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Author: William A. Stein
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\chapter{Modular algorithms}
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\label{chap:computing}
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\section{Computing the space of modular symbols}
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\begin{definition}
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Let~$W$ be a subspace of a finite-dimensional vector space~$V$.
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To \defn{compute} the quotient $V/W$ means to
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give a matrix representing the projection $V\ra V/W$, with
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respect to some basis for~$V$ and some basis~$B$ for $V/W$, along with
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a lift to~$V$ of each element of~$B$.
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\end{definition}
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In other words, to compute $V/W$ means to create a
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reduction function that assigns
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to each element of~$V$ its canonical representative
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on the ``free basis''~$B$.%, modulo the relations~$W$.
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Let~$N$ be a positive integer, fix a mod~$N$ Dirichlet
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character~$\eps$, let $K:=\Q[\eps]$ be the smallest extension
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containing the values of~$\eps$, and let $\O:=\Z[\eps]$.
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\begin{algorithm}\label{alg:MkNK}
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Given a positive integer~$N$, a Dirichlet character~$\eps$,
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and an integer $k\geq 2$
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this algorithm computes $\sM_k(N,\eps;K)$.
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It uses the Manin-symbols description of $\sM_k(N,\eps;K)$ given
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in Theorem~\ref{thm:maninsymbols}.
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We compute the quotient presentation of Theorem~\ref{thm:maninsymbols}
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in three steps.
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\begin{enumerate}
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\item Let $V_1$ be the finite-dimensional $K$-vector space
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generated by the Manin symbols
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$[X^iY^{k-2-i}, (u,v)]$
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for $i=0,\ldots, k-2$ and $0\leq u,v < N$ with $\gcd(u,v,N)=1$.
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Let~$W_1$ be the subspace of~$V_1$ generated by all differences
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$$[X^iY^{k-2-i}, (\lambda u,\lambda v)] -
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\eps(\lambda)[X^iY^{k-2-i}, (u, v)].$$
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Because all relations are two-term, it
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is easy to compute $V_2:=V_1/W_1$.
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In computing this quotient, we do not have to
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explicitly compute the {\em large} matrix representing
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the linear map $V_1\ra V_2$, as it can be replaced by a suitable
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``reduction procedure'' involving algebra over
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$\Z/N\Z$.
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\item
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Let~$\sigma$ act on Manin symbols as in Section~\ref{sec:maninsymbols}; thus
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$$[X^iY^{k-2-i}, (u, v)]\sigma = (-1)^i[Y^iX^{k-2-i}, (v,-u)].$$
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Let~$W_2$ be the subspace of~$V_2$ generated by the sums
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$x + x\sigma$ for $x\in V_2$.
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Because all relations are two-term relations, it is
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easy to compute $V_3 := V_2/W_2$.
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\item Let~$\tau$ act on Manin symbols as in
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Section~\ref{sec:maninsymbols};
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thus
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$$[X^iY^{k-2-i}, (u, v)]\tau =
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[(-Y)^i(X-Y)^{k-2-i}, (v,-u-v)].$$
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Note that the symbol on the right can be written as a sum
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of generating Manin symbols.
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Let~$W_3$ be the subspace of~$V_3$ generated by the sums
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$x + x\tau + x\tau^2$
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where~$x$ varies over the images of a basis of~$V_2$
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({\em not} just a basis for $V_3$!).
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Using some form of Gauss elimination, we compute $V_3/W_3$;
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we then have $\sM_k(N,\eps;K)\ncisom V_3/W_3$.
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\end{enumerate}
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\end{algorithm}
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\begin{proof}
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For $\lambda \in (\Z/N\Z)^*$, denote by $\langle \lambda \rangle$ the right
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action of~$\lambda$ on Manin symbols; thus
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$$[X^iY^{k-2-i}, (u,v)]\langle \lambda \rangle
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= [X^iY^{k-2-i}, (\lambda{}u,\lambda{}v)].$$
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By Theorem~\ref{thm:maninsymbols} the space $\sM_k(N,\eps;K)$ is isomorphic
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to the quotient of the vector spaces~$V_1$ of Step~1 modulo the relations
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$x+x\sigma=0$, $x+x\tau+x\tau^2=0$, and $x\langle \lambda \rangle=\lambda x$
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as~$x$ varies over all Manin symbols and~$\lambda$ varies over $(\Z/N\Z)^*$.
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As motivation, we note that a naive
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computation of $V_1$ modulo the $\sigma$, $\tau$, and
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$\langle \lambda\rangle$ relations using Gauss elimination
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is far too inefficient. This is why we compute the
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quotient in three steps. The complexity of Steps~1 and
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Steps~2 are negligible. The difficulty occurs in Step~3;
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at least the relations of this step occur in a space of dimension much
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smaller than that of~$V_1$.
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To see that the algorithm is correct, it is necessary only to observe
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that $\sigma$ and $\tau$ both commute with all diamond-bracket operators
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$\langle \lambda \rangle$; this is an immediate consequence of the
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above formulas. We remark that in Step~3 it is in
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general {\em necessary} to compute the quotient
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by all relations $x + x\tau + x\tau^2$ with~$x$ the image of a basis
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vector for $V_2$ instead
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of just~$x$ in~$V_3$ because~$\sigma$ and~$\tau$ do not commute.
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\end{proof}
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\begin{remark}
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In implementing the above algorithm, the reader should take care
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in Steps~1 and~2 because the relations can together force certain
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of the Manin symbols to equal~$0$.
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For example, there might be relations of the form $x_1+x_2=0$ and
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$x_1-x_2=0$ which together force $x_1=x_2=0$.
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\end{remark}
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\begin{remark}
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To compute the~$+1$ quotient
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$\sM_k(N,\eps;K)^{+}$, it
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is necessary to modify Step~2 of Algorithm~\ref{alg:MkNK}
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by including in~$W_2$ the differences $x - x I$
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where $I=\abcd{-1}{0}{\hfill 0}{1}$, and
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$$[X^iY^{k-2-i}, (u,v)] I
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= (-1)^i[X^iY^{k-2-i}, (-u,v)].$$
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Likewise, to compute the $-1$ quotient we include the sums $x + x I.$
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Note, as in the remarks in the proof of Algorithm~\ref{alg:MkNK},
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we can not add in the~$I$ relations in Step~1 because~$I$ and~$\sigma$
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do not commute.
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\end{remark}
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\begin{algorithm}\label{alg:MkNO}
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Given a positive integer~$N$, a Dirichlet character~$\eps$,
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and an integer $k\geq 2$,
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this algorithm computes the $\O$-modules $\sM_k(N,\eps)$
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and $\sS_k(N,\eps)$.
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\begin{enumerate}
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\item Using Algorithm~\ref{alg:MkNK}
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compute the $K$-vector space $V:=\sM_k(N,\eps;K)$.
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\item Compute the $\O$-lattice~$L$ in~$V$ generated
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by the classes of the finitely many symbols $[X^iY^{k-2-i}, (u,v)]$
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for $i=0,\ldots, k-2$ and $0\leq u,v < N$ with $\gcd(u,v,N)=1$.
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It is only necessary to take one symbol in each
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$\eps$-equivalence class, so there are
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$(k-2+1)\cdot\#\P^1(\Z/N\Z)$ generating symbols.
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This computes $\sM_k(N,\eps)$ which equals~$L$.
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\item To compute the submodule $\sS_k(N,\eps)$ of~$L$, we use
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the algorithm of Section~\ref{sec:computeboundary}
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to compute the
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boundary map $\delta:\sM_k(N,\eps;K)\ra B_k(N,\eps;K)$.
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Then $\sS_k(N,\eps)$ is the kernel of~$\delta$ restricted
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to the lattice~$L$.
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\end{enumerate}
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\end{algorithm}
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As a check, using the formulas of Section~\ref{sec:dimensionformulas},
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we compute the dimension of the space $S_k(N,\eps)$ of cusp
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forms and compare with the dimension of $\sS_k(N,\eps;K)$
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computed in Algorithm~\ref{alg:MkNO}.
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\begin{remark}
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[[To be removed.]] I have not thought through how to perform
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Steps 2 or 3 in case $\O\neq\Z$.
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In particular, I do not know how to compute
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the $\O$-module kernel of~$\delta$,
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except using a silly algorithm that reduces everything to working
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over~$\Z$.
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\end{remark}
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\section{Computing the Hecke algebra}
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\label{sec:computinghecke}
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In this chapter we give a formula for the dimension of
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$S_k(\Gamma_1(N),\eps)$ and an upper bound on the number
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of Hecke operators needed to generate the Hecke algebra
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as a $\Z$-module. We obtain the dimension formula by specializing
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Hijikata's generalization of the Eichler-Selberg trace formula
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to the case of the identity operator, and the bound on Hecke operators is obtained application of~\cite{sturm:cong}.
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Let~$\Gamma$ be a subgroup of $\SL_2(\Z)$ that contains
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$\Gamma_1(N)$ for some~$N$.
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Let $M_k(\Gamma)$ be the space of weight-$k$ modular
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forms for~$\Gamma$, and let $\T\subset\End(M_k(\Gamma))$
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be the corresponding Hecke algebra.
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This section contains a bound~$r$ such that
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the Hecke operators~$T_n$, with $n\leq r$, generate~$\T$ as
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a $\Z$-module.
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This result was suggested to the author by
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Ribet and Agash\'e.
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For any subring~$R$ of~$\C$,
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denote by $M_k(\Gamma;R)$ the space of modular forms
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for~$\Gamma$ with Fourier coefficients in~$R$.
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The following proposition is well-known.
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\begin{proposition}\label{prop:perfectpair}
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For any ring~$R$, there is a perfect pairing
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$$ \T_R\tensor_RM_k(N;R) \ra R,\qquad (T,f)\mapsto a_1(Tf),$$
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where $\T_R = \T\tensor_{\Z} R$.
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We have $(T_n,f)=a_n(f)$ where $T_n$ is the $n$th Hecke operator.
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\end{proposition}
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Let~$\mu$ denote the index of~$\Gamma$ in $\sltwoz$.
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\begin{theorem}[Sturm]
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\label{thm:sturm}
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Let~$\lambda$ be a prime ideal in the ring~$\O$
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of integers in some number field.
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If $f\in M_k(\Gamma;\O)$
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satisfies $a_n(f)\con 0\pmod{\lambda}$
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for $n\leq \frac{k}{12}\mu$, then $f\con 0\pmod{\lambda}$.
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\end{theorem}
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\begin{proof}
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Theorem 1 of \cite{sturm:cong}.
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\end{proof}
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Denote by $\lceil{}x\rceil$ the smallest integer $\geq x$.
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\begin{proposition}\label{prop:determine}
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If $f\in M_k(\Gamma)$ satisfies
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$a_n(f)=0$ for
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$n\leq r=\left\lceil\frac{k}{12}\mu\right\rceil$,
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then $f=0$.
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\end{proposition}
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\begin{proof}
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We must show that the composite map
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$M_k(\Gamma)\hookrightarrow\C[[q]]\into\C[[q]]/(q^{r+1})$
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is injective. Because~$\C$ is a flat $\Z$-module and
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$M_k(\Gamma;\Z)\tensor\C = M_k(\Gamma)$, it suffices
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to show that the map $F:M_k(\Gamma;\Z)\into\Z[[q]]/(q^{r+1})$ is injective.
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Suppose $F(f)=0$, and let~$p$ be a prime number.
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Then $a_n(f)=0$ for $n\leq r$, hence plainly
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$a_n(f)\con 0\pmod{p}$ for any such~$n$.
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Theorem~\ref{thm:sturm} implies that $f\con 0\pmod{p}$.
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Duplicating this argument shows that the coefficients
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of~$f$ are divisible by all primes~$p$, so they are~$0$.
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\end{proof}
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\begin{theorem}\label{thm:bound}
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As a $\Z$-module, $\T$ is generated by $T_1,\ldots,T_r$,
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where $r=\lceil \frac{k}{12}\mu \rceil $.
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\end{theorem}
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\begin{proof}
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Let~$Z$ be the submodule of~$\T$ generated by
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$T_1,T_2,\ldots,T_r$. Consider the exact sequence of
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additive abelian groups
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$0\into Z \xrightarrow{\,i\,} \T \into \T/Z \into 0.$
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Let~$p$ be a prime and tensor this sequence with~$\F_p$ to obtain
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the exact sequence
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$$Z\tensor \F_p\xrightarrow{\,\overline{i}\,}
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\T\tensor\F_p \into (\T/Z)\tensor\F_p\into 0.$$
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Put $R=\Fp$ in Proposition~\ref{prop:perfectpair}, and suppose that
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$f\in M_k(N,\Fp)$ pairs to~$0$ with each of $T_1,\ldots, T_r$.
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Then by Proposition~\ref{prop:perfectpair}, $a_m(f)=a_1(T_m f)=0$ in
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$\Fp$ for each~$m$, $1\leq m\leq r$. Theorem~\ref{thm:sturm}
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then asserts that $f = 0$. Thus the pairing, when restricted
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to the image of $Z\tensor\Fp$ in $\T\tensor\Fp$, is also perfect. Thus
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$\dim_{\Fp} \overline{i}(Z\tensor\Fp)
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= \dim_{\Fp} M_k(N,\Fp)= \dim_{\Fp} \T\tensor\Fp,$
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so $(\T/Z) \tensor \F_p = 0$; repeating this argument for
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all~$p$ shows that $\T/Z=0$.
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\end{proof}
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% dirichlet.tex
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\section{Representing and enumerating Dirichlet characters}
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A Dirichlet character is a homomorphism
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$\eps:(\Z/N\Z)^*\ra \C^*$. We have the
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following lemma, whose proof is well-known.
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\begin{lemma}
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If~$p$ is an odd prime, then $(\Z/p^n\Z)^*$ is a cyclic group.
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The group $(\Z/2^n\Z)^*$ is generated by~$-1$ and~$5$.
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\end{lemma}
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It is necessary to agree upon a representation of Dirichlet characters.
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Factor~$N$ as a product of prime powers: $N=\prod_{i=1}^r p_i^{e_i}$ with
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$p_i < p_{i+1}$ and each $e_i>0$; then
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$(\Z/N\Z)^* \isom \prod_{i=1}^r (\Z/p_i^{e_i}\Z)^*$.
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If $p_i$ is odd then the lemma implies that $(\Z/p_i^{e_i}\Z)^*$ is cyclic.
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If $p_1=2$, then $(\Z/p_1^{e_1}\Z)^*$ is a product
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$\langle -1 \rangle \cross \langle 5 \rangle$
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of two cyclic groups, both possibly trivial.
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For each~$i$, we let $a_i$ be the smallest generator of
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the $i$th factor $(\Z/p_i^{e_i}\Z)^*$. If $p_1=2$, let $a_1$ and $a_2$
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correspond to the two factors $\langle -1 \rangle$ and
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$\langle 5 \rangle$, respectively; then $a_3$ corresponds to $p_2$, etc.
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Here~$a_i$ is smallest in the
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sense that the minimal lift $\tilde{a}_i\in\Z_{>0}$ is smallest.
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Let~$n$ be the exponent of $(\Z/N\Z)^*$, and
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let $\zeta=e^{2\pi i /n}\in \C^*$.
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To give~$\eps$ is the same as giving the images of each generator
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of~$a_i$ as a power of~$\zeta$. We thus represent~$\eps$ as
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a matrix with respect to a canonically chosen, but unnatural,
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basis.
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The matrix representing a character~$\eps$ can be viewed as a vector with~$r$
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entries in $\Z/n\Z$, where~$m$ is the exponent of $(\Z/N\Z)^*$.
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Such a vector represents a character if and only if the $i$th component
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of the vector has additive order
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dividing $\vphi(p_i^{e_i})$. If $p_1=2$, then
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there are $r+1$ entries instead of~$r$ entries, and the condition
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is suitably modified.
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If a vector $v=[d_1,\ldots,d_r]$ represents a character~$\eps$,
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then each of the Galois conjugate characters is represented
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by $[md_1,\ldots,md_r]$ where~$m$ runs over $(\Z/n\Z)^*$.
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When performing actual machine computations, we work in the smallest
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field that contains all of the values of~$\eps$.
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Thus if $d=\gcd(d_1,\ldots,d_r,n)$, then we work in the subfield
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$\Q(\zeta^d)$, which is cheaper than working in $\Q(\zeta)$.
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It is sometimes important to work in characteristic~$\ell$.
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Then the notation is as above, except~$\zeta$ is replaced by a
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primitive $n'$th root of unity, where~$n'$ is the prime-to-$\ell$
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part of~$n$. Note that the primitive
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$n$th roots of unity in characteristic~$\ell$ need not be conjugate;
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for example, both~$2$ and~$3$ are square roots of~$-1$ in $\F_5$, but
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they are not conjugate. Thus we must specify~$\zeta$ as part
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of the notation when giving a mod~$\ell$ Dirichlet character.
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%In all cases, we extend~$\eps$ to a set-theoretic
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%mapping from~$\Z$ by setting $\eps(x)=0$ if $\gcd(x,N)\neq 1$.
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\begin{example}
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Suppose~$p$ is an odd prime.
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The group of mod~$p$ Dirichlet characters is isomorphic to
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$\Z/(p-1)\Z$, and two characters~$a$ and~$b$ are Galois
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conjugate if and only if there is an element $x\in(\Z/(p-1)\Z)^*$
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such that $xa=b$.
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A character is determined up to Galois conjugacy by its order,
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so the classes of mod~$p$ Dirichlet characters are
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in bijection with the divisors~$d$ of $p-1=\#(\Z/p\Z)^*$.
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The quadratic mod~$p$ character is denoted $[(p-1)/2]$.
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We denote the quadratic mod~$2p$ character by
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$[0,0,(p-1)/2]$; the quadratic mod~$4p$ character is denoted
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$[(p-1)/2,0,(p-1)/2]$. If $n\geq 2$, then the nontrivial
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mod~$2^n$ character that factors through $(\Z/4\Z)^*$ is denoted
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$[(2^{n-2}-1)/2,0]$.
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\end{example}
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\section{The dimension of $S_k(N,\eps)$}
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\label{sec:dimensionformulas}
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Explicit formulas for $\dim S_k(\Gamma)$,
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with~$\Gamma$ a congruence subgroup, were given by
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Shimura in \cite[Thms.~2.23--2.25]{shimura:intro}. I don't know
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if these methods generalize to give a formula for
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$\dim S_k(N,\eps)$. However, an extremely tedious specialization of
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Hijikata's trace formula (see \cite{hijikata:trace}) to the case $n=1$
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yields an explicit formula for $\tr(T_1)=\dim S_k(N,\eps)$; source
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code is available from the author.
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\section{Computing a $\Z$-basis of $q$-expansions}
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\label{sec:intbasis}
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Assume that $\eps^2=1$.
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To compute $S_k(N,\eps;\Z)$, first use modular symbols and the Sturm bound
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to compute a $\Q$-basis of $q$-expansions for $S_k(N,\eps;\Q)$.
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Construct a matrix~$A$ whose rows are the coefficients of the $q$-expansions.
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Let~$B$ be a matrix whose columns form a basis for the right kernel of~$A$.
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Then a basis for $S_k(N,\eps;\Z)$ is obtained by computing
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a basis for the integer kernel of the transpose of~$B$,
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which can be computed using standard algorithms. For example, we might
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have $A=[0,2]$; then $B=\begin{smallmatrix}1\\0\end{smallmatrix}$, and
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the integer kernel of the transposes of~$B$ is spanned by $[0,1]$.
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\section{Decomposing the space of modular symbols}
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\label{sec:decomposemodsym}
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Let $\sM_k(N,\eps)$ be the space of modular symbols
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of level $N$ and character $\eps$ over $K=\Q(\eps)$.
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In this section we describe how to decompose the
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new part of $\sM_k(N,\eps)$ as a direct sum of $\T$-modules
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corresponding to the Galois conjugacy classes
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of newforms with character $\eps$.
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As an application, we can compute the $q$-expansions
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of the normalized cuspidal newforms of level $N$ and
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character $\eps$. Using the theory of
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Atkin-Lehner~\cite{atkin-lehner}, it is
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then possible to construct a basis for $S$.
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The algorithm is, for the most part, a
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straightforward generalization of the method used
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by Cremona \cite{cremona:algs}
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to enumerate the $\Q$-rational weight two newforms
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corresponding to modular elliptic curves.
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Nevertheless, we present several nonobvious tricks which
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we learned in the course of doing computation and
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which greatly speed up the algorithm.
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One key idea is to work in the space dual to
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modular symbols as described in the next section.
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\subsection{Duality}
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Let $\sM_k(N,\eps)^\dual$ denote $\Hom(\sM_k(N,\eps),K)$
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equipped with its natural right $\T$-action:
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$$(\vphi T)(x) = \vphi(Tx).$$
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The natural pairing
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\begin{equation}\label{eqn:pairing}
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\langle\,,\,\rangle:\sM_k(N,\eps)^\dual \cross \sM_k(N,\eps) \ra K
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\end{equation}
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given by $\langle\vphi,x\rangle = \vphi(x)$
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satisfies
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$\langle\vphi{}T,x\rangle = \langle\vphi ,T{}x\rangle$.
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Viewing the elements $T\in\T$ as sitting inside
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$\sM_k(N,\eps)$,
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the transpose map $T\mapsto T^t$
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allows us to view $\sM_k(N,\eps)^\dual$
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as a left $\T$-module.
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\begin{proposition}\label{prop:heckeduality}
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Let $V\subset \sM_k(N,\eps)^{\new}$ be an irreducible
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new $\T$-submodule and set $I=\Ann_\T V$.
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Then $\sM_k(N,\eps)^{\dual}[I]$ is isomorphic to~$V$
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as a $\T\tensor\Q$-module.
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\end{proposition}
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The degeneracy maps $\alp_t$ and $\beta_t$ of
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Section~\ref{sec:degeneracymaps} give rise to
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maps $\alp_t^{\dual}$ and $\beta_t^{\dual}$
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between the dual spaces and having
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the dual properties to those of $\alp_t$ and $\beta_t$.
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In particular, they commute with the Hecke operators
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$T_p$ for $p$ prime to $N$.
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The new and old subspace of $\sM_k(N,\eps)^\dual$
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are defined as in Definition~\ref{def:newandoldsymbols}.
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\begin{algorithm}\label{alg:decompmknew}
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This algorithm computes a decomposition of $\sM_k(N,\eps)^{\dual\new}$.
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Using Algorithm~\ref{alg:MkNK} compute $\sM_k(N,\eps)$.
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Then compute the maps $\beta_t$ using Algorithm~\ref{alg:degenreps}
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and intersect the transposes of their kernels in order to
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obtain $\sM_k(N,\eps)^{\dual\new}$.
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Compute the boundary map $\delta:\sM_k(N,\eps)\ra B_k(N,\eps)$
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using Algorithm~\ref{alg:cusplist}. Using the Hecke operators,
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Algorithm~\ref{alg:efficienttpdual}, and
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Proposition~\ref{prop:heckeduality},
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cut out the cuspidal submodule $\sS_k(N,\eps)^{\dual\new}$.
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Set $p=2$ and perform the following steps.
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\begin{enumerate}
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\item Using Algorithm~\ref{alg:efficienttpdual},
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compute a matrix $A$ representing the Hecke operator $T_p$
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on $\sM_k(N,\eps)^{\dual\new}$.
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\item Compute and factor the characteristic polynomial $F$ of $A$.
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\item For each irreducible factor $f$ of $F$
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compute $V_f = \ker(f(A))$. Use the following criteria
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to determine if $V_f$ is irreducible:
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\begin{enumerate}
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\item If~$p$ is greater than the Sturm bound
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(see Theorem~\ref{thm:bound}) then $V_f$
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must be irreducible.
435
\item If some element $T\in \T$ has
436
characteristic polynomial on $V_f$ the
437
square of an irreducible polynomial, then $V_f$ is irreducible.
438
Determine if $V_f$ is cuspidal by checking
439
if $V_f$ is contained in $\sS_k(N,\eps)^{\dual\new}$
440
computed above.
441
If $V_f$ is cuspidal and some element $T\in \T$ has characteristic
442
polynomial on $V_f$ the square of an irreducible polynomial,
443
then $V_f$ is irreducible.
444
\end{enumerate}
445
\item If $V_f$ is irreducible, record $V_f$ in a list and continue
446
with the next factor of the characteristic polynomial. If
447
we can not show that $V_f$ is irreducible, repeat step 1 with
448
$\sM_k(N,\eps)^{\dual\new}$ replaced by $V_f$.
449
\end{enumerate}
450
\end{algorithm}
451
452
453
\subsection{Efficient computation of Hecke operators on the dual space}
454
In this section we give a method for computing the action
455
of the Hecke operators $T_p$ on an invariant subspace
456
$V\subset \sM_k(N,\eps)^{\dual}$.
457
A naive way to compute the right action of $T_p$ on $V$
458
is to compute a matrix representing $T_p$ on $\sM_k(N,\eps)$,
459
transpose to obtain $T_p$ on $\sM_k(N,\eps)^\dual$,
460
and then restrict to $V$ using Gaussian elimination.
461
To compute $T_p$ on $\sM_k(N,\eps)$, observe that $\sM_k(N,\eps)$
462
has a basis $e_1,\!\ldots,e_n$, where each $e_i$
463
is a Manin symbol $[P,(c,d)]$, and that the action
464
of $T_p$ on $[P,(c,d)]$ can be computed using
465
Section~\ref{subsec:heckeonmanin}.
466
In practice, $d=\dim V$ will often be much less than $n$;
467
it is then possible to compute $T_p$ on $V$ in $d/n$ of the
468
time it takes using the above naive method. This is a
469
substantial savings when $d$ is small.
470
Transposing the injection
471
$V\hookrightarrow \sM_k(N,\eps)^{\dual}$
472
we obtain a surjection $\sM_k(N,\eps)\ra V^\dual$.
473
There exists a subset $e_{i_1},\!\ldots, e_{i_d}$
474
of the $e_i$ whose image forms
475
a basis for $V^\dual$. With some care, it is then possible
476
to compute $T_p$ on $V^{\dual}$ by computing $T_p$ on each of
477
$e_{i_1},\!\ldots, e_{i_d}$. We describe in detail below
478
a definite way to carry out this computation using matrices.
479
480
Let $V$ be an $n\times m$ matrix whose rows
481
generate an $n$-dimensional subspace of an $m$-dimensional space of
482
row vectors. Let $T$ be an $m\times m$-matrix and
483
suppose that $V$ has rank $n$ and that $VT$ is contained
484
in the row space of $V$.
485
Let $E$ be an $m\times n$ matrix with the property that the
486
$n\times n$ matrix $VE$ is invertible, with inverse $D$.
487
\begin{proposition}
488
$VT = VTEDV.$
489
\end{proposition}
490
\begin{proof}
491
Observe that
492
$$V(EDV) = (VED)V = IV = V.$$
493
Thus right multiplication by $EDV$
494
$$ v \mapsto vEDV$$
495
induces the {\em identity map} on the row space of $V$.
496
Since $VT$ is contained in the row space of $V$, we have
497
$$(VT)EDV = VT,$$
498
as claimed.
499
\end{proof}
500
501
We have not computed $T$, but we can
502
compute $T$ on each basis element $e_1,\ldots,e_d$
503
of the ambient space--unfortunately, $d$ is extremely large.
504
Our problem: quickly compute the action of
505
$T^t$ on the invariant subspace spanned
506
by the rows of $V$. Can this be done without
507
having to compute $T$ on all $e_i$?
508
Yes, the following algorithm shows how using
509
a subset of only $n=\dim V$ of the $e_i$.
510
This results in a tremendous savings;
511
usually $\dim V$ is small and $d$ is large.
512
513
\begin{algorithm}\label{alg:efficienttpdual}
514
Let $T$ be any linear transformation [[of the ambient space]]
515
which leaves $V$ invariant and for
516
which we can compute $T(e_i)$ for $i=1,\ldots, d$.
517
This algorithm computes the matrix representing the
518
action of $T$ on $V$ while computing $T(e_i)$ for
519
only $\dim V$ of the~$i$.
520
521
Choose any $m\times n$ matrix $E$ whose columns are
522
sparse linear combinations of the $e_i$ and such that
523
$VE$ is invertible.
524
For this we find a set of positions so that elements of the space
525
spanned by the columns of $V$ are determined by the entries in
526
these spots. This is accomplished by row reducing, and setting $E$
527
equal to the pivot columns.
528
Using Gauss-elimination compute the inverse $D$ of the
529
$n\times n$ matrix $VE$.
530
The matrix representing the action of $T$ with respect
531
to $V$ is then
532
$$V(TE)D=V(TE)(VE)^{-1}.$$
533
\end{algorithm}
534
\begin{proof}
535
Let $A$ be any matrix so that
536
$VA$ is the $n\times n$ identity
537
matrix.
538
By the proposition we have
539
$$VTA = (VTEDV)A = VTED(VA) = VTED = V(TE)D.$$
540
To see that $VTA$ represents $T$,
541
observe that by the proposition,
542
\begin{eqnarray*}
543
VTAV &=& (VTEDV)AV=(VTEDV\!A)V\\
544
&=& (VTED)(V\!A)V=(VTED)V=VT
545
\end{eqnarray*}
546
so that $VTA$ gives the correct linear combination of the rows of $V$.
547
\end{proof}
548
549
\comment{
550
Compute $TE$, then multiply on the left by $V$ and on the right
551
by $DV$ to obtain $VT = V(TE)DV$.
552
Compute a right inverse $R$ of $V$ as follows: If the reduced
553
row echelon form of the augmented matrix $[V'\,|\,I]$ is $[I\,|\,A]$,
554
then $AV'=I$, where $I$ is the $n\times n$ identity matrix.
555
Taking the transpose reveals that $VA'=I$ and so we take $R=A'$.}
556
557
\subsection{Eigenvectors}\label{sec:eigenvector}
558
Once a $\T$-simple subspace of $\sM^*$ has been identified, the
559
following algorithm can be used to write down an eigenvector
560
defined over an extension of the base field.
561
562
\begin{algorithm}
563
Let $A$ be an $n\times n$ matrix over a field $K$ and
564
suppose that the characteristic polynomial $f(x)=x^n+\cdots+a_1 x + a_0$
565
of $A$ is irreducible. Let $\alpha$ be a root of $f(x)$
566
in an algebraic closure $\Kbar$ of $K$.
567
Factor $f(x)$ over $K(\alp)$ as
568
$f(x) = (x-\alp) g(x)$.
569
Then for any randomly chosen $v\in K^n$ the vector
570
$g(A)v$ is an eigenvector of $A$ with eigenvalue $\alp$.
571
Compute the vector $g(A)v$ by computing
572
$Av$, $A(Av)$, $A(A(Av))$ and then summing using that
573
$$g(x)=x^{n-1}+c_{n-2} x^{n-2}+\cdots+c_1 x+ c_0$$
574
where the coefficients $c_i$ are determined by the recurrence
575
$$c_0 = - a_0/\alp,\qquad c_i = (c_{i-1}-a_i)/\alp.$$
576
\end{algorithm}
577
\begin{proof}
578
By the Cayley-Hamilton theorem \cite[XIV.3]{lang:algebra}
579
we have that $f(A)=0$. Consequently, for any $v\in K^n$,
580
we have $(A-\alp)g(A)v=0$ so that $A g(A)v = \alp v$.
581
Since $f$ is irreducible it is the polynomial of least
582
degree satisfied by $A$ and so $g(A)\neq 0$.
583
Therefore $g(A)v\neq 0$ for all $v$ not in the proper
584
closed subset $\ker(g(A))$.
585
\end{proof}
586
587
\subsection{Eigenvalues}\label{sec:eigenvalues}
588
In this section we give an algorithm for
589
computing the $q$-expansion of one of the newforms
590
corresponding to a factor of $\sM_k(N,\eps)^{\new}$.
591
This is a generalization of the algorithm described
592
in \cite[\S2.9]{cremona:algs}.
593
594
\begin{algorithm}\label{alg:eigenvalues}
595
Given a factor $V\subset \sS_k(N,\eps)^{\dual\new}$
596
as computed by Algorithm~\ref{alg:decompmknew}
597
this algorithm computes the $q$-expansion of
598
one of the corresponding Galois conjugate newforms.
599
\begin{enumerate}
600
\item Using Algorithm~\ref{alg:efficienttpdual} compute
601
the action of the $*$-involution (Section~\ref{sec:heckeops})
602
on $V$. Then compute the $+1$ eigenspace $V^+\subset V$.
603
\item Find a Hecke operator $T\in\T$ such that the
604
characteristic polynomial of the matrix $A$
605
of $T$ acting on $V^+$ is irreducible.
606
Such a Hecke operator $T$ must exist by the primitive element theorem
607
\cite[V.4]{lang:algebra}.
608
\item Using Algorithm~\ref{sec:eigenvector} compute
609
an eigenvector $e$ for $A$ over an extension of $K$.
610
\item Because $e$ is an eigenvector and the pairing
611
given in Equation~\ref{eqn:pairing} respects the
612
Hecke action, we have that for any Hecke operator
613
$T_n$ and element $w\in \sM_k(N,\eps)$, that
614
$$a_n \langle e, w \rangle
615
= \langle e T_n, w\rangle
616
= \langle e, T_n w \rangle.$$
617
Choose $w$ so that 1) $w$ is a freely generating
618
Manin symbols (see Algorithm~\ref{alg:MkNK})
619
and 2) $\langle e,w\rangle \ne 0$. Then
620
$$a_n =\frac {\langle e, T_n w \rangle}
621
{\langle e, w \rangle}.$$
622
The $a_n$ can now be computed by
623
computing $\langle e, w \rangle$ once and for all,
624
and then computing $\langle e, T_n w \rangle$ for
625
each $n$.
626
\end{enumerate}
627
\end{algorithm}
628
The beauty of this algorithm is that when $w$
629
is a freely generating Manin symbol the computation
630
of $T_p w = \sum_{x\in R_p} wx$ is very quick, requiring
631
only summing over the Heilbronn matrices of determinant
632
$p$ {\em once}.
633
634
In practice we compute only the eigenvalues $a_p$ using
635
the above algorithm, then use the following
636
recurences to obtain the $a_n$:
637
\begin{eqnarray*}
638
a_{nm} &=& a_n a_m \qquad\text{if $(n,m)=1$, and}\\
639
a_{p^r}&=& a_{p^{r-1}}a_p - \eps(p) p^{k-2} a_{p^{r-2}}.
640
\end{eqnarray*}
641
642
643
\subsection{Sorting and labeling eigenforms}
644
In Section~\ref{sec:eigenvalues} we saw how to associate
645
to each new factor a sequence $a_n$ of Hecke eigenvalues.
646
These can be used to sort and hence label the factors.
647
This is essential so that we can refer to the factors in our
648
explicit investigations.
649
650
Except in the case of weight~$2$ and trivial character,
651
we use the following ordering. To each eigenvector
652
associate the following sequence of integers
653
$$\tr(a_1), \tr(a_2), \,\tr(a_3),\, \tr(a_5),\, \ldots$$
654
where the trace is from $K[f]$ down to $\Q$.
655
Sort the eigenforms by ordering the
656
sequences in dictionary order with
657
minus coming before plus.
658
Note that we included $\tr(a_1)$ so that
659
this ordering would gather together factors of the same dimension.
660
661
When $k=2$ and the character is trivial we
662
use a different and somewhat complicated ordering
663
because it extends the notation for elliptic curves
664
that was introduced in the second edition of \cite{cremona:algs}
665
and has since become standard.
666
Sort the factors of $\sS_k(N,\eps)^{\new}$ as follows.
667
First by dimension, with smallest dimension first.
668
Within each dimension, sort in binary order,
669
by the signs of the Atkin-Lehner involutions
670
with - corresponding to 0 and + to 1, thus if there
671
are three Atkin-Lehner involutions then the sign patterns
672
are sorted as follows:
673
$$+++, -++, +-+, --+, ++-, -+-, +--, ---.$$
674
Finally, let $p$ be the smallest prime not dividing $N$.
675
Within each of the Atkin-Lehner classes, sort by
676
the magnitudes of the $K(f)/\Q$-trace of
677
$a_p$ breaking ties by letting the positive trace be first.
678
If there are still any ties, repeat the final step with the
679
next smallest prime not dividing $N$, etc.
680
681
682
683
\section{Intersections and congruences}
684
685
Consider a complex torus $J=V/\Lambda$, and let
686
$A=V_A/\Lambda_A$ and $B=V_B/\Lambda_B$ be subtori whose
687
intersection $A\intersect B$ is finite.
688
\begin{proposition}\label{prop:intersection}
689
There is a natural isomorphism of groups
690
$$A\intersect B \isom
691
\left(\frac{\Lambda}{\Lambda_A + \Lambda_B}\right)_{\tor}.$$
692
\end{proposition}
693
\begin{proof}
694
There is an exact sequence
695
$$0\ra A\intersect B \ra A \oplus B \ra J.$$
696
Consider the diagram
697
$$\xymatrix{
698
& {\Lambda_A \oplus\Lambda_B}\ar[d] \ar[r] & {\Lambda} \ar[r]\ar[d]&
699
{\Lambda/(\Lambda_A + \Lambda_B)}\ar[d]\\
700
& {V_A \oplus V_B}\ar[d] \ar[r] & V \ar[r]\ar[d] & {V/(V_A+V_B)}\ar[d]\\
701
{A\intersect B}\ar[r] & A\oplus B\ar[r] & J \ar[r] & J/(A+ B)}$$
702
The snake lemma gives an exact sequence
703
$$0 \ra
704
A\intersect B \ra
705
\Lambda/(\Lambda_A + \Lambda_B) \ra
706
V/(V_A+V_B).$$
707
Since $V/(V_A+V_B)$ is a $\C$-vector space, the torsion
708
part of $\Lambda/(\Lambda_A + \Lambda_B)$ must map to~$0$.
709
No nontorsion in $\Lambda/(\Lambda_A + \Lambda_B)$ could
710
map to~$0$, because if it did then $A\intersect B$ wouldn't
711
be finite. The lemma follows.
712
\end{proof}
713
714
The following formula for the intersection of~$n$ subtori is obtained
715
in a similar way.
716
\begin{proposition}
717
For $i=1,\ldots,n$ let $A_i = V_i/\Lambda_i$ be a subtorus of
718
$J=V/\Lambda$, and assume that each pairwise intersection
719
$A_i \intersect A_j$ is finite.
720
Then
721
$$A_1\intersect \cdots \intersect A_n
722
\isom
723
\left(\frac{\Lambda\oplus \cdots \oplus \Lambda}
724
{f(\Lambda_1\oplus\cdots\oplus \Lambda_n)}\right)$$
725
where $f(x_1,\ldots,x_n)=(x_1-x_2,x_2-x_3,x_3-x_4,\ldots,x_{n-1}-x_n)$.
726
\end{proposition}
727
728
\begin{remark}
729
Using this proposition the author constructed the
730
t-shirt design in Figure~\ref{cap:tshirt}.
731
\begin{figure}
732
\begin{center}
733
\includegraphics{shirt.eps}
734
\end{center}
735
\caption{T-shirt design}
736
\label{cap:tshirt}
737
\end{figure}
738
\end{remark}
739
740
741
Let~$N$ be a positive integer, $k\geq 2$ an integer, and~$\eps$
742
a mod~$N$ Dirichlet character.
743
Suppose~$f$ and~$g$ are newforms in $S_k(N,\eps;\Qbar)$.
744
The following proposition gives rise to an algorithm for
745
computing congruences between infinite Fourier expansions;
746
the key advantage of the algorithm is that it only
747
involves finite exact computations and avoids any need
748
to compute $q$-expansions.
749
\begin{proposition}\label{prop:degcong}
750
Suppose~$f$ and~$g$ are newforms in $S_k(N,\eps;\Qbar)$.
751
Let $I_f$ and $I_g$ be the corresponding annihilators in the Hecke
752
algebra~$\T$.
753
Let $\Lambda = \sS_k(N,\eps;\O)$, and set
754
$\Lambda_f=\Lambda[I_f]$
755
and $\Lambda_g=\Lambda[I_g]$.
756
If $p\mid \#\left(\frac{\Lambda}{\Lambda_f + \Lambda_g}\right)_{\tor}$
757
then there is a prime~$\wp$ of residue characteristic~$p$ such
758
that $f\con g \pmod{\wp}$.
759
\end{proposition}
760
\begin{proof}
761
Consider the exact sequence
762
$$0 \ra \Lambda_f \oplus \Lambda_g \ra \Lambda \ra
763
\Lambda/(\Lambda_f+\Lambda_g) \ra 0$$
764
where the first map is $(a,b)\mapsto a-b$.
765
Upon tensoring this sequence with~$\F_p$ we obtain:
766
$$Z
767
\ra (\Lambda_f \tensor\F_p) \oplus (\Lambda_g \tensor\F_p)
768
\ra \Lambda\tensor\F_p \ra (\Lambda/(\Lambda_f+\Lambda_g))\tensor\F_p\ra 0,$$
769
where $Z=\Tor_1(\Lambda/(\Lambda_f+\Lambda_g),\F_p)$.
770
Denote by $\im(\Lambda_f)$ the image of $\Lambda_f\tensor\F_p$
771
in $\Lambda\tensor\F_p$
772
and likewise for $\Lambda_g$.
773
Our assumption that~$p$ divides the torsion part
774
of $\Lambda/(\Lambda_f+\Lambda_g)$ implies that~$Z$
775
is nonzero, so $\im(\Lambda_f)$ and $\im(\Lambda_g)$
776
have nonzero intersection inside the $\F_p$-vector
777
space $\Lambda\tensor\F_p$.
778
The Hecke algebra~$\T$ acts on $\im(\Lambda_f)$ is through
779
its action on~$f$, that is, through the quotient $\T/I_f$;
780
similarly,~$\T$ acts on $\im(\Lambda_g)$ through $\T/I_g$.
781
Thus~$\T$ acts on the nonzero $\T\tensor\F_p$-module
782
$\im(\Lambda_f)\intersect \im(\Lambda_g)$
783
through $\T/(I_f+I_g+p)$. This implies that
784
$I_f+I_g+p$ is not the unit ideal, which is equivalent
785
to the assertion of the proposition.
786
\end{proof}
787
788
\comment{ %HEY, AMOD'S ALREADY DONE IT...
789
The method of proof can also be used to show that
790
``if a prime divides the modular degree, then it is
791
a congruence prime'' (see Ribet's proof
792
in~\cite[\S5]{zagier:parametrizations}). Here's
793
how the argument goes, with a slight generalization
794
to arbitrary weight. Let~$f$ be a weight-$k$ newform
795
on $\Gamma_0(N)$ or $\Gamma_1(N)$,
796
and denote by~$A$ the corresponding abelian
797
subvariety of the appropriate self-dual abelian variety $J$.
798
For example, if~$k$ is~$2$, then~$J$ is $J_0(N)$ or $J_1(N)$.
799
We have the following diagram of abelian varieties (over~$\C$):
800
$$\xymatrix{
801
& B\ar[d]\\
802
*++{A}\[email protected]{^(->}[r]\ar[dr]^{\vphi}& J\ar[d]\\
803
& {\Adual}}$$
804
where~$B$ is the kernel of the natural map $J\ra \Adual$.
805
If $p\mid \deg(\vphi)$, then
806
Proposition~\ref{prop:intersection}
807
implies that~$p$ also divides
808
$\#(\Lambda/(\Lambda_A+\Lambda_B))_{\tor}$.
809
Arguing as in the proof of Proposition~\ref{prop:degcong},
810
we see that $I_A+I_B+p$ is not the unit ideal in~$\T$.
811
There is a perfect pairing between $S_k=S_k(N;\Z)$ and~$\T$,
812
which [[I HOPE!]] induces a non-canonical isomorphism
813
$$\frac{S_k}{S_k[I_A]+ S_k[I_B]}
814
\isom
815
\frac{\T}{I_A + I_B}.$$
816
Thus~$r$, which equals the order of the left hand quotient,
817
is also equal to $\T/(I_A + I_B)$. Since $I_A+I_B+p$ is
818
not the unit ideal, it follows that $p\mid r$.
819
820
\begin{selfnote}
821
FIX THE LAST BIT!!!!!!!!!!!!
822
It seems Amod has given the weight~$2$ proof of
823
this in his note on the Manin constant.
824
Mention Amod's results on this question.
825
\end{selfnote}
826
}
827
828
829
% ratperiod.tex
830
\section{The rational period mapping}
831
\label{sec:ratperiod}
832
833
Consider a triple $(N,k,\eps)$, and let $K=\Q[\eps]$.
834
Let~$I$ be an ideal in the Hecke algebra~$\T$ associated to
835
$(N,k,\eps)$.
836
The rational period mapping associated
837
to~$I$ is a map from the space $\sM_k(N,\eps;K)$ of
838
modular symbols to a finite dimensional $K$-vector space.
839
It is a computable analogue of the classical integration
840
pairing, and is of great value in extracting the rational parts
841
of analytic invariants; e.g., of special values of $L$-functions.
842
In the next section we use it to computing the
843
image of cuspidal points on $J(N,k,\eps)$.
844
845
\begin{definition}
846
Let $D:=\Hom(\sM_k(N,\eps;K),K)[I]$; the
847
\defn{rational period mapping} is the natural
848
quotient map
849
$$\Theta_I : \sM_k(N,\eps;K) \ra \frac{\sM_k(N,\eps;K)}
850
{\bigcap\, \{\ker(\vphi) : \vphi \in D\}}.$$
851
\end{definition}
852
853
\begin{algorithm}\label{alg:ratperiod}
854
This algorithm computes $\Phi_I$.
855
Choose a basis for $W=\sM_k(N,\eps;K)$ and use it
856
to view~$W$ as a space of column vectors equipped with
857
a left action of~$\T$.
858
View $W^*=\Hom(\sM_k(N,\eps;K),K)$ as the
859
space of row vectors of length equal to $\dim \sM_k(N,\eps;K)$;
860
thus~$W^*$ is dual to~$W$ via the natural pairing between
861
row and column vectors. The Hecke operators acts
862
on $W^*$ on the right.
863
Compute a basis $\vphi_1,\ldots,\vphi_{n}$ for
864
the $K$-vector space $W^*[I]$.
865
Then the rational period mapping with respect to
866
this basis is $\vphi_1\cross \cdots \cross \vphi_n$;
867
it is given by the matrix whose rows
868
are $\vphi_1,\ldots,\vphi_n$.
869
\end{algorithm}
870
\begin{proof}
871
The kernels of $\vphi_1\cross \cdots \cross \vphi_n$
872
and $\Phi_I$ are the same.
873
\end{proof}
874
875
876
\begin{example}\label{ex:ratperiod1}
877
Let~$I$ be the annihilator of the newform $f=q-2q^2+\cdots \in M_2(37,1;\Q)$
878
corresponding to the elliptic curve {\bf 37k2A}.
879
There is a basis for $W=\sM_2(37,1;\Q)$ such that
880
$$T_2 = \left(\begin{matrix}
881
-1& 1 &1&-1& 0\\
882
1&-1& 1& 0& 0\\
883
0& 0&-2& 1& 0\\
884
0& 0& 0& 0& 0\\
885
0& 0& 0& 1& 3\\
886
\end{matrix}\right)$$
887
The characteristic polynomial of $T_2$ is $x^2(x+2)^2(x-3)$.
888
Thus $W[I]=\ker(T_2+2)$ is spanned by the column
889
vectors $(1,-1,0,1/2,0)^t$ and
890
$(0,0,1,-1/2,0)^t$, and $W^*[I]=\ker(T_2^{t}+2)$ is spanned by the row vectors
891
$(1,0,-1,0,0)$ and $(0,1,-1,0,0)$. The rational period mapping
892
is $\Phi_I((a,b,c,d,e)^t) = (a-c,b-c)$.
893
%Using $\Phi_I$ we can compute the image of the modular
894
%symbol $\{0,\infty\}$.
895
\end{example}
896
897
\begin{lemma}\label{lem:ratperiodlemma}
898
$$\dim \sM_k(N,\eps;K)[I] = \dim \Hom(\sM_k(N,\eps;K),K)[I].$$
899
\end{lemma}
900
\begin{proof}
901
Let $W=\sM_k(N,\eps;K)$ and $W^*$ be its dual.
902
Let $a_1,\ldots,a_n$ be a set of generators for~$I$.
903
Choose a basis for~$W$ that is compatible with the following filtration:
904
$$0\subset (\ker(a_1)\intersect\cdots\intersect\ker(a_n))
905
\subset (\ker(a_1)\intersect\cdots\intersect\ker(a_{n-1}))
906
\subset\cdots \subset \ker(a_1)\subset W.$$
907
The rank of a square matrix
908
equals the rank of its transpose, so the dimension of $\ker(a_1)$ is
909
the same as the dimension of $\ker(a_1^t)$, that is,
910
$\dim W[(a_1)] = \dim W^*[(a_1)]$.
911
Since~$\T$ is commutative,~$a_2$ leaves $\ker(a_1)$ invariant;
912
because of how we chose our basis for~$W$,
913
the transpose of $a_2|_{\ker(a_1)}$ is $a_2^t|_{\ker(a_1^t)}$.
914
Thus again, $\dim (\ker(a_2|_{\ker(a_1)}))$ equals
915
$\dim (\ker(a_2^t|_{\ker(a_1^t)}))$.
916
Proceeding inductively, we prove the lemma.
917
\end{proof}
918
919
\begin{corollary}
920
Suppose $\sM_k(N,\eps;K)[I]\subset \sS_k(N,\eps;K)$, and
921
let $P:\sM_k(N,\eps;K)\ra \Hom(S_k(N,\eps;\C)[I],\C)$
922
be the classical period map induced by the integration
923
pairing. Then $\ker(P) = \ker(\Phi_I)$.
924
\end{corollary}
925
\begin{proof}
926
Since $P(\sM_k(N,\eps;\O)$ is known to be an $\O$-lattice in
927
the complex vector space $\Hom(S_k(N,\eps;\C)[I],\C)$,
928
the $K$-dimension of $\im(P)$ equals
929
$2\cdot \dim_\C S_k(N,\eps;\C)[I]$,
930
which in turn equals
931
$\dim_K \sM_k(N,\eps;K)[I]$.
932
Thus by Lemma~\ref{lem:ratperiodlemma} the
933
images $\im(P)$ and $\im(\Phi_I)$ have the same
934
dimension, hence $\ker(P)$ and $\ker(\Phi_I)$ also have the
935
same dimension. It thus suffices to prove the inclusion
936
$\ker(\Phi_I)\subset\ker(P)$.
937
Suppose $\Phi_I(x)=0$; then $\vphi(x)=0$ for all
938
$x\in W^*[I]$, where $W=\sM_k(N,\eps;K)$.
939
Thus $\vphi(x)=0$ for all $\vphi\in (W\tensor\C)^*[I]$.
940
Since the integration pairing that defines~$P$ respects
941
the action of~$\T$, the composition of~$P$ with any linear
942
functional lies in $(W\tensor\C)^*[I]$. Thus $P(x)=0$,
943
as required.
944
\end{proof}
945
946
947
% cuspdiff.tex
948
\section{The images of cuspidal points}
949
\label{sec:cuspdiff}
950
951
Consider a triple $(N,k,\eps)$, and let $K=\Q[\eps]$.
952
Recall that integration defines a period mapping
953
$$P : \sM_k(N,\eps;K)\ra \Hom(S_k(N,\eps;\C),\C).$$
954
A \defn{cuspidal point} of
955
$$J=J(N,k,\eps):=
956
\frac{\Hom(S_k(N,\eps;\C),\C)}
957
{P(\sS_k(N,\eps;\O))}$$
958
is a point that is in the image under~$P$ of $\sM_k(N,\eps;\O)$.
959
It is of great interest to compute the structure of
960
the cuspidal subgroup of~$J$ and of the quotients of~$J$.
961
For example, when $k=2$ and $\eps=1$ Manin proved
962
(see~\cite{manin:parabolic}) that the cuspidal
963
point $\{0,\infty\}$ is a torsion point in $J(\Q)$, so
964
its order gives a lower bound on $J(\Q)_{\tor}$.
965
966
\begin{algorithm}[Cuspidal subgroup]
967
Let~$I$ be an ideal in the Hecke algebra~$\T$.
968
This algorithm computes the cuspidal subgroup
969
of the quotient $A_I$ of~$J$.
970
Using Algorithm~\ref{alg:MkNO}
971
compute $\sM_k(N,\eps;\O)$ and $\sS_k(N,\eps;\O)$.
972
Using Algorithm~\ref{alg:ratperiod},
973
compute the rational period mapping~$\Phi_I$.
974
Then the cuspidal subgroup is the
975
subgroup of $\Phi_I(\sS_k(N,\eps;\O))$ generated
976
by the elements $\Phi_I(x)$ for $x\in \sM_k(N,\eps;\O)$.
977
In particular, the point of $A_I(\C)$ corresponding
978
to $X^iY^{k-2-i}\{\alp,\beta\}$ is
979
the image of $\Phi_I(X^iY^{k-2-i}\{\alp,\beta\})$ in the quotient
980
of $\Phi_I(\sM_k(N,\eps;\O)$ by $\Phi_I(\sS_k(N,\eps;\O))$.
981
\end{algorithm}
982
983
\begin{example}
984
This example continues Example~\ref{ex:ratperiod1}.
985
The basis chosen is also a basis for $\sM_2(37,1;\Z)$,
986
so by computing the boundary map, or the integer
987
kernel of $T_2(T_2+2)$, we find that $\sS_2(37,1;\Z)$
988
is spanned by $(1,0,0,0,0)$, $(0,1,0,0,0)$,
989
$(0,0,1,0,0)$, and $(0,0,0,1,0)$.
990
Thus $\Phi_I(\sS_2(37,1;\Z))$ is generated by
991
$(1,0)$ and $(0,1)$.
992
The modular symbols $\{0,\infty\}$ is represented
993
by $(0,0,0,0,-1)$, so the image of the cusp $(0)-(\infty)\in J_0(37)$
994
is~$0$ in {\bf 37k2A}.
995
996
The rational period mapping associated to {\bf 37k2B} (with respect
997
to some basis) is
998
$$\Phi_I((a,b,c,d,e)^t) = (a-c-2d+\frac{2}{3}e,\,\,
999
b+c+2d-\frac{2}{3}e).$$
1000
Thus $\Phi_I(\sS_2(37,1;\Z))$ is generated by
1001
$(1,0)$ and $(0,1)$.
1002
The image of $\{0,\infty\}$ is
1003
is $\frac{2}{3}(1,-1)$, so the image of
1004
$(0)-(\infty)$ in {\bf 37k2B} has order~$3$.
1005
\end{example}
1006
1007
1008
1009
\section{The modular degree}
1010
Let~$f$ be a newform of level~$N$, weight
1011
$k\geq 2$ and character~$\eps$ such that $\eps^2=1$.
1012
In this section we define and compute the modular degree of
1013
the torus $A_f$ attached to~$f$.
1014
1015
\begin{definition}\label{defn:modulardegree}
1016
The \defn{modular map} is the map
1017
$\theta_f:\Adual_f \ra A_f$
1018
of Diagram~\ref{dgm:uniformization}.
1019
The \defn{modular degree}~$m_f$ of~$f$ (or of~$A_f$) is
1020
the degree of this map.
1021
\end{definition}
1022
1023
When $k=2$, $\theta_f$ is a polarization so
1024
(\cite[Thm.~13.3]{milne:abvars}) the degree of~$\theta$
1025
is a square.
1026
\begin{proposition}
1027
Let $E/\Q$ be a modular elliptic curve of conductor~$N$
1028
which is an optimal quotient of $J_0(N)$.
1029
Then $\delta_f$ is the square of the usual modular degree, which is
1030
the least degree of a map $X_0(N)\ra E$.
1031
\end{proposition}
1032
When $k\neq 2$, the degree of~$\theta$ need not be a perfect square.
1033
For example, their is a one-dimensional quotient $A_f$
1034
associated to form $f\in S_4(10)$ such that $m_f=2\cdot 5$.
1035
1036
\begin{algorithm}
1037
The modular kernel is the cokernel of the natural map
1038
$\sS[I_f] \ra \Phi_f(\sS)$ of
1039
Diagram~\ref{dgm:uniformization}.
1040
\end{algorithm}
1041
\begin{proof}
1042
Use the snake lemma.
1043
\end{proof}
1044
1045
1046
\section{The modular degree}
1047
Let~$f$ be a newform of level~$N$, weight
1048
$k\geq 2$ and character~$\eps$ such that $\eps^2=1$.
1049
In this section we define and compue the modular degree of
1050
the torus $A_f$ attached to~$f$.
1051
1052
\begin{definition}
1053
The \defn{modular map} is the map
1054
$\theta_f:\Adual_f \ra A_f$
1055
of Diagram~\ref{dgm:uniformization}.
1056
The \defn{modular degree} $m_f$ of~$f$ (or of~$A_f$) is
1057
the degree of this map.
1058
\end{definition}
1059
1060
When $k=2$, $\theta_f$ is a polarization so
1061
(\cite[Thm.~13.3]{milne:abvars}) the degree of~$\theta$
1062
is a square.
1063
\begin{proposition}
1064
Let $E/\Q$ be a modular elliptic curve of conductor~$N$
1065
which is an optimal quotient of $J_0(N)$.
1066
Then $\delta_f$ is the square of the usual modular degree, which is
1067
the least degree of a map $X_0(N)\ra E$.
1068
\end{proposition}
1069
When $k\neq 2$, the degree of~$\theta$ need not be a perfect square.
1070
For example, their is a one-dimensional quotient $A_f$
1071
associated to form $f\in S_4(10)$ such that $m_f=2\cdot 5$.
1072
1073
\begin{algorithm}
1074
The modular kernel is the cokernel of the natural map
1075
$\sS[I_f] \ra \Phi_f(\sS)$ of
1076
Diagram~\ref{dgm:uniformization}.
1077
\end{algorithm}
1078
\begin{proof}
1079
Use the snake lemma.
1080
\end{proof}
1081
1082
1083
% vals.tex
1084
\section{Rational part of the $L$-function}
1085
\label{sec:rationalvals}
1086
1087
Let $\sM(\Q)=\sM(N,\Q)$ and extend $\Phi_f$ to a map $\sM(\Q) \ra \C$.
1088
Then $\Phi_f$ has a rational structure in the following sense.
1089
1090
\begin{lemma}\label{algphi}
1091
Let $\vphi_1,\ldots, \vphi_n$ be a $\Q$-basis for
1092
$\Hom(\sM(\Q),\Q)[I_f]$ and set
1093
$$\Psi=\vphi_1\cross\cdots\cross\vphi_n : \sM(\Q) \ra \Q^n.$$
1094
Then $n=2d$ and $\ker(\Psi)=\ker(\Phi_f)$.
1095
\end{lemma}
1096
\begin{proof}
1097
This result is due to Shimura (\cite{shimura:onperiods}),
1098
but we sketch a proof.
1099
To compute the dimension of $\Hom(\sM(\Q),\Q)[I_f]$ we may first tensor with $\C$.
1100
Let $\Sbar_2$ denote the weight 2 anti-holomorphic cusp forms
1101
and $E_2$ the weight $2$ Eisenstein series for $\Gamma_0(N)$.
1102
Then $\sM(\C)$ is isomorphic as a $\T$-module to
1103
$S_2\oplus \Sbar_2\oplus E_2$ (prop. 9 of \cite{merel:1585} and
1104
the Eichler-Shimura embedding). Because of the Peterson inner product,
1105
the dual $\Hom(\sM(\C),\C)$ is also isomorphic as a $\T$-module
1106
to $S_2\oplus \Sbar_2\oplus E_2$. Since $f$ is new, by
1107
the Atkin-Lehner theory,
1108
$$(S_2\oplus \Sbar_2\oplus E_2)[I_f] = S_2[I_f]\oplus \Sbar_2[I_f]$$
1109
has complex dimension $2d$, which gives the first assertion.
1110
1111
Next note that $\ker(\Phi_f)\tensor\C\subset\ker(\Psi)\tensor\C$
1112
because each map $x\mapsto \langle f_i, x \rangle$ lies in
1113
$\Hom(\sM(\Q),\C)[I_f]$ and $\ker(\Psi)\tensor\C$ is the intersection of
1114
the kernels of {\em all} maps in $\Hom(\sM(\Q),\C)[I_f]$.
1115
By Theorem~\ref{Af} the image of $\Phi_f$ is a lattice, so
1116
$\dim_\Q \ker(\Phi_f)=\dim_{\Q} \sM(\Q) - 2d$. Since $\Psi$ is the
1117
intersection of the kernels of $n=2d$ independent
1118
linear functionals $\vphi_1,\ldots, \vphi_n$,
1119
$\ker(\Psi)$ also has dimension $\dim\sM(\Q)-2d$. Since
1120
the dimensions are the same and there is an inclusion,
1121
we have an equality
1122
$\ker(\Phi_f)\tensor\C = \ker(\Psi)\tensor\C$ which forces
1123
$\ker(\Phi_f)=\ker(\Psi)$.
1124
\end{proof}
1125
1126
1127
Let $V$ be a finite dimensional
1128
vector space over $\R$. A \defn{lattice} $L\subset V$
1129
is a free abelian group of rank $=\dim V$ such that
1130
$\R L=V$.
1131
If $L, M\subset V$ are lattices, the \defn{lattice
1132
index}\label{pg:latticeindex}
1133
$[L:M]$ is the absolute value of the
1134
determinant of an automorphism of $V$
1135
taking $L$ isomorphically onto $M$.
1136
Extend the definition to the case
1137
when $M$ has rank strictly smaller than $\dim V$
1138
by defining $[L:M]=0$.
1139
\begin{lemma}\label{latticeker}
1140
Suppose $\tau_i : V\ra W_i$, $i=1,2$ are surjective linear maps such that
1141
$\ker(\tau_1)=\ker(\tau_2)$. Then
1142
$$[\tau_1(L):\tau_1(M)] = [\tau_2(L):\tau_2(M)].$$
1143
\end{lemma}
1144
\begin{proof}
1145
Surjectivety and equality of kernels insures that there is a unique
1146
isomorphism $\iota:W_1\ra W_2$ such that $\iota\tau_1 = \tau_2$.
1147
Let $\sigma$ be an automorphism of $W_1$ such that $\sigma(\tau_1(L))=\tau_1(M)$.
1148
Then
1149
$$\iota\sigma\iota^{-1}(\tau_2(L)) = \iota\sigma\tau_1(L)=\iota\tau_1(M)=\tau_2(M).$$
1150
Since conjugation doesn't change the determinant,
1151
$$[\tau_2(L):\tau_2(M)]=|\det(\iota\sigma\iota^{-1})|
1152
=|\det(\sigma)| = [\tau_1(L):\tau_1(M)].$$
1153
\end{proof}
1154
1155
Let $S_2(N,\Z)$ be the space of cusp forms whose $q$-expansion
1156
at infinity hass integer coefficients.
1157
Let $\Omega_f^0$ be the measure of the identity component of
1158
$A_f(\R)$ with respect to an integral basis for
1159
$S_f(\Z)=S_2(N,\Z)[I_f]$.
1160
Let $\e=\{0,i\infty\}\in\sM(N,\Z)$ denote the \defn{winding
1161
element\label{defn:windingelement}}.
1162
1163
\begin{theorem}\label{ratpart}
1164
Let $\Psi$ be as in Lemma~\ref{algphi}. Then
1165
$$\pm \frac{L(A_f,1)}{\Omega_f^0} = [\Psi(\sS(N,\Z)^+) : \Psi(\T{}\e)]$$
1166
\end{theorem}
1167
\begin{proof}
1168
Let $\Phi=\Phi_f$ be the period map defined
1169
by a basis $f_1,\ldots,f_d$ of conjugate newforms.
1170
The codomain of $\Phi$, which we identify with $\C^d$, is an algebra
1171
with unit element $\mathbf{1}=(1,\ldots,1)$ equipped with an action
1172
of the Hecke operators:
1173
$T_p$ acts as $(a_p^{(1)},\ldots,a_p^{(d)})$
1174
where the components are the Galois conjugates of $a_p$.
1175
Let $\Z^d\subset\R^d\subset\C^d$ be the usual submodules.
1176
Let $\Vol(\sS^+)$ be the volume of a fundamental domain
1177
for the real lattice $\Phi(\sS^+)=\Phi(\sS(N,\Z)^+)$.
1178
Observe that
1179
$\Vol(\sS^+)=[\Z^d:\Phi(\sS^+)]$ and $|L(A_f,1)|=[\Z^d:\Phi(\e)\Z^d]$.
1180
Let $W\subset\C^d$ be the $\Z$-module spanned by the columns of a basis
1181
for $S_f(\Z)$. Because $\Omega_f^0$ is computed with respect to
1182
a basis for $S_f(\Z)$,
1183
$$\Vol(\sS^+)=[W:\T\mathbf{1}]\cdot \Omega_f^0.$$
1184
Because $S_2(N,\Z)$ is saturated,
1185
$[\Z^d:W]=1$ so $[\Z^d:\T\mathbf{1}]=[W:\T\mathbf{1}]$.
1186
The following calculation involves lattices in $\R^d$:
1187
\begin{eqnarray*}
1188
[\Phi(\sS^+):\Phi(\T{}\e)]
1189
&=& [\Phi(\sS^+):\Z^d]\cdot[\Z^d:\Phi(\T{}\e)]\\
1190
&=& \frac{1}{[\Z^d:\Phi(\sS^+)]} \cdot [\Z^d:\Phi(\T\e)]\\
1191
&=&\frac{1}{\Vol(\sS^+)}\cdot [\Z^d:\Phi(\e)\Z^d]\cdot [\Phi(\e)\Z^d:\Phi(\T\e)]\\
1192
&=&\frac{|L(A_f,1)|}{\Vol(\sS^+)}\cdot[\Phi(\e)\Z^d:\Phi(\T\e)]\\
1193
&=&\frac{|L(A_f,1)|}{\Vol(\sS^+)}\cdot[\Phi(\e)\Z^d:\Phi(\e)\T{}\mathbf{1}]\\
1194
&=&\frac{|L(A_f,1)|}{\Omega_f^0\cdot [W:\T\mathbf{1}]}\cdot[\Z^d:\T{}\mathbf{1}]\\
1195
% &=&\frac{|L(A_f,1)|}{\Omega_f^0}\cdot [\Z^d:W]\\
1196
&=&\frac{|L(A_f,1)|}{\Omega_f^0}.\\
1197
\end{eqnarray*}
1198
The theorem now follows from lemmas \ref{algphi}, \ref{latticeker},
1199
and the fact that $f$ has real Fourier coefficients so $L(A_f,1)\in\R$
1200
hence $|L(A_f,1)|=\pm L(A_f,1)$.
1201
1202
\end{proof}
1203
1204
\begin{corollary}
1205
Let $n_f$ be the order of the image in $A_f(\Q)$ of the point
1206
$(0)-(\infty)\in J_0(N)(\Q)$. Then
1207
$$ \frac{L(A_f,1)}{\Omega_f^0}\in \frac{1}{n_f}\Z.$$
1208
\end{corollary}
1209
\begin{proof}
1210
Let $x$ denote the image of $(0)-(\infty)\in A_f(\Q)$
1211
and set $I=\Ann(x)\subset\T$. Since $f$ is a {\em newform}
1212
the Hecke operators $T_p$ for $p|N$ act as $0$ or $\pm 1$ on
1213
$A_f(\Q)$ (end of section 6 of \cite{diamond-im}).
1214
If $p\nmid N$ a standard calculation (section 2.8 of \cite{cremona:algs})
1215
combined with the Abel-Jacobi theorem shows that $T_p(x) = (p+1)x$.
1216
Let $C=\Z{}x$ denote the (finite, by Manin-Drinfeld) cyclic subgroup of
1217
$A_f(\Q)$ generated by $x$, so $n_f$ is the order of $C$.
1218
There is an injection
1219
$ \T/I\hookrightarrow C$
1220
sending $T_p$ to $T_p(x)$.
1221
By the theorem, we have
1222
\begin{eqnarray*}
1223
\pm L(A_f,1)/\Omega_f^0 &=& [\Psi(\sS^+):\Psi(\T{}e)]\\
1224
&=& [\Psi(\sS^+):\Psi(I\e)]\cdot [\Psi(I\e):\Psi(\T{}\e)]\\
1225
&=& [\Psi(\sS^+):I\Psi(\e)]\cdot [I\Psi(\e):\T{}\Psi(\e)]\\
1226
&=& \frac{[\Psi(\sS^+):I\Psi(\e)]}{[\T{}\Psi(\e):I\Psi(\e)]} \in \frac{1}{n_f}\Z.
1227
\end{eqnarray*}
1228
The final inclusion follows from two observations.
1229
By Abel-Jacobi, $I$ is exactly those
1230
elements of $\T$ which send $\Psi(\e)$ into $\Psi(\sS^+)$, so
1231
$[\Psi(\sS^+):I\Psi(\e)]\in\Z$. Second, there is a surjective map
1232
$$\T/I \ra \frac{\T\Psi(\e)}{I\Psi(\e)}$$
1233
sending $t$ to $t \Psi(\e)$, so $[\T{}\Psi(\e):I\Psi(\e)]$
1234
divides $n_f=|C|=|\T/I|$.
1235
\end{proof}
1236
1237
1238
% analytic.tex
1239
1240
\section{Analytic invariants}
1241
Let
1242
$$f =\sum_{n\geq 1} a_n q^n\in S_k(N,\eps)$$ be a newform,
1243
{\bf and assume that $\eps^2=1$.}
1244
Let $K_f = \Q(\ldots a_n \ldots)$ and
1245
let $f_1,\ldots,f_d$ be the Galois conjugates of~$f$,
1246
where $d=[K_f:\Q]$.
1247
As in Section~\ref{sec:tori},
1248
we consider the complex torus $A_f$ attached to~$f$.
1249
In this section we describe how to compute the torus $A_f$ and
1250
the special values at the critical integers $1,2,\ldots,k-1$
1251
of the~$L$ function associated to~$f$.
1252
1253
Recall that the $L$-series associated to~$f$ is
1254
$$L(f,s) \defeq \sum_{n=1}^{\infty} a_n n^{-s},$$
1255
and that Hecke proved that $L(f,s)$ has an analytic
1256
continuation to the whole complex plane.
1257
Set
1258
$$L(A_f,s) \defeq \prod_{i=1}^d L(f_i,s).$$
1259
When $k=2$ and $\eps=1$, this is the canonical $L$-series
1260
associated to the abelian variety $A_f/\Q$.
1261
1262
Let
1263
$$g = \sum_{n\geq 1} a_n q^n \in M_k(N,\eps)$$
1264
be a modular form (we do not assume that~$g$ is an eigenform).
1265
We recall the integration pairing of Theorem~\ref{thm:perfectpairing}:
1266
$$
1267
\langle\,, \,\rangle:\, M_k(N,\eps) \cross \sM_k(N,\eps)
1268
\lra \C$$
1269
$$\langle f , P\{\alp,\beta\}\rangle =
1270
\twopii \int_{\alp}^{\beta} f(z)P(z,1) dz.$$
1271
Let $I_f\subset \T$ be the kernel of the
1272
map $\T\ra K_f$ sending~$T_n$ to~$a_n$.
1273
The integration pairing
1274
gives rise to the period mapping\label{defn:periodmapping}
1275
$$\Phi_f : \sM_k(N,\eps) \ra \Hom(S_k(N,\eps)[I_f],\C),$$
1276
and $A_f = \Hom(S_k(N,\eps)[I_f],\C)/\Phi_f(\sS_k(N,\eps))$
1277
is the cokernel.
1278
1279
\subsection{Extended modular symbols}\label{defn:extendedmodsyms}
1280
For the purposes of computing periods, it
1281
is advantageous to extend the notion of modular
1282
symbols to allows symbols of the form
1283
$P\{z,w\}$ where~$z$ and~$w$ are now
1284
arbitrary elements of $\h^*=\h\union\P^1(\Q)$.
1285
The free abelian group $\esM_k$
1286
of \defn{extended modular symbols}
1287
is spanned by such symbols, and is of uncountable
1288
rank over~$\Z$. However, it is still equipped with
1289
an action of $\gzero$ and we can form the
1290
largest torsion-free quotient
1291
$\esM_k(N,\eps)$ of $\esM_k$ by
1292
the relations $\gam x = \eps(\gam)x$ for
1293
$\gam\in\gzero$.
1294
1295
The integration pairing
1296
extends to $\esM_k(N,\eps)$. There is a natural
1297
embedding
1298
$\iota: \sM_k(N,\eps)\hookrightarrow \esM_k(N,\eps)$
1299
which respect the pairing in the sense that
1300
$\langle f, \iota(x)\rangle = \langle f , x\rangle.$
1301
In many cases it is advantageous to replace
1302
$x\in\sM_k(N,\eps)$ first by $\iota(x)$, and then
1303
by an equivalent sum $\sum y_i$ of symbols
1304
$y_i\in \esM_k(N,\eps)$.
1305
The period
1306
$\langle f, x\rangle$
1307
is then replaced by the equivalent
1308
sum of periods $\sum \langle f , y_i\rangle$.
1309
The latter is frequently {\em much} easier to approximate
1310
numerically.
1311
1312
1313
\subsection{Numerically computing period integrals}
1314
Consider a point~$\alp$ in the upper half plane
1315
and any one of the (extended) modular symbols
1316
$X^mY^{k-2-m}\{\alp,\infty\}$.
1317
Given a cuspform $g =\sum_{n\geq 1} b_n q^n\in S_k(N,\eps)$
1318
and an integer $m\in \{0,1,\ldots,k-2\}$, we find that
1319
\begin{equation}\label{intsum}
1320
\langle g, \, X^mY^{k-2-m}\{\alpha,\infty\}\rangle =
1321
\twopii \int_{\alpha}^{i\infty} g(z)z^m dz =
1322
\twopii \sum_{n=1}^{\oo} b_n \int_{\alpha}^{i\infty} e^{2\pi i n z} z^m dz.
1323
\end{equation}
1324
The reversal of summation and integration is justified because
1325
the imaginary part of~$\alp$ is positive so that the sum
1326
converges absolutely. This is made explicit in the following
1327
lemma, which can be proved using repeated integration by parts.
1328
\begin{lemma}\label{lem:intexp}
1329
\begin{equation}\label{intexp}
1330
\int_{\alpha}^{i\infty} e^{2\pi i n z} z^m dz
1331
\,\,=\,\, e^{2\pi i n \alpha}
1332
\sum_{s=0}^m \left(
1333
\frac{(-1)^s \alpha^{m-s}}
1334
{(2\pi i n)^{s+1}}
1335
\prod_{j=(m+1)-s}^m j\right).
1336
\end{equation}
1337
\end{lemma}
1338
1339
The following proposition is the higher weight
1340
analogue of \cite[Prop. 2.1.1(5)]{cremona:algs}.
1341
\begin{proposition}\label{modsym-errorterm}
1342
For any $\gam\in \Gamma_0(N)$, $P\in V_{k-2}$ and $\alp\in\h^*$
1343
the following holds:
1344
\begin{eqnarray}
1345
P\{\oo, \gam(\oo)\}
1346
&=& P\{\alp,\gam(\alp)\} + (P - \eps(\gam)\gam^{-1}P)\{\oo,\alp\}\\
1347
&=& \eps(\gam)(\gam^{-1}P)\{\alp, \oo\} - P\{\gamma(\alp),\oo\}.
1348
\end{eqnarray}
1349
\end{proposition}
1350
\begin{proof}
1351
By definition, if $x\in\sM_k(N,\eps)$ is a modular symbol
1352
and $\gam\in\Gamma_0(N)$ then $\gam{}x=\eps(\gam)x$;
1353
in particular, $\eps(\gam)\gam^{-1}x=x$, so
1354
\begin{eqnarray*}
1355
P\{\oo, \gam(\oo)\}
1356
&=& P\{\oo,\alp\} + P\{\alp,\gam(\alp)\} + P\{\gam(\alp),\gam(\oo)\}\\
1357
&=& P\{\oo,\alp\} + P\{\alp,\gam(\alp)\} + \eps(\gam)\gam^{-1}(P\{\gam(\alp),\gam(\oo)\})\\
1358
&=& P\{\oo,\alp\} + P\{\alp,\gam(\alp)\} + \eps(\gam)(\gam^{-1}P)\{\alp, \oo\}\\
1359
&=& P\{\alp,\gam(\alp)\} + P\{\oo,\alp\} - \eps(\gam)(\gam^{-1}P)\{\oo, \alp\}\\
1360
&=& P\{\alp,\gam(\alp)\} + (P - \eps(\gam)\gam^{-1}P)\{\oo,\alp\}.
1361
\end{eqnarray*}
1362
The second equality in the statement of the proposition now follows easily.
1363
\end{proof}
1364
In the classical case of weight two and trivial character,
1365
the error term $(P - \eps(\gam)\gam^{-1}P)\{\oo,\alp\}$
1366
vanishes. In general this term does
1367
not vanish, instead perturbing
1368
(or rendering apparently unrepairable) the
1369
analogues of the formulas
1370
found in \cite[2.10]{cremona:algs}.
1371
1372
\begin{algorithm}
1373
Given a triple $\gam\in\Gamma_0(N)$, $P\in V_{k-2}$ and
1374
$g\in S_k(N,\eps)$
1375
(as a $q$-expansion to some precision) this algorithm computes
1376
the period integral
1377
$\langle g, \,P\{\oo, \gamma(\oo)\}\rangle.$
1378
Express $\gam$ as $\abcd{\hfill a}{b}{cN}{d}\in\gzero$ and take
1379
$\alp = \frac{-d+i}{cN}$ in Proposition~\ref{modsym-errorterm}.
1380
Replacing~$\gam$ by $-\gam$ if necessary,
1381
we find that the imaginary parts of~$\alp$ and
1382
$\gam(\alp)=\frac{a+i}{cN}$
1383
are both $1/cN>0$.
1384
Equation~\ref{intsum} and Lemma~\ref{lem:intexp} can now be
1385
used to compute the period integrals of
1386
Proposition~\ref{modsym-errorterm}.
1387
\end{algorithm}
1388
1389
With the goal of computing period lattices in mind, it is
1390
reassuring to know that every element of $\sS_k(N,\eps)$
1391
can be written as a linear combination of symbols of the form
1392
$P\{\oo,\gamma(\oo)\}$.
1393
In the special case of weight two and trivial character,
1394
this is the assertion (proved by Manin~\cite{manin:parabolic})
1395
that the group homomorphism $\Gamma_0(N)\ra H_1(X_0(N),\Z)$
1396
sending~$\gamma$ to $\{0,\gamma(0)\}$ is surjective. When the
1397
weight is greater than two, we have not found any similar
1398
group-theoretic statement.
1399
1400
\begin{proposition}\label{onlyoo}
1401
Any element of $\sS_k(N,\eps)$ can be written in the form
1402
$$\sum_{i=1}^n P_{i}\{\infty,\gam_i(\infty)\}$$
1403
with $P_i\in V_{k-2}$ and $\gam_i\in\gzero.$
1404
Moreover, $P_i$ and $\gamma_i$ can be chosen so that
1405
$\sum \eps(\gamma_i) P_i = \sum \gamma_i^{-1} P_i$.
1406
\end{proposition}
1407
\begin{proof}\footnote{Helena Verrill found this proof!}
1408
First recall
1409
the definition of the spaces~$\sM$,
1410
$\sM_k=V_{k-2}\tensor\sM$ and
1411
$\sM_k(N,\eps)=\sM_k/I$ (see Section~\ref{sec:defnofmodsyms}).
1412
Let $I=I_{N,\eps}$ be the ideal in the
1413
group ring of~$\gzero$ generated by all
1414
elements of the form $\eps(\gamma) -\gamma$
1415
for $\gam\in\gzero$.
1416
1417
Suppose $v\in\sS_k(N,\eps)$. Use the relation
1418
$\{\alp,\beta\}=\{\oo,\beta\}-\{\oo,\alp\}\in\sM$
1419
to see that any~$v$ is the image
1420
of an element $\tilde{v}\in \sM_k$ of the form
1421
$$\tilde{v} = \sum_{\beta\in\Q}P_\beta\tensor \{\oo,\beta\}\in \sM_k$$
1422
with only finitely many $P_\beta$ nonzero.
1423
The boundary map~$\delta$ lifts in a natural way
1424
to $V_{k-2}\tensor\sM$, as illustrated.
1425
$$\xymatrix{
1426
&I(V_{k-2}\otimes\sM)\ar[r]\ar[d]&
1427
I(V_{k-2}\otimes\sB)\ar[d] \\
1428
&V_{k-2}\otimes\sM\ar[r]^{\tilde{\delta}}\ar[d] & V_{k-2}\otimes\sB\ar[d] \\
1429
*++{\sS_k(N,\eps)}\[email protected]{^{(}->}[r]
1430
&\sM_k(N,\eps)\ar[r]^{\delta}
1431
&\sB_k(N,\eps)\\
1432
}\qquad\qquad\mbox{}$$
1433
Our assumption that $\delta(v)=0$ implies that
1434
$\tilde{\delta}(\tilde{v})\in I(V_{k-2}\otimes\sB)$.
1435
So there are $Q_{\gam,\beta}\in V_{k-2}$,
1436
for $\gam\in\gzero$ and $\beta\in\P^1(\Q)$, only
1437
finitely many nonzero, such that
1438
$$\tilde{\delta}(\tilde{v})
1439
= \sum_{\gam,\beta}(\eps(\gamma)-\gamma)
1440
(Q_{\gamma,\beta}\tensor\{\beta\}).$$
1441
We now use a summation trick.
1442
\begin{eqnarray*}
1443
\sum_{\beta\in\Q}
1444
\tilde{\delta}(\tilde{v})
1445
= P_\beta\tensor \{\beta\}-P_\beta\tensor \{\oo\}
1446
&=& \sum_{\gam, \beta}
1447
\eps(\gamma) Q_{\gam,\beta}\tensor \{\beta\}
1448
-(\gam Q_{\gam,\beta})\tensor \{\gam\beta\}\\
1449
&=&
1450
\sum_{\gam, \beta} \eps(\gamma) Q_{\gam,\beta}\tensor \{\beta\}
1451
-(\gam{}Q_{\gam,\gam^{-1}\beta})\tensor \{\beta\}\\
1452
&=&
1453
\sum_{ \gam, \beta}\Bigl( \eps(\gamma)Q_{\gam,\beta}
1454
-\gam{}Q_{\gam,\gam^{-1}\beta}\Bigr)\tensor \{\beta\}.\\
1455
\end{eqnarray*}
1456
Equating terms we deduce that for $\beta\not=\infty$,
1457
$$P_\beta=\sum_{\gam}
1458
\eps(\gam)Q_{\gam,\beta}-\gam{}Q_{\gam,\gam^{-1}\beta}.$$
1459
Using this expression for $P_\beta$ and
1460
that $\eps(\gamma)\gamma^{-1}$ acts trivially
1461
on $\sM_k(N,\eps)$ we find that
1462
\begin{eqnarray*}
1463
v = \sum_{\beta}
1464
P_\beta
1465
\{\oo,\beta\}
1466
&=&
1467
\sum_{\gam,\beta}
1468
\Bigl(\eps(\gam)Q_{\gam,\beta}
1469
-\gam{}Q_{\gam{},\gam^{-1}\beta}\Bigr)
1470
\{\oo,\beta\}\\
1471
&=&
1472
\sum_{\gam,\beta}
1473
\eps(\gam)Q_{\gam,\beta}
1474
-\eps(\gamma)\gamma^{-1}
1475
\Bigl(\gam{}Q_{\gam{},\gam^{-1}\beta}\Bigr)
1476
\{\oo,\beta\}\\
1477
&=&
1478
\sum_{\gam,\beta}
1479
\eps(\gam)Q_{\gam,\beta}\{\oo,\beta\}
1480
-\eps(\gam)Q_{\gam,\gam^{-1}\beta}\{\gam^{-1}\oo,\gam^{-1}\beta\}\\
1481
&=&
1482
\sum_{\gam,\beta}
1483
\eps(\gam)Q_{\gam,\beta}\{\oo,\beta\}
1484
-\eps(\gam)Q_{\gam,\beta}\{\gam^{-1}\oo,\beta\}\\
1485
&=&
1486
\sum_{\gam,\beta}
1487
\eps(\gam)Q_{\gam,\beta}
1488
\{\oo,\gam^{-1}\oo\}.\\
1489
\end{eqnarray*}
1490
This is of the desired form.
1491
\end{proof}
1492
1493
Unlike the case of weight two and trivial character,
1494
Proposition~\ref{onlyoo} does not give generators for $\sS_k(N,\eps)$.
1495
This is because not every element of the form $P\{\oo,\gam(\oo)\}$
1496
must lie in $\sS_k(N,\eps)$. However, if $\gam P = P$ then
1497
$P\{\oo,\gam(\oo)\}$ does lie in $\sS_k(N,\eps)$. It would be
1498
interesting to know whether $\sS_k(N,\eps)$ is generated by symbols of
1499
the form $P\{\oo,\gam(\oo)\}$ with $\gam P = P$. When $k$ is odd this
1500
is clearly not the case: when $k=3$ the condition $\gamma P = P$
1501
implies that $\gamma\in\gzero$ has an eigenvector with eigenvalue~$1$,
1502
hence is of finite order. When~$k$ is even the author can see no
1503
obstruction to generating $\sS_k(N,\eps)$ using such symbols.
1504
1505
\subsection{The $W_N$-trick}\label{sec:wntrick}
1506
{\bf In this section we assume that~$k$ is even.}
1507
Consider the involution $W_N$ defined in
1508
Section~\cite{atkin-lehner}. This is an involution that
1509
acts on both modular symbols and modular forms.
1510
The follow proposition shows how to compute
1511
$\langle g, P\{\oo,\gam(\oo)\}$ under
1512
certain restrictive assumptions.
1513
It generalizes the main result of~\cite{cremona:periods} to
1514
higher weight.
1515
1516
\begin{proposition}\label{wntrick}
1517
Let $g \in S_k(N,\eps)$ be a cuspform which is
1518
an eigenform for the Atkin-Lehner involution~$W$
1519
having eigenvalue $w\in \{\pm 1\}$.
1520
Then for any $\gamma\in\Gamma_0(N)$ and any
1521
$P\in V_{k-2}$, with the property that $\gamma P = \eps(\gamma)P$, we have
1522
for any $\alp\in\h$ the following formula:
1523
$$\langle g, P\{\oo,\gamma(\oo)\}\rangle = \hspace{4.5in}$$
1524
$$ \langle g, w \frac{P(Y,-NX)}{N^{k/2-1}}\{W(\alp),\oo\}
1525
+(P - w \frac{P(Y,-NX)}{N^{k/2-1}})\{i/\sqrt{N},\oo\}
1526
-P\{\gamma(\alp),\oo\} \rangle.$$
1527
Here $W(\alp) = -1/(N\alp)$.
1528
\end{proposition}
1529
\begin{proof}
1530
By Proposition~\ref{modsym-errorterm} our condition on~$P$
1531
implies that $P\{\oo,\gamma(\oo)\}= P\{\alp,\gamma(\alp)\}$.
1532
The steps of the following computation are described below.\vspace{1ex}\\
1533
$\langle g, P\{\alp,\gamma(\alp)\}\rangle $\vspace{-1ex}
1534
\begin{eqnarray*}
1535
&=&\langle g, P\{\alp,i/\sqrt{N}\} + P\{i/\sqrt{N},W(\alp)\}+P\{W(\alp),\gamma(\alp)\}
1536
\rangle \\
1537
&=&\langle g, w \frac{W(P)}{N^{k/2-1}}
1538
\{W(\alp),i/\sqrt{N}\} + P\{i/\sqrt{N},W(\alp)\}+P\{W(\alp),\gamma(\alp)\}
1539
\rangle \\
1540
&=&\langle g, (w \frac{W(P)}{N^{k/2-1}}-P)
1541
\{W(\alp),i/\sqrt{N}\} +P\{W(\alp),\oo\} - P\{\gamma(\alp),\oo\}\rangle\\
1542
&=& \langle g, w \frac{W(P)}{N^{k/2-1}}\{W(\alp),\oo\}
1543
+(P - w \frac{W(P)}{N^{k/2-1}})\{i/\sqrt{N},\oo\}
1544
-P\{\gamma(\alp),\oo\} \rangle.\\
1545
\end{eqnarray*}
1546
For the first step, we break the path into three paths.
1547
In the second step, we apply the $W$-involution to the first
1548
term, and use that the action of~$W$ is compatible with
1549
the pairing $\langle \,,\, \rangle$. The third step involves
1550
combining the first two terms and breaking up the third.
1551
In the final step, we replace $\{ W(\alp), i/\sqrt{N}\}$
1552
by $\{W(\alp),\infty\}+\{\infty,i/\sqrt{N}\}$ and regroup.
1553
\end{proof}
1554
1555
A good choice for~$\alp$ is
1556
$\alp=\gamma^{-1}\left(\frac{b}{d}+\frac{i}{d\sqrt{N}}\right)$,
1557
so that $W(\alp) = \frac{c}{d}+\frac{i}{d\sqrt{N}}$.
1558
This maximizes the minimum of the imaginary parts
1559
of~$\alp$ and~$W(\alp)$.
1560
1561
Let $\gamma=\abcd{a}{b}{c}{d}\in \Gamma_0(N)$.
1562
A polynomial~$P$ for which $\gamma (P)=P$ is given by
1563
$$P(X,Y) = (cX^2 + (d-a)XY - bY^2)^{\frac{k-2}{2}}.$$
1564
This formula was obtained by viewing $V_{k-2}$ as
1565
the $k-2$th symmetric product of the two-dimensional
1566
space on which $\gzero$ acts naturally. For example,
1567
observe that since $\det(\gamma)=1$
1568
the symmetric product of two eigenvectors for~$\gamma$ is an eigenvector
1569
in $V_{2}$ having eigenvalue~$1$.
1570
For the same reason, if $\eps(\gamma)\neq 1$, there is
1571
often no polynomial $P(X,Y)$ such that $\gamma(P)=\eps(\gamma) P$.
1572
When this is the case, first choose~$\gamma$ so that $\eps(\gamma)=1$.
1573
1574
Since the imaginary parts of the terms
1575
$i/\sqrt{N}$, $\alp$ and $W(\alp)$ in the proposition
1576
are all relatively large, the sums appearing in
1577
Equation~\ref{intsum} converge quickly if~$d$ is small.
1578
Let us emphasize, that {\em it
1579
is {\bf extremely} important to choose~$\gamma$
1580
in Proposition~\ref{wntrick} with~$d$ small, otherwise
1581
the series will converge {\em very} slowly.}
1582
1583
1584
\subsection{Computing the period map}\label{computephi}
1585
Let $I\subset \T$ be the kernel of the
1586
map $\T\ra K_f$ sending~$T_n$ to~$a_n$.
1587
As in Section~\ref{sec:ratperiod},
1588
let $\Theta_f$ be the rational period mapping associated to~$f$.
1589
We have a commutative diagram
1590
$$\xymatrix{
1591
{\sM_k(N,\eps)}\ar[dr]_{\Theta_I}\ar[rr]^{\Phi_f}
1592
& & \Hom(S_k(N,\eps)[I],\C) \\
1593
& {\displaystyle\frac{\sM_k(N,\eps)}{\ker(\Phi_f)}}\ar[ur]^{\iota}
1594
}$$
1595
Using Algorithm~\ref{alg:ratperiod}, we can
1596
compute $\Theta_f$ so to compute $\Phi_f$ we need to compute~$\iota$.
1597
Let $g_1,\ldots,g_d$ be a basis for the $\Q$-vector space $S_k(N,\eps;\Q)[I]$.
1598
We will compute the period mapping with respect to the basis of
1599
$\Hom(S_k(N,\eps;\Q)[I],\C)$ dual to this basis.
1600
Choose elements $x_1,\ldots,x_d\in \sM_k(N,\eps)$
1601
with the following properties:
1602
\begin{enumerate}
1603
\item Using Proposition~\ref{modsym-errorterm} or Proposition~\ref{wntrick}
1604
it is possible to efficiently compute the period integrals
1605
$\langle g_i, x_j \rangle$, $i,j\in\{1,\ldots d\}$.
1606
\item The $2d$ elements $v+*v$ and $v-*v$ for $v=\Theta_I(x_1),\ldots,\Theta_I(x_d)$
1607
span a space of dimension $2d$.
1608
\end{enumerate}
1609
Given this data, we can compute
1610
$$\iota (v+*v) =
1611
2\Re(\langle g_1, x_i\rangle, \ldots, \langle g_d, x_i\rangle)$$
1612
and
1613
$$\iota (v-*v) =
1614
2i\Im(\langle g_1, x_i\rangle, \ldots, \langle g_d, x_i\rangle).$$
1615
We break the integrals into real and imaginary parts because this
1616
increases the precision of our answers.
1617
Since the vectors $v_n+*v_n$ and $v_n-*v_n$, $n=1,\ldots,d$ span
1618
$\frac{\sM_k(N,\eps)}{\ker(\Phi_f)}$ we have computed~$\iota$.
1619
1620
It is advantageous when possible to find symbols~$x_i$ satisfying the
1621
conditions of Proposition~\ref{wntrick}. This is usually possible
1622
when~$d$ is very small, but in practice we have had problems doing
1623
this when~$d$ is large, for example with \text{\bf 131k2B},
1624
in which case the dimension is~$10$.
1625
1626
\subsection{Computing special values}
1627
\label{sec:compspecval}
1628
For $s = 1,\ldots,k-1$ we have
1629
\begin{eqnarray}\label{specialvalueformula}
1630
L(f,s) &=&
1631
\frac{-2\pi^{s-1}i^{s-1}}{(s-1)!}\cdot
1632
\langle f, X^{s-1}Y^{k-1-s}\{0,\oo\}\rangle,\\
1633
L(A_I,s) &=& \prod_{i=1}^d L(f_i,s).
1634
\end{eqnarray}
1635
For ease of notation let
1636
$$\e_i := X^{s-1}\{0,\oo\}$$
1637
denote the $i$th \defn{winding element}.
1638
In section~\ref{computephi} we computed the period map
1639
$\Phi_f$ with respect to a basis $g_1,\ldots,g_d$
1640
for $S_k(N,\eps;\Q)[I]$.
1641
Upon writing~$f$ as a $K_f$-linear combination
1642
$\alp_1 g_1+ \cdots+ \alp_d g_d$ we find that
1643
\begin{eqnarray*}
1644
\langle f, \e_i\rangle
1645
&=& \langle \alp_1 g_1+ \cdots+ \alp_d g_d, \e_i\rangle\\
1646
&=& \alp_1 \langle g_1, \e_i \rangle + \cdots
1647
\alp_d \langle g_d, \e_i \rangle\\
1648
&=& \alp_1 \Phi_f(\e_i)_1 + \cdots
1649
\alp_d \Phi_f(\e_i)_d
1650
\end{eqnarray*}
1651
Here $\Phi_f(\e_i)_j$ denotes the $j$th coordinate of
1652
$\Phi_f(\e_i)$.
1653
Finally using Equation~\ref{specialvalueformula} we compute
1654
the special value.
1655
1656
\subsection{Computing the real and pure imaginary volume}
1657
\label{sec:realvolume}
1658
[[THIS SECTION IS REALLY BAD; WHAT HAPPENED? See
1659
\begin{verbatim}
1660
/home/was/papers/periods/old/periods.tex
1661
\end{verbatim}
1662
for something that is much better.
1663
]]
1664
1665
By Proposition~\ref{prop:starpairing}, for any $x\in\sS_k(N,\eps)$
1666
we have that
1667
\begin{eqnarray*}
1668
\overline{\Phi_f(x)} &=& (\overline{\langle g_1, x\rangle},
1669
\ldots,\overline{\langle g_d,x\rangle})\\
1670
&=& (\langle g_1^*, x^*\rangle,
1671
\ldots, \langle g_d^*,x^*\rangle)\\
1672
&=& (\langle g_1, x^*\rangle,
1673
\ldots, \langle g_d,x^*\rangle)
1674
\in \Phi_f(\sS_k(N,\eps)).
1675
\end{eqnarray*}
1676
Therefore complex conjugation leaves the period
1677
lattice $\Lambda_f=\Phi_f(\sS_k(N,\eps))$ invariant, which
1678
means that $A_f = \C^d/\Lambda_f$
1679
is equipped with an action of complex conjugation.
1680
We can thus define the real and imaginary volumes of $A_f$.
1681
Let $g_1,\ldots,g_d$ be a
1682
$\Z$-basis for $S_k(N,\eps;\Z)[I_f]$. This choice
1683
equips $A_f=\C^d/\Lambda_f$ with a real-valued measure~$\mu$.
1684
We define the \defn{real volume} $\Omega_f^+=\mu(A_f(\R))$ to be the measure
1685
of the points of $A_f$ invariants under complex conjugation.
1686
The \defn{imaginary volume} $\Omega_f^-=\mu(A_f(\C)^{-})i^d$ is the
1687
measure of those points anti-invariant under complex conjugation,
1688
multiplied by the power $d=\dim A_f$ of~$i$.
1689
1690
Suppose~$s$ is an integer in the set $\{1,\ldots,k-1\}$.
1691
If~$s$ is odd then the ratio $L(A_f,s)/\Omega_f^+$ is a
1692
rational number. When~$s$ is even the ratio
1693
$L(A_f,s)/\Omega_f^-$ is an integer times a power of~$2$.
1694
This is proved in Section~\ref{sec:rationalvals}.
1695
1696
When $k=2$ and~$\eps$ is trivial, $A_f$
1697
has the structure of abelian variety over~$\Q$.
1698
The quantity $\Omega_f^+$ above is
1699
related to the quantity $\Omega_A$\label{defn:omega} appearing in the Birch
1700
and Swinnerton-Dyer conjecture \cite{tate:bsd}
1701
for $A_f$. The latter quantity is the volume
1702
of $A_f(\R)$ with respect to a basis of integral
1703
differentials on the N\'{e}ron model of $A_f$ over $\Spec(\Z)$.
1704
The two quantities are related by the
1705
Manin constant, which the author conjectures is always~$1$
1706
(see Section~\ref{sec:maninconstant}).
1707
1708
\subsection{Examples}
1709
\subsubsection{Jacobians of genus two curves}
1710
\label{sec:analytic-empirical}
1711
The authors of~\cite{empirical}
1712
gather empirical evidence for the BSD conjecture for
1713
Jacobian of genus two curves. Of the~$32$ Jacobians considered, all but
1714
four are optimal quotients of $J_0(N)$ for some~$N$. The methods
1715
of this section can be used to compute $\Omega_f^{+}$ for the
1716
Jacobians of these~$28$ curves. Using explicit models
1717
for the genus two curves, the authors of \cite{empirical}
1718
computed the volume of~$A$ with respect to a basis for the N\'eron
1719
differentials of~$A$. In all~$28$ cases our answers agreed
1720
to the precision computed. Thus in these cases we have numerically
1721
verified that the Manin constant equals one.
1722
1723
The first example considered in \cite{empirical} is the Jacobian
1724
$A=J_0(23)$ of the modular curve $X_0(23)$. This curve has as a model
1725
$$y^2+(x^3+x+1)y = -2x^5-3x^2+2x-2$$
1726
from which one can compute the BSD $\Omega_A = 2.7328...$.
1727
The following integral basis of cusp forms for $S_2(23)$ can
1728
be found using the method described in
1729
Section~\ref{sec:intbasis}:
1730
\begin{eqnarray*}
1731
g_1 &=& q - q^3 - q^4 - 2q^6 + 2q^7 + \cdots \\
1732
g_2 &=& q^2 - 2q^3 - q^4 + 2q^5 + q^6 + 2q^7 +\cdots
1733
\end{eqnarray*}
1734
The space $\sM_2(23;\Q)$ of modular symbols has dimension five and is spanned
1735
by $\{-1/19,0\}$, $\{-1/17,0\}$, $\{-1/15,0\}$, $\{-1/11,0\}$
1736
and $\{\oo,0\}$. The submodule $\sS_2(23;\Z)$ has rank four and
1737
has as basis the first four of the above five symbols.
1738
Choose $\gamma_1 = \abcd{8}{1}{23}{3}$ and
1739
$\gamma_2=\abcd{6}{1}{23}{4}$ and let
1740
$x_i = \{\oo,\gamma_i(\oo)\}$.
1741
Using the $W_N$-trick (see Section~\ref{sec:wntrick}) we compute
1742
the period integrals $\langle g_i, x_j\rangle $ using $97$ terms
1743
of the $q$-expansions of $g_1$ and $g_2$, and obtain
1744
$$\begin{array}{ll}
1745
\langle g_1, x_1 \rangle \almost -1.3543+1.0838i,\qquad
1746
&\langle g_1, x_2 \rangle \almost -0.5915+ 1.6875i\\
1747
\langle g_2, x_1 \rangle \almost -0.5915 - 0.4801i,\qquad
1748
&\langle g_2, x_2 \rangle \almost -0.7628 + 0.6037i
1749
\end{array}$$
1750
Using $97$ terms we already obtain about 14 decimal digits
1751
of accuracy, but we don't reproduce them all here.
1752
We next find that
1753
$$\langle g_1, x_1 + x_1^*\rangle \sim 2\Re(-1.3543+1.0838i) = 2.7086,$$
1754
and so on.
1755
Upon writing each generator of $\sS_2(23)$ in terms
1756
of $x_1 + x_1^*$, $x_1 - x_1^*$, $x_2 + x_2^*$ and $x_2 - x_2^*$
1757
we discover that the period mapping with respect to the
1758
basis dual to $g_1$ and $g_2$ is (approximately)
1759
$$\begin{array}{rcll}
1760
\{-1/19,0\}&\mapsto&(\hspace{.8em}0.5915 - 1.6875i,& \hspace{.8em}0.7628 - 0.6037i)\\
1761
\{-1/17,0\}&\mapsto&(-0.5915 - 1.6875i,& -0.7628 - 0.6037i)\\
1762
\{-1/15,0\}&\mapsto&(-1.3543 - 1.0838i,& -0.5915 + 0.4801i)\\
1763
\{-1/11,0\}&\mapsto&(-1.5256,& \hspace{.8em}0.3425)
1764
\end{array}$$
1765
Working in $\sS_2(23)$ we find $\sS_2(23)^+$ is spanned by
1766
$\{-1/19,0\}-\{-1/17,0\}$ and $\{-1/11,0\}$. Using
1767
the algorithm of Section~\ref{sec:realvolume},
1768
we find that there is only one real component so
1769
$$\Omega_I^+ \sim
1770
\left|\begin{array}{cc}
1771
1.1831 & 1.5256 \\
1772
-1.5256 & 0.3425
1773
\end{array}\right| = 2.7327...$$
1774
To greater precision we find that $\Omega_f^+\sim 2.7327505324965$.
1775
This agrees with the value in \cite{empirical}; since the Manin constant
1776
is an integer, it must equal~$1$.
1777
1778
\subsubsection{Level one cusp forms}
1779
In the following two sections we consider several specific examples
1780
of tori attached to modular forms of weight greater than two.
1781
1782
Let $k\geq 12$ be an even integer. Associated to each Galois
1783
conjugacy class of normalized eigenforms~$f$, there is a
1784
torus $A_f$ over~$\R$.
1785
The real and pure imaginary volumes of the first few of
1786
these tori are displayed in
1787
Table~\ref{table:vols}\footnote{It didn't take more than three minutes
1788
to compute any number in this table}. (For weights~$24$ and~$28$ we
1789
give $\Omega^-/i$ so that the columns will line up nicely.)
1790
In each case, $97$ terms of the $q$-expansion were used.
1791
1792
\begin{table}
1793
\begin{center}
1794
\caption{Volumes of level one cusp forms.\label{table:vols}}
1795
\begin{tabular}{|c|}\hline
1796
\vspace{-2ex}\\
1797
$\begin{array}{clll}
1798
\hspace{2em}k\hspace{2em} & \hspace{4em}\Omega^+ \hspace{4em}&
1799
\hspace{4em}\Omega^-\hspace{4em}\\
1800
12& 0.002281474899 & 0.000971088287i \\
1801
16 &0.003927981492 & 0.000566379403i \\
1802
18& 0.000286607497 & 0.023020042428i \\
1803
20& 0.008297636952 & 0.0005609325015i \\
1804
22& 0.002589288079 & 0.0020245743816i \\
1805
24 &0.000000002968& 0.0000000054322i& \\
1806
26 &0.003377464512 & 0.3910726132671i \\
1807
28& 0.000000015627 & 0.0000000029272i&
1808
\end{array}$
1809
\vspace{-2ex}\\
1810
\\\hline
1811
\end{tabular}\end{center}
1812
\end{table}
1813
1814
The volumes appear to be {\em much} smaller than
1815
the volumes of weight two abelian varieties.
1816
The dimension of each $A_f$ is~$1$, except for
1817
weights $24$ and $28$ when the dimension is~$2$.
1818
%The invariants $c_4$, $c_6$, and~$j$ of the elliptic curves
1819
%can be calculated from the period lattice using the
1820
%algorithm described in \cite{cremona:algs}.
1821
1822
1823
\subsubsection{CM elliptic curves of weight greater than two}
1824
\label{cmellipticcurves}
1825
Let~$f$ be a rational newform with complex multiplication.
1826
Experimentally, it appears that the
1827
associated elliptic~$A_f$ has rational $j$-invariant.
1828
As evidence for this we present Table~\ref{table:cmcurves},
1829
which includes the analytic data about every
1830
rational CM form of weight four and level $\leq 197$.
1831
The computations of Table~\ref{table:cmcurves} were done
1832
using at least~$97$ terms of the $q$-expansion of~$f$.
1833
The rationality of $j$ could probably be proved by observing
1834
that the CM forces $A_f$ to have extra automorphisms.
1835
1836
\begin{table}
1837
\begin{center}
1838
\caption{CM elliptic curves of weight $>2$.\label{table:cmcurves}}
1839
\begin{tabular}{|c|}\hline
1840
$\begin{array}{rcccrr}
1841
E & j &\Omega^+&\Omega^- & c_4\hspace{2em} & c_6 \hspace{2em} \\
1842
\text{\bf 9k4A} & 0 & 0.2095 & 0.1210i & 0.0000 & -56626421686.2951\\
1843
\text{\bf 32k4A} &1728 & 0.2283 & 0.2283i & -3339814.8874 & 0.0000\\
1844
\text{\bf 64k4D} &1728 & 0.1614 & 0.1614i& 53437038.1988 & 0.0000\\
1845
\text{\bf 108k4A} & 0 & 0.0440 & 0.0762i& -14699.2655 & 24463608892439.7456\\
1846
\text{\bf 108k4C}& 0 & 0.0554 & 0.0960i& 1608.7743 & 6115643810955.1724\\
1847
\text{\bf 121k4A}&-2^{15}& 0.0116 & 0.0385i &
1848
85659519816.8841 & 25723073306989527.1216\\
1849
\text{\bf 144k4E}& 0 & 0.0454 & 0.0262i& 81.1130&
1850
-549788016394046.1396\\
1851
\text{\bf 27k6A} & 0 & 0.0110 & 0.0191i& 0.0000 & 97856189971744203.7795\\
1852
\text{\bf 32k6A} &1728 & 0.0199 & 0.0199i& -58095643136.7658&8.0094\\
1853
\end{array}$
1854
\vspace{-2ex}\\
1855
\\\hline
1856
\end{tabular}\end{center}
1857
\end{table}
1858
1859
In these examples, the invariants $c_4$ and $c_6$ are unreckognizable
1860
to the author; in contrast, in weight~$2$ these invariants are (expected to
1861
be) integers (see \cite[2.14]{cremona:algs}).
1862
1863
\comment{
1864
\begin{remark}
1865
For the curves {\bf 32k4A} and {\bf 32k6A},
1866
we have $\Omega^+=\Omega^-\pmod{i}$.
1867
This is because each lattice admits complex multiplication by~$i$
1868
and is hence invariant under rotation by $90$ degrees.
1869
The same thing happens with the next few higher weights at
1870
level~$32$. For how many weights does it persists?
1871
\end{remark}
1872
1873
1874
function ssinvs(N,k)
1875
p := 100;
1876
ss := 37;
1877
np := #[p : p in [1..ss] | IsPrime(p)];
1878
M:=ModularSymbols(N,k);
1879
D:=Decomposition(M);
1880
LabelFactors(D);
1881
L := [* *];
1882
for A in D do
1883
if IsNew(A) and IsCuspidal(A) then
1884
f := qEigenform(A,ss);
1885
ns := #[i : i in [1..Degree(f)] | IsPrime(i) and Coefficient(f,i) eq 0];
1886
if ns/np gt 0.3 then
1887
e := EllipticInvariants(A,p);
1888
omr := RealVolume(A,p);
1889
omi := ImaginaryVolume(A,p);
1890
printf "%ok%o%o\nc4=\t%o\nc6=\t%o\nj=\t%o\no=\t%o\ni=\t%o\n\n",
1891
N,k,ToIsogenyCode(IsogenyClass(A)),e[3],e[4],e[5],omr,omi;
1892
Append(~L,A);
1893
end if;
1894
end if;
1895
end for;
1896
return L;
1897
end function;
1898
}
1899
1900
\subsubsection{Some abelian varieties of large dimension}
1901
In Table~\ref{table:bigvols}, we give the volumes of five abelian
1902
varieties of dimension greater than~$1$. In each case, at least
1903
$200$ terms of the $q$-expansions were used.
1904
\begin{table}
1905
\begin{center}
1906
\caption{Volumes of high dimensional abelian varieties.\label{table:bigvols}}
1907
\begin{tabular}{|c|}\hline
1908
\vspace{-2ex}\\
1909
$\begin{array}{rcrr}
1910
A & \dim & \hspace{2em}\Omega^+ & \hspace{2em}\Omega^- \\
1911
\text{\bf 79k2B} & 5 & 10 & 209i\\
1912
\text{\bf 83k2B} & 6 & 22 & 41\\
1913
\text{\bf 131k2B} & 10 & 51 & 615\\
1914
\text{\bf 11k4A} & 2 & 0.0815 & 0.0212\,\,\\
1915
\text{\bf 17k4B} & 3 & 0.0047 & 0.0007i\\
1916
\end{array}$
1917
\vspace{-2ex}\\
1918
\\\hline
1919
\end{tabular}\end{center}
1920
\end{table}
1921
1922
1923
1924
1925