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