Math 252: Modular Abelian Varieties

Universal Fourier expansions of modular forms

Merel, Loïc

What it is about

This paper contains a systematic description of modular symbols for Gamma1(N) of weight at least 2. The main application is a discussion of how to compute Hecke and other operators directly on Manin symbols using Herbrand matrices.

Electronic Version

Here is the paper (300KB).

MathSciNet


This paper is concerned with an interesting way of describing (in terms of Fourier expansions at infinity) the space $S_k(N)$ of cusp forms of integral weight $k\geq 2$ for the subgroup $\Gamma_1(N)$ of ${\rm SL}_2(Z)$. (The subspaces $S_k(N,\chi)$, where the $\chi$ are Dirichlet characters, are also considered, as are spaces of newforms and not necessarily cuspidal forms.)
The author uses the fact that if $\alpha$ is a linear map from the Hecke algebra to $\bold C$ then $\sum _{i=1}^{\infty}\alpha(T_n)q^n$ is the Fourier expansion of a cusp form. Using results of Shokurov on modular symbols he shows how to produce such an $\alpha$ given a linear map $\phi\colon \bold C_{k-2}[X,Y][(\bold Z/N\bold Z)^2]\rightarrow \bold C$ and an element $x\in \bold C_{k-2}[X,Y][(\bold Z/N\bold Z)^2]$, both satisfying certain special conditions. (The subscript is the weight of homogeneous polynomials.) In combination with all possible $\phi$, "most" $x$ will then produce the whole of $S_k(N)$, and there is a natural special choice, though it is difficult to find such $x$ explicitly.

Given $\phi$ and $x$, it remains to compute $\alpha(T_n)$. For this the author examines in detail the actions of the Hecke operators on modular symbols, showing how to make the computation whenever one has an element $\sum_Mu_MM$ of $\bold C[M_2(\bold Z)_n]$ satisfying a special condition $C_n$. (Here $M_2(\bold Z)_n$ is the set of $2\times 2$ matrices of determinant $n$ with integer coefficients.) Then he actually produces four separate families of elements satisfying the conditions $C_n$, and suggests that there may be other interesting examples.

Reviewed by Neil Dummigan