modular symbols are furnished with analogues of each of the standard
45
operators that one finds on spaces of modular forms, and in
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Section~\ref{sec:degeneracymaps} we see that the same is true of the
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degeneracy maps. Section~\ref{sec:maninsymbols} describes Manin
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symbols, which supply a convenient finite presentation for the space of
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modular symbols. Finally, Section~\ref{sec:tori} introduces the
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complex torus attached to a newform, which appears in various guises
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throughout this thesis.
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Before continuing, we offer an apology. We will only consider modular
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symbols that are already equipped with a fixed Dirichlet character.
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Though fixing a character complicates the formulas, the resulting increase
57
in efficiency is of extreme value in computational applications.
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Fixing a character allows us to compute in just the part of the space
59
of modular symbols for $\Gamma_1(N)$ that interests us. We apologize
60
for any inconvenience this may cause the less efficiency minded
61
reader.
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{\bf Acknowledgment.} This chapter and the next were greatly
64
influenced by the publications of Cremona~\cite{cremona:gammaone,
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cremona:algs}\index{Cremona} and Merel~\cite{merel:1585}\index{Merel},
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along with the foundational contributions of
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Manin~\cite{manin:parabolic}, Mazur~\cite{mazur:arithmetic_values,
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mazur:symboles}, and Shokurov~\cite{sokurov:modsym}. Cremona's
69
book~\cite{cremona:algs} provides a motivated roadmap that guides the
70
reader who wishes to compute with modular symbols in the familiar
71
context of elliptic curves, and Merel's\index{Merel} article provides an accessible
72
overview of the action of Hecke operators on higher weight modular
73
symbols, and the connection between modular symbols and related
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cohomology theories.
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\section{The definition of modular symbols}
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\label{sec:defnofmodsyms}
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Fix a positive integer~$N$, an integer $k\geq 2$, and a continuous
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homomorphism
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$$\eps:(\Z/N\Z)^*\ra\C^*$$
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such that $\eps(-1)=(-1)^k$.
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We call~$N$ the \defn{level}\index{Level of modular symbols|textit},~$k$ the
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\defn{weight}\index{Weight of modular symbols|textit},
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and~$\eps$ the \defn{Dirichlet character}.\index{Dirichlet character|textit}
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Consider the quotient of the abelian group generated by all formal symbols
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$\{\alp,\beta\}$, with $\alp, \beta\in\P^1(\Q)=\Q\union\{\infty\}$,
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by the following relations:
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$$\{\alp,\beta\}+\{\beta,\gamma\}+\{\gamma,\alp\} = 0,$$
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for all $\alp,\beta,\gamma\in\P^1(\Q)$.
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Let $\sM$ be the torsion-free quotient of this group by its torsion
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subgroup. Because $\sM$ is torsion free, $\{\alp,\alp\}=0$ and
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$\{\alp,\beta\} = -\{\beta,\alp\}$.
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\index{Modular symbols!relations satisfied by}
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\begin{remark}
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One is motivated to consider these relations by viewing
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$\{\alp,\beta\}$ as the homology class of an appropriate
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path from~$\alpha$ to~$\beta$ in the upper half plane.
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\end{remark}
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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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all 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 $\overline{a}\in\Z/N\Z$ is the reduction modulo~$N$ of~$a$.
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Let
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$$\Z[\eps] := \Z[\eps(a) : a \in \Z/N\Z]$$
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be the subring of~$\C$ generated by the values of the
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character~$\eps$.
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\begin{definition}[Modular symbols]\label{defn:modsym}
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\index{Modular symbols|textit}%
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The space of \defn{modular symbols} $\sM_k(N,\eps)$
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of level~$N$, weight~$k$ and character~$\eps$ is
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the largest torsion-free quotient of $\sM_k\tensor\Z[\eps]$ by the
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$\Z[\eps]$-submodule generated 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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$$\sM_k(N,\eps;R) := \sM_k(N,\eps)\tensor_{Z[\eps]} R.$$
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See Section~\ref{sec:computingmk} for an algorithm which
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can be used to compute $\sM_k(N,\eps;\Q(\eps))$.
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\section{Cuspidal modular symbols}
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\label{cuspidalsymbols}
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\index{Cuspidal modular symbols|textit}
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Let~$\sB$ be the free abelian group generated by the symbols
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$\{\alp\}$ for all $\alp\in\P^1(\Q)$.
153
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 $\sB_k(N,\eps)$ of
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\index{Boundary modular symbols|textit}%
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\defn{boundary modular symbols}
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is the largest 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 $P\tensor\{\alp\}$
167
in $\sB_k(N,\eps)$.
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The \defn{boundary map}
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$$\delta: \sM_k(N,\eps) \ra \sB_k(N,\eps)$$
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is defined by
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$$\delta(P\{\alp,\beta\}) =
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P\{\beta\}-P\{\alp\}.$$
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\begin{definition}[Cuspidal modular symbols]%
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\label{defn:cuspidalmodularsymbols}%
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\index{Cuspidal modular symbols|textit}%
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The space $\sS_k(N,\eps)$ of
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\defn{cuspidal modular symbols}
178
is the kernel of~$\delta$.
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\end{definition}
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The three spaces defined above fit together in the
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following exact sequence:
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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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\section{Duality between modular symbols and modular forms}%
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\label{sec:duality}
189
\index{Modular symbols!duality with modular forms}%
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\index{Modular forms!duality with modular symbols}%
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\index{Integration pairing}%
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For any positive integer~$k$, any $\C$-valued function~$f$ on
193
the complex upper half plane
194
$$\h:=\{z \in \C : \im(z) > 0\},$$
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and any matrix $\gamma\in\GL_2(\Q)$, define a function
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$f|[\gamma]_k$ 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]\index{Cusp forms|textit}
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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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$$f|[\gamma]_k = \eps(\gamma)f$$
203
for all $\gamma\in\Gamma_0(N)$, and such that~$f$
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is holomorphic and vanishes 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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\index{Cusp forms!antiholomorphic|textit}%
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\index{Antiholomorphic cusp forms|textit}
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Let $\Sbar_k(N,\eps)$ be the space of
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\defn{antiholomorphic cusp forms};
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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 $\gamma\in\Gamma_0(N)$.\footnote{The $\overline{\eps}$
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should be replaced by~$\eps$ in this formula, as
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in \cite[\S2.5]{merel:1585}.}
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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 antiholomorphic cusp form defined by the function
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$z\mapsto \overline{f(z)}$.
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\begin{theorem}[Merel]\label{thm:perfectpairing}\index{Merel}
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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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$$\langle f\oplus g, P\{\alp,\beta\}\rangle =
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\int_{\alp}^{\beta} f(z)P(z,1) dz
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+ \int_{\alp}^{\beta} g(z)P(\zbar,1) d\zbar,$$
235
where the path from~$\alp$ to~$\beta$ is,
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except for the endpoints, contained in~$\h$.
237
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}.
241
\end{proof}
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\section{Linear operators}
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\label{sec:heckeops}
246
\subsection{Hecke operators}\label{heckeops:modsym}
247
\index{Hecke operators}\index{Operators!Hecke}
248
For each positive integer~$n$ and each space~$V$ of modular symbols or modular
249
forms, there is a \defn{Hecke operator}~$T_n$, which acts
250
as a linear endomorphism of~$V$.
251
For the definition of $T_n$ on modular symbols,
252
see~\cite[\S2]{merel:1585}.
253
Alternatively, because we consider only modular symbols
254
with character, the following
255
recipe completely determines the Hecke operators.
256
First, when $n=p$ is prime, we have
257
$$T_p(x) = \left[ \mtwo{p}{0}{0}{1} + \sum_{r \md p}
258
\mtwo{1}{r}{0}{p}\right] x,$$
259
where the first matrix is omitted if $p\mid N$.
260
If~$m$ and~$n$ are coprime, then $T_{mn} = T_mT_n$.
261
Finally, if~$p$ is a prime, $r\geq 2$ is an integer,~$\varepsilon$ is
262
the Dirichlet character of associated to~$V$, and~$k$ is the weight
263
of~$V$, then
264
$$T_{p^r} =
265
T_p T_{p^{r-1}} - \varepsilon(p) p^{k-1} T_{p^{r-2}}.$$
266
267
\begin{definition}\index{Hecke algebra|textit}
268
The \defn{Hecke algebra associated to $V$} is the subring
269
$$\T=\T_V = \Z[\ldots T_n \ldots]$$
270
of $\End(V)$ generated by all Hecke operators $T_n$, with $n=1,2,3,\ldots$.
271
\end{definition}
272
273
\begin{proposition}\label{prop:modsympairing}
274
The pairing of Theorem~\ref{thm:perfectpairing} respects the
275
action of the Hecke operators\index{Hecke operators!respect pairing},
276
in the sense that $\langle f T, x \rangle = \langle f , T x \rangle$
277
for all $T\in \T$, $x\in\sM_k(N,\eps)$,
278
and $f\in S_k(N,\eps)\oplus \Sbar_k(N,\eps)$.
279
\end{proposition}
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\begin{proof}
281
See~\cite[Prop.~10]{merel:1585}.
282
\end{proof}
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284
\subsection{The $*$-involution}\label{sec:starinvolution}
285
\index{Star involution|textit}\index{Operators!$*$-involution|textit}
286
The matrix $j=\abcd{-1}{0}{\hfill0}{1}$ defines
287
an involution~$*$ of $\sM_k(N,\eps)$ given by
288
$x\mapsto x^*=j(x)$. Explicitly,
289
\begin{equation*}
290
(P(X,Y)\{\alp,\beta\})^* = P(X,-Y)\{-\alp,-\beta\}.
291
\end{equation*}
292
Because the space of modular symbols is constructed as a quotient,
293
it is not obvious that the $*$-involution is well defined.%
294
\index{Star involution!is well defined}
295
\begin{proposition}
296
The $*$-involution is well defined.
297
\end{proposition}
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\begin{proof}
299
Recall that $\sM_k(N,\eps)$ is the largest torsion-free quotient of the
300
free $\Z[\eps]$-module generated by symbols
301
$x=P\{\alp,\beta\}$ by the submodule generated by
302
relations $\gamma x - \eps(\gamma)x$ for
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all $\gamma\in \Gamma_0(N)$.
304
In order to check that the operator~$*$ is well defined, it
305
suffices to check, for any $x\in\sM_k$, that
306
$*(\gamma x - \eps(\gamma)x)$ is of
307
the form $\gamma' y - \eps(\gamma') y$, for some~$y$ in $\sM_k$.
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Note that if $\gamma=\abcd{a}{b}{c}{d}\in \Gamma_0(N)$, then
309
$j\gamma j^{-1} = \abcd{\hfill a}{-b}{-c}{\hfill d}$ is also in $\Gamma_0(N)$
310
and $\eps(j\gamma j^{-1}) = \eps(\gamma)$. We have
311
\begin{align*}
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j(\gamma x - \eps(\gamma) x) &=
313
j \gamma x - j \eps(\gamma) x \\
314
&= 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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319
If~$f$ is a modular form\index{Modular forms}, let $f^*$ be the holomorphic
320
function $\overline{f(-\overline{z})}$, where the bar
321
denotes complex conjugation.
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The Fourier coefficients\index{Fourier coefficients}
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of $f^*$ are the complex conjugates of those of~$f$; though $f^*$
324
is again a holomorphic modular form\index{Modular forms}, its character
325
is $\overline{\eps}$ instead of~$\eps$.
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The pairing of Theorem~\ref{thm:perfectpairing}
327
is the restriction of a pairing on the full spaces without
328
character, and we have the following proposition.
329
\index{Star involution!and integration pairing}
330
\begin{proposition}\label{prop:starpairing}\footnote{G. Weber pointed
331
out that this isn't correct. It is correct if the pairing is replaced
332
by $(f,x) = -2\pi i\langle f, x\rangle$ and $x$ is
333
restricted to modular symbols that are fixed by complex
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conjugation.}
335
We have
336
\begin{equation*}
337
\langle f^*, x^* \rangle = \overline{\langle f, x\rangle}.
338
\end{equation*}
339
\end{proposition}
340
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\begin{definition}[Plus-one quotient]\index{Plus-one quotient|textit}%
342
\index{Modular symbols!plus-one quotient of}
343
\index{Modular symbols!minus-one quotient of}
344
The \defn{plus-one quotient} $\sM_k(N,\eps)_+$ is the
345
largest torsion-free quotient of $\sM_k(N,\eps)$ by the relations
346
$x^*-x=0$ for all $x\in \sM_k(N,\eps)$.
347
Similarly, the \defn{minus-one quotient}\index{Minus-one quotient}
348
is the quotient of $\sM_k(N,\eps)$ by all relations
349
$x^*+x=0$, for $x\in\sM_k(N,\eps)$.
350
\end{definition}
351
352
\begin{warning} We were forced to make
353
a choice in our definition of the operator~$*$.
354
Fortunately, it agrees with that of~\cite[\S2.1.3]{cremona:algs},
355
but {\em not} with the choice made in~\cite[\S1.6]{merel:1585}.
356
\end{warning}
357
358
\subsection{The Atkin-Lehner involutions}\label{sec:atkin-lehner}
359
\index{Operators!Atkin-Lehner|textit}
360
\index{Atkin-Lehner involution|textit}
361
In this section we assume
362
that~$k$ is even and $\eps^2=1$.
363
The assumption on~$\eps$ is necessary only so that
364
the involution we are about to define preserves
365
$\sM_k(N,\eps)$. More generally, it is possible to define
366
a map which sends $\sM_k(N,\eps)$ to $\sM_k(N,\overline{\eps})$.
367
368
To each divisor~$d$ of~$N$ such that $(d,N/d)=1$ there is an
369
\defn{Atkin-Lehner involution}~$W_d$ of $\sM_k(N,\eps)$,
370
which is defined as follows. Using the Euclidean algorithm, choose
371
integers $x,y,z,w$ such that
372
$$dxw - (N/d)yz = 1.$$
373
Next let $g=\abcd{dx}{y}{Nz}{dw}$ and define
374
$$W_d(x) \define \frac{1}{d^{\frac{k-2}{2}}}\cdot g(x).$$
375
For example, when $d=N$ we have $g=\abcd{0}{-1}{N}{\hfill 0}$.
376
The factor of $d^{\frac{k-2}{2}}$ is necessary to normalize
377
$W_d$ so that it is an involution.
378
379
On modular forms there is an Atkin-Lehner involution,
380
also denoted $W_d$,\index{Modular forms!and Atkin-Lehner involution}
381
which acts by $W_d(f) = f|[W_d]_k$. These two like-named involutions
382
are compatible with the integration pairing:
383
$$\langle W_d(f), x\rangle = \langle f, W_d(x)\rangle.$$
384
\index{Atkin-Lehner involution!and integration pairing}
385
386
\section{Degeneracy maps}
387
\label{sec:degeneracymaps}
388
\label{pg:degeneracymaps}
389
\index{Degeneracy maps}
390
In this section, we describe natural maps between spaces of
391
modular symbols of different levels.
392
393
Fix a positive integer~$N$ and a Dirichlet
394
character\index{Dirichlet character}
395
$\eps : (\Z/N\Z)^*\ra \C^*$. Let~$M$ be a positive divisor
396
of~$N$ that is divisible by the conductor of~$\eps$, in the sense
397
that~$\eps$ factors through $(\Z/M\Z)^*$ via the natural map
398
$(\Z/N\Z)^*\ra (\Z/M\Z)^*$ composed with some uniquely defined
399
character $\eps':(\Z/M\Z)^*\ra\C^*$. For any positive divisor~$t$ of
400
$N/M$, let $T=\abcd{1}{0}{0}{t}$ and fix a choice $D_t=\{T\gamma_i :
401
i=1,\ldots, n\}$ of coset representatives for $\Gamma_0(N)\backslash
402
T\Gamma_0(M)$.
403
404
\begin{warning}
405
There is a mistake in \cite[\S2.6]{merel:1585}:
406
The quotient ``$\Gamma_1(N)\backslash\Gamma_1(M)T$'' should be replaced
407
by ``$\Gamma_1(N)\backslash T\Gamma_1(M)$''.
408
\end{warning}
409
\begin{proposition}
410
For each divisor~$t$ of $N/M$ there are well-defined linear maps
411
\begin{align*}
412
\alp_t : \sM_k(N,\eps) \ra \sM_k(M,\eps'),&\qquad
413
\alp_t(x) = (tT^{-1})x = \mtwo{t}{0}{0}{1} x\\
414
\beta_t : \sM_k(M,\eps') \ra \sM_k(N,\eps),&\qquad
415
\beta_t(x) = \sum_{T\gam_i\in D_t} \eps'(\gam_i)^{-1}T\gam_i{} x.
416
\end{align*}
417
Furthermore,
418
$\alp_t\circ \beta_t$ is multiplication by
419
$t^{k-2}\cdot [\Gamma_0(M) : \Gamma_0(N)].$
420
\end{proposition}
421
\begin{proof}
422
To show that~$\alp_t$ is well defined, we must show that for
423
each $x\in\sM_k(N,\eps)$ and $\gam=\abcdmat\in\Gamma_0(N)$, that we
424
have
425
$$\alp_t(\gamma x -\eps(\gamma) x)=0\in\sM_k(M,\eps').$$
426
We have
427
$$\alp_t(\gam x) = \mtwo{t}{0}{0}{1}\gam x
428
= \mtwo{a}{tb}{c/t}{d}\mtwo{t}{0}{0}{1} x
429
= \eps'(a)\mtwo{t}{0}{0}{1} x,$$
430
so
431
$$\alp_t(\gamma x -\eps(\gamma) x)
432
= \eps'(a)\alp_t(x) - \eps(\gamma)\alp_t(x) = 0.$$
433
434
We next verify that~$\beta_t$ is well defined.
435
Suppose that $x\in\sM_k(M,\eps')$ and $\gamma\in\Gamma_0(M)$;
436
then $\eps'(\gam)^{-1}\gam x = x$, so
437
\begin{align*}
438
\beta_t(x)
439
&= \sum_{T\gam_i\in D_t}
440
\eps'(\gam_i)^{-1}T\gam_i{}\eps'(\gam)^{-1}\gam{} x\\
441
&= \sum_{T\gam_i\gam\in D_t}
442
\eps'(\gam_i\gam)^{-1}T\gam_i{}\gam{} x.
443
\end{align*}
444
This computation shows that the definition of~$\beta_t$
445
does not depend on the choice~$D_t$ of coset representatives.
446
To finish the proof that~$\beta_t$ is well defined
447
we must show that, for $\gam\in\Gamma_0(M)$, we have
448
$\beta_t(\gam x) = \eps'(\gam)\beta_t(x)$ so that $\beta_t$
449
respects the relations that define $\sM_k(M,\eps)$.
450
Using that~$\beta_t$ does not depend on the choice of
451
coset representative, we find that for $\gamma\in\Gamma_0(M)$,
452
\begin{align*}
453
\beta_t(\gam x)
454
&= \sum_{T\gam_i\in D_t} \eps'(\gam_i)^{-1}T\gam_i{} \gam{} x\\
455
&= \sum_{T\gam_i\gam^{-1}\in D_t}
456
\eps'(\gam_i\gam^{-1})^{-1}T\gam_i{}\gam{}^{-1} \gam{} x\\
457
&= \eps'(\gam)\beta_t(x).\\
458
\end{align*}
459
To compute $\alp_t\circ\beta_t$, we use
460
that $\#D_t = [\Gamma_0(N) : \Gamma_0(M)]$:
461
\begin{align*}
462
\alp_t(\beta_t(x)) &=
463
\alp_t \left(\sum_{T\gamma_i}
464
\eps'(\gam_i)^{-1}T\gam_i x\right)\\
465
&= \sum_{T\gamma_i}
466
\eps'(\gam_i)^{-1}(tT^{-1})T\gam_i x\\
467
&= t^{k-2}\sum_{T\gamma_i}
468
\eps'(\gam_i)^{-1}\gam_i x\\
469
&= t^{k-2}\sum_{T\gamma_i} x \\
470
&= t^{k-2} \cdot [\Gamma_0(N) : \Gamma_0(M)] \cdot x.
471
\end{align*}
472
The scalar factor of $t^{k-2}$ appears instead
473
of~$t$, because~$t$ is acting on~$x$ as an element of $\GL_2(\Q)$
474
{\em not} as an an element of~$\Q$.
475
\end{proof}
476
477
\begin{definition}[New and old modular symbols]%
478
\label{def:newandoldsymbols}%
479
\index{New modular symbols|textit}%
480
\index{Old modular symbols|textit}%
481
\index{Modular symbols!new and old subspace of|textit}%
482
The subspace $\sM_k(N,\eps)^{\new}$
483
of \defn{new modular symbols} is the
484
intersection of the kernels of the $\alp_t$ as~$t$
485
runs through all positive divisors of $N/M$ and~$M$
486
runs through positive divisors of~$M$ strictly less than~$N$
487
and divisible by the conductor of~$\eps$.
488
The subspace $\sM_k(N,\eps)^{\old}$
489
of \defn{old modular symbols}
490
is the subspace generated by the images of the $\beta_t$
491
where~$t$ runs through all positive divisors of $N/M$ and~$M$
492
runs through positive divisors of~$M$ strictly less than~$N$
493
and divisible by the conductor of~$\eps$.
494
\end{definition}
495
496
{\bf WARNING:} The new and old subspaces need not be disjoint, as
497
the following example illustrates!
498
This is contrary to the statement on page~80 of~\cite{merel:1585}.
499
\begin{example}
500
We justify the above warning.
501
Consider, for example, the case $N=6$, $k=2$, and trivial character.
502
The spaces $\sM_2(2)$ and $\sM_2(3)$ are each of dimension~$1$, and
503
each is generated by the modular symbol $\{\infty,0\}$.
504
The space $\sM_2(6)$ is of dimension~$3$, and is generated by
505
the~$3$ modular symbols $\{\infty, 0\}$, $\{-1/4, 0\}$,
506
and $\{-1/2, -1/3\}$.
507
The space generated by the~$2$ images
508
of $\sM_2(2)$ under the~$2$ degeneracy
509
maps has dimension~$2$, and likewise for $\sM_2(3)$.
510
Together these images generate $\sM_2(6)$, so $\sM_2(6)$ is
511
equal to its old subspace.
512
However, the new subspace is nontrivial because
513
the two degeneracy maps $\sM_2(6) \ra \sM_2(2)$ are equal,
514
as are the two degeneracy maps $\sM_2(6) \ra \sM_2(3)$.
515
In particular, the intersection of the kernels of the degeneracy
516
maps has dimension at least~$1$ (in fact, it equals~$1$).
517
518
Computationally, it appears that something similar to this happens
519
if and only if the weight is~$2$, the character is trivial,
520
and the level is composite. This behavior is probably related
521
to the nonexistence of a characteristic~$0$ Eisenstein series
522
of weight~$2$ and level~$1$.
523
\end{example}
524
525
The following tempting argument is incorrect;
526
the error lies in the fact that
527
an element of the old subspace
528
is a {\em linear combination} of $\beta_t(y)$'s
529
for various~$y$'s and~$t$'s:
530
``If~$x$ is in both the new and old subspace,
531
then $x=\beta_t(y)$ for some modular symbol~$y$
532
of lower level. This implies $x=0$ because
533
$$0 = \alp_t(x) = \alp_t(\beta_t(y))=
534
t^{k-2}\cdot[\Gamma_0(N):\Gamma_0(M)] \cdot{}y.\text{''}$$
535
536
537
\begin{remark}
538
The map $\beta_t\circ\alp_t$ cannot in general be multiplication by
539
a scalar since $\sM_k(M,\eps')$
540
usually has smaller dimension than $\sM_k(N,\eps)$.
541
\end{remark}
542
543
\comment{
544
\begin{example}
545
The proposition implies that $\beta_t$ is injective in
546
characteristic~$0$. This need not be the case in positive
547
characteristic, as the following example illustrates.
548
Let~$p$ be any prime, and let $\eps:(\Z/N\Z)^* \ra
549
\Fbar_p^*$ be the reduction to characteristic~$p$
550
of a Dirichlet character.
551
There is again a map $\beta_{t,p}:\sM_k(M,\eps';\Fbar_p) \ra
552
\sM_k(N,\eps;\Fbar_p)$, where the space $\sM_k(N,\eps;\Fbar_p)$ is
553
defined by choosing a maximal ideal $\wp$ lying over~$p$ in an
554
appropriate extension $\O$ of~$\Z$, and letting~$R=\Fbar_p$
555
be an algebraic closure of the finite field~$\O/\wp$.
556
When~$p$
557
does not divide $t^{k-2}\cdot [\Gamma_0(M) : \Gamma_0(N)]$, the
558
proposition shows that $\beta_{t,p}$ is injective. However,
559
$\beta_t\tensor\F_p$ need not be injective for all~$p$. For example,
560
suppose $M=14$, $N=28$, and $\eps=1$. Then there are bases with
561
respect to which the matrix of $\beta_1$ is the transpose of
562
$$\left(
563
\begin{matrix}
564
1&0&0&1&0&0&0&0&0\\
565
0&1&0&0&1&0&0&0&0\\
566
0&0&1&0&0&1&0&0&0\\
567
0&0&0&0&0&0&2&1&-1\\
568
0&0&0&0&0&0&0&1&1
569
\end{matrix}
570
\right),$$
571
and the row vector $(0,0,0,1,1)$ is in the kernel of the mod~$2$
572
reduction of this matrix.
573
\end{example}
574
}
575
576
\subsection{Computing coset representatives}%
577
\index{Coset representatives}
578
\begin{definition}[Projective line mod~$N$]%
579
\index{Projective line modulo~$N$|textit}%
580
Let~$N$ be a positive integer.
581
Then the \defn{projective line}
582
$\P^1(N)$ is the set of
583
pairs $(a,b)$, with $a, b\in\Z/N\Z$ and $\gcd(a,b,N)=1$, modulo
584
the eqivalence relation which identifies $(a,b)$ and $(a',b')$ if and only
585
if $ab'\con ba'\pmod{N}$.
586
\end{definition}
587
588
Let~$M$ be a positive divisor of~$N$ and~$t$ a
589
divisor of~$N/M$. The following {\em random} algorithm
590
computes a set~$D_t$ of representatives for the orbit space
591
$\Gamma_0(M)\backslash T\Gamma_0(N).$
592
There are deterministic algorithms for computing
593
$D_t$, but all of the ones the author has found are
594
{\em vastly} less efficient than the following random algorithm.
595
\begin{algorithm}\label{alg:degenreps}%
596
\index{Algorithm for computing!coset representatives}
597
Let $\Gamma_0(N/t,t)$ denote the subgroup of $\SL_2(\Z)$
598
consisting of matrices that are upper triangular modulo $N/t$ and lower
599
triangular modulo~$t$. Observe that two right cosets
600
of $\Gamma_0(N/t,t)$ in $\SL_2(\Z)$, represented by
601
$\abcd{a}{b}{c}{d}$ and $\abcd{a'}{b'}{c'}{d'}$,
602
are equivalent if and only if
603
$(a,b)=(a',b')$ as points of $\P^1(t)$
604
and $(c,d)=(c',d')$ as points of $\P^1(N/t)$.
605
Using the following algorithm, we compute right coset
606
representatives for $\Gamma_0(N/t,t)$
607
inside~$\Gamma_0(M)$.
608
\begin{enumerate}
609
\item Compute the number $[\Gamma_0(M):\Gamma_0(N)]$ of cosets.
610
\item Compute a random element $x \in \Gamma_0(M)$.
611
\item If~$x$ is not equivalent to anything generated so
612
far, add it to the list.
613
\item Repeat steps (2) and (3) until the list is as long
614
as the bound of step (1).
615
\end{enumerate}
616
There is a natural bijection between
617
$\Gamma_0(N)\backslash T \Gamma_0(M)$
618
and $\Gamma_0(N/t,t)\backslash \Gamma_0(M)$,
619
under which~$T\gamma$ corresponds to~$\gamma$.
620
Thus we obtain coset representatives for
621
$\Gamma_0(N)\backslash T\Gamma_0(M)$
622
by left multiplying each
623
coset representative of $\Gamma_0(N/t,t)\backslash\Gamma_0(M)$ by~$T$.
624
\end{algorithm}
625
626
\subsection{Compatibility with modular forms}%
627
\index{Degeneracy maps!compatibility}%
628
The degeneracy maps defined above
629
are compatible with the corresponding degeneracy maps
630
$\tilde{\alp}_t$ and $\tilde{\beta}_t$
631
on modular forms\index{Modular forms}. This is because the degeneracy
632
maps on modular forms are defined by summing over the
633
same coset representatives $D_t$.
634
Thus we have the following compatibilities.
635
\begin{align*}
636
\langle \tilde{\alp}_t(f), x \rangle &= \langle f, \alp_t(x)\rangle,\\
637
\langle \tilde{\beta}_t(f), x\rangle &= \langle f, \beta_t(x) \rangle .
638
\end{align*}
639
If~$p$ is prime to~$N$, then $T_p\alp_t = \alp_t T_p$
640
and $T_p\beta_t = \beta_t T_p$.
641
642
\section{Manin symbols}%
643
\label{sec:maninsymbols}%
644
\index{Manin symbols}%
645
From the definition given in
646
Section~\ref{sec:defnofmodsyms}, it is not obvious
647
that $\sM_k(N,\eps)$ is of finite rank. The Manin
648
symbols provide a finite presentation of~$\sM_k(N,\eps)$
649
that is vastly more useful from a computational point of view.
650
\index{Modular symbols!finite presentation of}
651
652
\begin{definition}[Manin symbols]\label{defn:maninsymbols}%
653
\index{Manin symbols|textit}%
654
The \defn{Manin symbols} are the set of pairs
655
$$[P(X,Y),(u,v)]$$
656
where $P(X,Y)\in V_{k-2}$ and
657
$0\leq u,v < N$ with $\gcd(u,v,N)=1$.
658
\end{definition}
659
Define a {\em right} action of $\GL_2(\Q)$ on
660
the free $\Z[\eps]$-module~$M$ generated by the Manin
661
symbols as follows. The element $g=\abcd{a}{b}{c}{d}$ acts by
662
\begin{equation*}
663
[P,(u,v)]g=[g^{-1}P(X,Y),(u,v) g]
664
= [P(aX+bY,cX+dY),(au+cv,bu+dv)].
665
\end{equation*}
666
Let $\sigma=\abcd{0}{-1}{1}{\hfill 0}$ and $\tau=\abcd{0}{-1}{1}{-1}$\label{defn:sigmatau}.
667
Let $\sM_k(N,\eps)'$ be the largest torsion-free quotient
668
of~$M$ by the relations
669
\begin{align*}
670
\mbox{}x + x\sigma &= 0,\\
671
\mbox{}x + x\tau+ x\tau^2 &= 0,\\
672
\eps(\lambda) [P,(u,v)]- [P,(\lambda u, \lambda v)] &=0.
673
\end{align*}
674
675
\begin{theorem}\label{thm:maninsymbols}
676
There is a natural isomorphism
677
$\vphi:\sM_k(N,\eps)'\lra\sM_k(N,\eps)$ given by
678
$$[X^iY^{2-k-i},(u,v)] \mapsto g(X^iY^{k-2-i}\{ 0,\infty\})$$
679
where $g=\abcd{a}{b}{c}{d}\in\SL_2(\Z)$ is any matrix
680
such that $(u,v)\con (c,d) \pmod{N}$.
681
\end{theorem}
682
\begin{proof}
683
In~\cite[\S1.2, \S1.7]{merel:1585} it is proved that
684
$\vphi\tensor_{\Z[\eps]}\C$ is
685
an isomorphism, so~$\vphi$ is injective and well defined.
686
The discussion in Section~\ref{sec:modmanconv} below
687
(``Manin's trick'')\index{Manin's trick}\index{Manin symbols!and Manin's trick}
688
shows that every element in $\sM_k(N,\eps)$ is a $\Z[\eps]$-linear
689
combination of elements in the image, so~$\vphi$ is surjective as well.
690
\end{proof}
691
692
\subsection{Conversion between modular and Manin symbols}%
693
\index{Manin symbols!conversion to modular symbols}%
694
\index{Modular symbols!conversion to Manin symbols}%
695
\label{sec:modmanconv}%
696
For some purposes it is better to work with modular symbols, and for
697
others it is better to work with Manin symbols. For example, there
698
are descriptions of the Atkin-Lehner involution\index{Atkin-Lehner involution}
699
in terms of both Manin
700
and modular symbols; it appears more efficient to compute this
701
involution using modular symbols. On the other hand, practically
702
Hecke operators can be computed more efficiently using Manin symbols.
703
It is thus essential to be able to convert between these two
704
representations. The conversion from Manin to modular symbols is
705
straightforward, and follows immediately from
706
Theorem~\ref{thm:maninsymbols}. The conversion back is accomplished
707
using the method used to prove Theorem~\ref{thm:maninsymbols}; it is
708
known as ``Manin's trick'',\index{Manin's trick|textit}\index{Manin!trick of|textit} and involves continued fractions\index{Continued fractions}.
709
710
Given a Manin symbol $[X^iY^{k-2-i},(u,v)]$\index{Manin symbols},
711
we write down a corresponding modular symbol\index{Modular symbols}
712
as follows.
713
Choose $\abcd{a}{b}{c}{d}\in\SL_2(\Z)$ such that
714
$(c,d)\con (u,v)\pmod{N}$. This is possible
715
by Lemma~1.38 of~\cite[pg.~20]{shimura:intro}; in practice,
716
finding $\abcd{a}{b}{c}{d}$ is not completely trivial, but
717
can be accomplished using the extended Euclidean
718
algorithm.
719
Then
720
\begin{eqnarray*}
721
[X^iY^{k-2-i},(u,v)] &\corrto&
722
\abcd{a}{b}{c}{d}(X^iY^{k-2-i}\{ 0,\infty\})\\
723
&&= (dX-bY)^i(-cX+aY)^{2-k-i}
724
\left\{\frac{b}{d},\,\frac{a}{c}\right\}.\\
725
\end{eqnarray*}
726
727
In the other direction, suppose that we are given a modular
728
symbol $P(X,Y)\{\alp,\beta\}$ and wish to represent it as a
729
sum of Manin symbols. Because
730
$$P\{a/b,c/d\} = P\{a/b,0\}+P\{0,c/d\},$$
731
it suffices to write $P\{0,a/b\}$ in
732
terms of Manin symbols.
733
Let
734
$$0=\frac{p_{-2}}{q_{-2}} = \frac{0}{1},\,\,
735
\frac{p_{-1}}{q_{-1}}=\frac{1}{0},\,\,
736
\frac{p_0}{1}=\frac{p_0}{q_0},\,\,
737
\frac{p_1}{q_1},\,\,
738
\frac{p_2}{q_2},\,\ldots,\,\frac{p_r}{q_r}=\frac{a}{b}$$
739
denote the continued fraction convergents of the
740
rational number $a/b$.
741
Then
742
$$p_j q_{j-1}
743
- p_{j-1} q_j = (-1)^{j-1}\qquad \text{for }-1\leq j\leq r.$$
744
If we let
745
$g_j = \mtwo{(-1)^{j-1}p_j}{p_{j-1}}{(-1)^{j-1}q_j}{q_{j-1}}$,
746
then $g_j\in\sltwoz$ and
747
\begin{align*}
748
P(X,Y)\{0,a/b\}
749
&=P(X,Y)\sum_{j=-1}^{r}\left\{\frac{p_{j-1}}{q_{j-1}},\frac{p_j}{q_j}\right\}\\
750
&=\sum_{j=-1}^{r} g_j((g_j^{-1}P(X,Y))\{0,\infty\})\\
751
&=\sum_{j=-1}^{r} [g_j^{-1}P(X,Y),((-1)^{j-1}q_j,q_{j-1})].
752
\end{align*}
753
Note that in the $j$th summand, $g_j^{-1}P(X,Y)$, replaces $P(X,Y)$.
754
Since $g_j\in\sltwoz$ and $P(X,Y)$ has integer coefficients,
755
the polynomial $g_j^{-1}P(X,Y)$ also has integer coefficients,
756
so no denominators are introduced.
757
758
The continued fraction expansion $[c_1,c_2,\ldots,c_n]$
759
of the rational number $a/b$ can be computed
760
using the Euclidean algorithm.
761
The first term, $c_1$, is the ``quotient'': $a = bc_1+r$,
762
with $0\leq r < b$.
763
Let $a'=b$, $b'=r$ and compute $c_2$ as
764
$a'=b'c_2+r'$, etc., terminating when the
765
remainder is $0$. For example, the expansion
766
of $5/13$ is $[0,2,1,1,2]$.
767
The numbers $$d_i=c_1+\frac{1}{c_2+\frac{1}{c_3+\cdots}}$$
768
will then be the (finite) convergents.
769
For example if $a/b=5/13$, then the convergents are
770
$$0/1,\,\, 1/0,\,\, d_1=0,\,\, d_2=\frac{1}{2},\,\, d_3=\frac{1}{3},\,\,
771
d_4=\frac{2}{5},\,\, d_5=\frac{5}{13}.$$
772
773
774
775
\subsection{Hecke operators on Manin symbols}%
776
\index{Hecke operators!on Manin symbols}%
777
\index{Manin symbols!and Hecke operators}%
778
\label{subsec:heckeonmanin}%
779
Thoerem~2 of \cite{merel:1585} gives a description of
780
the Hecke operators~$T_n$
781
directly on the space of Manin symbols.
782
This avoids the expense of first converting a Manin
783
symbol to a modular symbol, computing~$T_n$ on the modular symbol,
784
and then converting back. For the reader's convenience, we very
785
briefly recall Merel's\index{Merel} theorem here, along with
786
an enhancement due to Cremona\index{Cremona}.
787
788
As in~\cite[\S2.4]{cremona:algs}, define~$R_p$ as follows.
789
When $p=2$,
790
$$R_2 := \left\{\mtwo{1}{0}{0}{2},
791
\mtwo{2}{0}{0}{1}, \mtwo{2}{1}{0}{1},
792
\mtwo{1}{0}{1}{2}\right\}.$$
793
When~$p$ is odd,~$R_p$ is the set of $2\times 2$ integer
794
matrices $\abcd{a}{b}{c}{d}$ with determinant~$p$, and either
795
\begin{enumerate}
796
\item $a>|b|>0$, $d>|c|>0$, and $bc<0$; or
797
\item $b=0$, and $|c|<d/2$; or
798
\item $c=0$, and $|b|<a/2$.
799
\end{enumerate}
800
\begin{proposition}
801
For $[P(X,Y),(u,v)]\in\sM_k(N,\eps)$ and~$p$ a prime, we have
802
\begin{align*}T_p([P(X,Y),(u,v)])
803
&= \sum_{g\in R_p} [P(X,Y),(u,v)].g \\
804
&= \sum_{\abcd{a}{b}{c}{d}\in R_p} [P(aX+bY,cX+dY),(au+cv,bu+dv)],
805
\end{align*}
806
where the sum is restricted to matrices $\abcd{a}{b}{c}{d}$
807
such that $\gcd(au+cv,bu+dv,N)=1$.
808
\end{proposition}
809
\begin{proof}
810
For the case $k=2$ and an algorithm to compute $R_p$,
811
see \cite[\S2.4]{cremona:algs}.
812
The general case follows from~\cite[Theorem 2]{merel:1585} applied
813
to the set~$\sS$ of~\cite[\S3]{merel:1585} by observing that
814
when~$p$ is an odd {\em prime} $\sS_p'$ is empty.
815
\end{proof}
816
817
\subsection{The cuspidal and boundary spaces in terms of Manin symbols}%
818
\index{Manin symbols!and cuspidal subspace}%
819
\index{Manin symbols!and boundary space}%
820
\index{Cuspidal modular symbols!and Manin symbols}%
821
\index{Boundary modular symbols!and Manin symbols}%
822
This section is a review of Merel's\index{Merel} explicit description
823
of the boundary map in terms of Manin symbols\index{Manin symbols}
824
for $\Gamma=\Gamma_1(N)$
825
(see~\cite[\S1.4]{merel:1585}). In the next section, we
826
describe a very efficient way to compute the boundary map.
827
828
Let~$\cR$ be the equivalence relation
829
on $\Gamma\backslash\Q^2$ which identifies
830
the element
831
$[\Gamma\smallvtwo{\lambda u}{\lambda v}]$
832
with $\sign(\lambda)^k[\Gamma\smallvtwo{u}{v}]$,
833
for any $\lambda\in\Q^*$. Denote by $B_k(\Gamma)$
834
the finite dimensional $\Q$-vector space
835
with basis the equivalence classes
836
$(\Gamma\backslash\Q^2)/\cR$.
837
The dimension of this space is $\#(\Gamma\backslash\P^1(\Q))$.
838
\begin{proposition}
839
The map
840
$$\mu:\sB_k(\Gamma)\ra B_k(\Gamma),
841
\qquad P\left\{\frac{u}{v}\right\}\mapsto
842
P(u,v)\left[\Gamma\vtwo{u}{v}\right]$$
843
is well defined and injective.
844
Here $u$ and $v$ are assumed coprime.
845
\end{proposition}
846
Thus the kernel of $\delta:\sS_k(\Gamma)\ra \sB_k(\Gamma)$
847
is the same as the kernel of $\mu\circ \delta$.
848
\begin{proposition}\label{prop:boundary}
849
Let $P\in V_{k-2}$ and $g=\abcd{a}{b}{c}{d}\in\sltwoz$. We have
850
$$\mu\circ\delta([P,(c,d)])
851
= P(1,0)[\Gamma\smallvtwo{a}{c}]
852
-P(0,1)[\Gamma\smallvtwo{b}{d}].$$
853
\end{proposition}
854
855
856
\subsection{Computing the boundary map}%
857
\index{Boundary map}%
858
\label{sec:computeboundary}%
859
In this section we describe how to compute the
860
map $\delta:\sM_k(N,\eps)\ra B_k(N,\eps)$
861
given in the previous section.
862
The algorithm presented here
863
generalizes the one in~\cite[\S2.2]{cremona:algs}.
864
To compute the image of $[P,(c,d)]$, with
865
$g=\abcd{a}{b}{c}{d}\in\sltwoz$,
866
we must compute the class of $[\smallvtwo{a}{c}]$ and of
867
$[\smallvtwo{b}{d}]$.
868
Instead of finding a canonical form for cusps, we
869
use a quick test for equivalence modulo scalars.
870
In the following algorithm, by the $i$th standard
871
cusp\index{Cusps!and boundary map} we mean
872
the $i$th basis vector for a basis of $B_k(N,\eps)$. The
873
basis is constructed as the algorithm is called successively.
874
We first give the algorithm, then prove the facts
875
used by the algorithm in testing equivalence.
876
877
\begin{algorithm}\label{alg:cusplist}
878
\index{Algorithm for computing!cusps}
879
Given a cusp $[\smallvtwo{u}{v}]$ this algorithm computes an
880
integer~$i$ and a scalar~$\alp$ such that $[\smallvtwo{u}{v}]$ is
881
equivalent to~$\alp$ times the $i$th standard cusp. First, using
882
Proposition~\ref{prop:cusp1} and Algorithm~\ref{alg:cusp1}, check
883
whether or not $[\smallvtwo{u}{v}]$ is equivalent, modulo scalars, to
884
any cusp found so far. If so, return the index of the representative
885
and the scalar. If not, record $\smallvtwo{u}{v}$ in the
886
representative list. Then, using Proposition~\ref{prop:cuspdies},
887
check whether or not $\smallvtwo{u}{v}$ is forced to equal zero by the
888
relations. If it does not equal zero, return its position in the list
889
and the scalar~$1$. If it equals zero, return the scalar~$0$ and the
890
position~$1$; keep $\smallvtwo{u}{v}$ in the list, and record that it
891
is zero.
892
\end{algorithm}
893
894
In the case considered in Cremona's book \cite{cremona:algs}, the
895
relations between cusps involve only the trivial character, so they do
896
not force any cusp classes to vanish. Cremona gives the following two
897
criteria for equivalence.
898
\begin{proposition}[Cremona]\label{prop:cusp1}\index{Cremona}
899
Let $\smallvtwo{u_i}{v_i}$, $i=1,2$ be written so that
900
$\gcd(u_i,v_i)=1$.
901
\begin{enumerate}
902
\item There exists $g\in\Gamma_0(N)$ such that
903
$g\smallvtwo{u_1}{v_1}=\smallvtwo{u_2}{v_2}$ if and only if
904
$$s_1 v_2 \con s_2 v_1 \pmod{\gcd(v_1 v_2,N)},\,
905
\text{ where $s_j$ satisfies $u_j s_j\con 1\pmod{v_j}$}.$$
906
\item There exists $g\in\Gamma_1(N)$ such that
907
$g\smallvtwo{u_1}{v_1}=\smallvtwo{u_2}{v_2}$ if and only if
908
$$v_2 \con v_1 \pmod{N}\text{ and } u_2 \con u_1 \pmod{\gcd(v_1,N)}.$$
909
\end{enumerate}
910
\end{proposition}
911
\begin{proof}
912
The first is Proposition 2.2.3 of \cite{cremona:algs}, and
913
the second is Lemma 3.2 of \cite{cremona:gammaone}.
914
\end{proof}
915
916
\begin{algorithm}\label{alg:cusp1}%
917
\index{Algorithm for computing!equivalent cusps}%
918
Suppose $\smallvtwo{u_1}{v_1}$ and
919
$\smallvtwo{u_2}{v_2}$
920
are equivalent modulo $\Gamma_0(N)$.
921
This algorithm computes a matrix $g\in\Gamma_0(N)$ such
922
that $g\smallvtwo{u_1}{v_1}=\smallvtwo{u_2}{v_2}$.
923
Let $s_1, s_2, r_1, r_2$ be solutions to
924
$s_1 u_1 -r_1 v_1 =1$ and
925
$s_2 u_2 -r_2 v_2 =1$.
926
Find integers $x_0$ and $y_0$ such
927
that $x_0v_1v_2+y_0N=1$.
928
Let $x=-x_0(s_1v_2-s_2v_1)/(v_1v_2,N)$
929
and $s_1' = s_1 + xv_1$.
930
Then $g=\mtwo{u_2}{r_2}{v_2}{s_2}
931
\cdot \mtwo{u_1}{r_1}{v_1}{s_1'}^{-1}$
932
sends $\smallvtwo{u_1}{v_1}$ to $\smallvtwo{u_2}{v_2}$.
933
\end{algorithm}
934
\begin{proof}
935
This follows from the proof of Proposition~\ref{prop:cusp1} in
936
\cite{cremona:algs}.
937
\end{proof}
938
939
940
To see how the~$\eps$ relations, for nontrivial~$\eps$,
941
make the situation more complicated, observe that it is
942
possible that $\eps(\alp)\neq \eps(\beta)$ but
943
$$\eps(\alp)\left[\vtwo{u}{v}\right] =\left[\gamma_\alp \vtwo{u}{v}\right]=
944
\left[\gamma_\beta \vtwo{u}{v}\right]=\eps(\beta)\left[\vtwo{u}{v}\right];$$
945
One way out of this difficulty is to construct
946
the cusp classes for $\Gamma_1(N)$, then quotient
947
out by the additional~$\eps$ relations using
948
Gaussian elimination. This is far too
949
inefficient to be useful in practice because the number of
950
$\Gamma_1(N)$ cusp classes can be unreasonably large.
951
Instead, we give a quick test to determine whether or not
952
a cusp vanishes modulo the $\eps$-relations.
953
954
\begin{lemma}\label{lem:canlift}
955
Suppose $\alp$ and $\alp'$ are integers
956
such that $\gcd(\alp,\alp',N)=1$.
957
Then there exist integers $\beta$ and $\beta'$,
958
congruent to $\alp$ and $\alp'$ modulo $N$, respectively,
959
such that $\gcd(\beta,\beta')=1$.
960
\end{lemma}
961
\begin{proof}
962
By \cite[1.38]{shimura:intro} the map
963
$\SL_2(\Z)\ra\SL_2(\Z/N\Z)$ is surjective.
964
By the Euclidean algorithm, there exist
965
integers $x$, $y$ and $z$ such that
966
$x\alp + y\alp' + zN = 1$.
967
Consider the matrix
968
$\abcd{y}{-x}{\alp}{\hfill\alp'}\in \SL_2(\Z/N\Z)$
969
and take $\beta$, $\beta'$ to be the bottom
970
row of a lift of this matrix to $\SL_2(\Z)$.
971
\end{proof}
972
973
\begin{proposition}\label{prop:cuspdies}\index{Cusps!criterion for vanishing}
974
Let~$N$ be a positive integer and~$\eps$ a Dirichlet
975
character\index{Dirichlet character!and cusps} of modulus~$N$.
976
Suppose $\smallvtwo{u}{v}$ is a cusp with $u$ and $v$ coprime.
977
Then $\smallvtwo{u}{v}$ vanishes modulo the relations
978
$$\left[\gamma\smallvtwo{u}{v}\right]=
979
\eps(\gamma)\left[\smallvtwo{u}{v}\right],\qquad
980
\text{all $\gamma\in\Gamma_0(N)$}$$
981
if and only if there exists $\alp\in(\Z/N\Z)^*$,
982
with $\eps(\alp)\neq 1$, such that
983
\begin{align*}
984
v &\con \alp v \pmod{N},\\
985
u &\con \alp u \pmod{\gcd(v,N)}.
986
\end{align*}
987
\end{proposition}
988
\begin{proof}
989
First suppose such an~$\alp$ exists.
990
By Lemma~\ref{lem:canlift}
991
there exists $\beta, \beta'\in\Z$ lifting
992
$\alp,\alp^{-1}$ such that $\gcd(\beta,\beta')=1$.
993
The cusp $\smallvtwo{\beta u}{\beta' v}$
994
has coprime coordinates so,
995
by Proposition~\ref{prop:cusp1} and our
996
congruence conditions on~$\alp$, the cusps
997
$\smallvtwo{\beta{}u}{\beta'{}v}$
998
and $\smallvtwo{u}{v}$ are equivalent by
999
an element of $\Gamma_1(N)$.
1000
This implies that $\left[\smallvtwo{\beta{}u}{\beta'{}v}\right]
1001
=\left[\smallvtwo{u}{v}\right]$.
1002
Since $\left[\smallvtwo{\beta{}u}{\beta'{}v}\right]
1003
= \eps(\alp)\left[\smallvtwo{u}{v}\right]$,
1004
our assumption that $\eps(\alp)\neq 1$
1005
forces $\left[\smallvtwo{u}{v}\right]=0$.
1006
1007
Conversely, suppose $\left[\smallvtwo{u}{v}\right]=0$.
1008
Because all relations are two-term relations, and the
1009
$\Gamma_1(N)$-relations identify $\Gamma_1(N)$-orbits,
1010
there must exists $\alp$ and $\beta$ with
1011
$$\left[\gamma_\alp \vtwo{u}{v}\right]
1012
=\left[\gamma_\beta \vtwo{u}{v}\right]
1013
\qquad\text{ and }\eps(\alp)\ne \eps(\beta).$$
1014
Indeed, if this did not occur,
1015
then we could mod out by the $\eps$ relations by writing
1016
each $\left[\gamma_\alp \smallvtwo{u}{v} \right]$
1017
in terms of $\left[\smallvtwo{u}{v}\right]$, and there would
1018
be no further relations left to kill
1019
$\left[\smallvtwo{u}{v}\right]$.
1020
Next observe that
1021
$$
1022
\left[\gamma_{\beta^{-1}\alp}
1023
\vtwo{u}{v}\right]
1024
= \left[\gamma_{\beta^{-1}}\gamma_\alp
1025
\vtwo{u}{v}\right]
1026
= \eps(\beta^{-1})\left[\gamma_\alp
1027
\vtwo{u}{v}\right]
1028
= \eps(\beta^{-1})\left[\gamma_\beta
1029
\vtwo{u}{v}\right]
1030
= \left[\vtwo{u}{v}\right].$$
1031
Applying Proposition~\ref{prop:cusp1} and
1032
noting that $\eps(\beta^{-1}\alp)\neq 1$ shows
1033
that $\beta^{-1}\alp$ satisfies the properties
1034
of the ``$\alp$'' in the statement of the
1035
proposition we are proving.
1036
\end{proof}
1037
1038
We enumerate the possible~$\alp$ appearing
1039
in Proposition~\ref{prop:cuspdies} as follows.
1040
Let $g=(v,N)$ and list the
1041
$\alp=v\cdot\frac{N}{g}\cdot{}a+1$, for $a=0,\ldots,g-1$,
1042
such that $\eps(\alp)\neq 0$.
1043
1044
{\vspace{3ex}\em\par\noindent Working in the
1045
plus one\index{Plus-one quotient} or
1046
minus one quotient\index{Minus-one quotient}.}
1047
Let~$s$ be a sign, either~$+1$ or~$-1$.
1048
To compute $\sS_k(N,\eps)_s$ it is necessary
1049
to replace $B_k(N,\eps)$ by its quotient modulo the
1050
additional relations
1051
$\left[ \smallvtwo{-u}{\hfill v}\right]
1052
= s \left[\smallvtwo{u}{v}\right]$
1053
for all cusps $\smallvtwo{u}{v}$.
1054
Algorithm~\ref{alg:cusplist} can be modified to deal
1055
with this situation as follows.
1056
Given a cusp $x=\smallvtwo{u}{v}$, proceed as
1057
in Algorithm~\ref{alg:cusplist} and check if
1058
either $\smallvtwo{u}{v}$ or $\smallvtwo{-u}{\hfill v}$
1059
is equivalent (modulo scalars) to any cusp seen so far. If not,
1060
use the following trick to determine whether
1061
the $\eps$ and $s$-relations
1062
kill the class of $\smallvtwo{u}{v}$:
1063
use the unmodified Algorithm~\ref{alg:cusplist}
1064
to compute the scalars $\alp_1, \alp_2$ and
1065
standard indices $i_1$, $i_2$ associated to
1066
$\smallvtwo{u}{v}$ and $\smallvtwo{-u}{\hfill v}$, respectively.
1067
The $s$-relation kills the class of $\smallvtwo{u}{v}$
1068
if and only if $i_1=i_2$ but $\alp_1\neq s\alp_2$.
1069
1070
1071
\section{The complex torus attached to a modular form}%
1072
\index{Complex torus}%
1073
\index{Modular forms!associated complex torus}%
1074
\label{sec:tori}%
1075
Fix integers $N\geq 1$, $k\geq 2$, and let~$\eps$ be a mod~$N$
1076
Dirichlet character\index{Dirichlet character}.
1077
For the rest of this section assume that $\eps^2=1$.
1078
1079
We construct a lattice in $\Hom(S_k(N,\eps),\C)$ that is invariant
1080
under complex conjugation and under the action of the Hecke
1081
operators.\index{Hecke operators} The quotient of
1082
$\Hom(S_k(N,\eps),\C)$ by this lattice is a complex torus
1083
$J_k(N,\eps)$, which is equipped with an action of the Hecke operators
1084
and of complex conjugation.
1085
1086
The reader may wish to compare our construction with a closely related
1087
construction of Shimura\index{Shimura}~\cite{shimura:surles}. Shimura
1088
observes that the Petersson pairing\index{Petersson pairing} gives his
1089
torus the structure of an abelian variety over~$\C$. Note that his
1090
torus is, a priori, different than our torus. We do not know if
1091
our torus has the structure of abelian variety over~$\C$.
1092
1093
When $k=2$, the torus $J_2(N,\eps)$ is the set of complex points of an
1094
abelian variety, which is actually defined over $\Q$; when $k>2$,
1095
the study of these complex tori is of interest in trying to understand the
1096
conjectures of Bloch and Kato (see \cite{bloch-kato})%
1097
\index{Conjecture!Bloch and Kato}%
1098
\index{Bloch and Kato conjecture} on motifs\index{Motifs} attached
1099
to modular forms\index{Modular forms}.
1100
1101
Let $\sS=\sS_k(N,\eps)$ (respectively, $S=S_k(N,\eps)$)
1102
be the space of cuspidal modular symbols (respectively, cusp forms)
1103
of weight~$k$, level~$N$, and character~$\eps$.
1104
The Hecke algebra~$\T$\index{Hecke algebra!and integration pairing}
1105
acts in a way compatible with the
1106
integration pairing\index{Integration pairing!and complex torus}
1107
$\langle\,,\,\rangle
1108
: S \cross \sS \ra \C$.
1109
This pairing induces a $\T$-module
1110
homomorphism $\Phi:\sS\ra S^*=\Hom_\C(S,\C)$,
1111
called the \defn{period mapping}.%
1112
\index{Period mapping|textit}
1113
Because $\eps^2=1$, the $*$-involution\index{Star involution} preserves~$S$.
1114
\begin{proposition}
1115
The period mapping~$\Phi$\index{Period mapping!is injective}
1116
is injective and $\Phi(\sS)$ is a lattice in~$S^*$.
1117
\end{proposition}
1118
\begin{proof}
1119
By Theorem~\ref{thm:perfectpairing},
1120
$$\sS\tensor_{\R}\C\isom
1121
\Hom_\C(S\oplus \Sbar,\C).$$
1122
Because $\eps^2=1$, we have $S = S_k(N,\eps;\R)\tensor_{\R}\C$.
1123
Set $S_\R := S_k(N,\eps;\R)$ and likewise define $\Sbar_\R$.
1124
We have
1125
$$\Hom_\C(S\oplus \Sbar,\C) =
1126
\Hom_\R(S_\R \oplus \Sbar_\R,\R)\tensor_\R \C.$$
1127
Let $\sS_{\R} = \sS_k(N,\eps;\R)$ and $\sS_{\R}^+$ be the
1128
subspace fixed under~$*$. By Proposition~\ref{prop:starpairing}
1129
we have maps
1130
$$\sS_{\R}^+ \ra \Hom_{\R}(S_{\R}\oplus\Sbar_\R,\R)
1131
\ra \Hom_{\R}(S_{\R},\R)$$
1132
and
1133
$$\sS_{\R}^- \ra \Hom_{\R}(S_{\R}\oplus\Sbar_\R,i\R)
1134
\ra \Hom_{\R}(S_{\R},i\R).$$
1135
The map $\sS_{\R}^+\ra \Hom_{\R}(S_{\R},\R)$ is
1136
an isomorphism: the point is that if
1137
$\langle \bullet, x\rangle$, for $x\in \sS_{\R}^+$,
1138
vanishes on $S_\R$ then it vanishes on the
1139
whole of $S\oplus \Sbar$. Likewise, the map
1140
$\sS_{\R}^-\ra \Hom_{\R}(S_{\R},i\R)$
1141
is an isomorphism. Thus
1142
$$\sS\tensor\R = \sS_{\R} \isom \Hom_{\R}(S_{\R},\R)
1143
\oplus \Hom_{\R}(S_{\R},i\R)
1144
\isom \Hom_{\C}(S,\C).$$
1145
Finally, we observe that~$\sS$ is by definition
1146
torsion free, which completes the proof.
1147
\end{proof}
1148
1149
The torus $J_k(N,\eps)$ fits into an exact sequence
1150
$$0\lra \sS \xrightarrow{\quad\Phi\quad}
1151
\Hom_\C(S,\C) \lra J_k(N,\eps) \lra 0.$$
1152
Let $f\in S$ be a newform and $S_f$ the complex vector
1153
space spanned by the Galois conjugates of~$f$.
1154
The period map $\Phi_f$ associated to~$f$ is the map
1155
$\sS\ra \Hom_\C(S_f,\C)$
1156
obtained by composing~$\Phi$ with restriction to $S_f$.
1157
Set
1158
$$A_f := \Hom_\C(S_f,\C) / \Phi_f(\sS).$$
1159
1160
We associate\label{pg:dual} to~$f$ a subtorus of~$J$ as follows.
1161
\index{Complex torus!dual of}%
1162
\index{Modular forms!associated subtorus}%
1163
Let $I_f = \Ann_{\T}(f)$ be the annihilator
1164
of~$f$ in the Hecke algebra\index{Hecke algebra}, and set
1165
$$\Adual_f := \Hom_\C(S,\C)[I_f]/\Phi(\sS[I_f])$$
1166
where $\Hom_\C(S,\C)[I_f] = \intersect_{t \in I_f} \ker(t)$.
1167
1168
The following diagram summarizes the tori just defined;
1169
its columns are exact but its rows need not be.
1170
\begin{equation}\label{dgm:uniformization}
1171
\[email protected]=.9pc{
1172
0\ar[d] & 0\ar[d] & 0\ar[d] \\
1173
\sS[I_f]\ar[r]\ar[dd] & \sS\ar[r]\ar[dd]&\Phi_f(\sS)\ar[dd] \\
1174
& & \\
1175
\Hom_\C(S,\C)[I_f]\ar[r]\ar[dd] &\Hom_\C(S,\C)\ar[r]\ar[dd] &\Hom_\C(S[I_f],\C)\ar[dd]\\
1176
& & \\
1177
{\Adual_f}\ar[r]\ar[d]
1178
& J_k(N,\eps) \ar[r]\ar[d]& A_f \ar[d]\\
1179
0 & 0 & 0 \\
1180
}\end{equation}
1181
1182
1183
\subsection{The case when the weight is $2$}%
1184
\index{Complex torus!in weight two}%
1185
When $k=2$ and $\eps=1$ the above is just Shimura's\index{Shimura}
1186
classical association of an abelian variety to a modular
1187
form\index{Modular forms}; see~\cite[Thm.~7.14]{shimura:intro}
1188
and~\cite{shimura:factors}. In this case $A_f$ and $\Adual_f$ are
1189
abelian varieties that are defined over~$\Q$. Furthermore $A_f$ is an
1190
\defn{optimal quotient}\index{Optimal quotient|textit} of~$J$, in the sense
1191
that the kernel of the map $J\ra A_f$ is connected.
1192
For a summary of the main results in this situation,
1193
see Section~\ref{sec:optquoj0n}.
1194
1195
1196