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.
Here is the paper (300KB).
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
