Summation methods and distribution of eigenvalues of Hecke operators

Abstract

Let $p$ be a fixed prime number. Let $S_k(N)$ be the space of cusp forms of weight $k$ and level $N$. We prove a weighted equidistribution theorem for the eigenvalues of the $p$-th Hecke operator $T_p$ acting on $S_k(N)$. This is a variant of a celebrated theorem of Serre.