Effective equidistribution of eigenvalues of Hecke operators

Abstract

In 1997, Serre proved an equidistribution theorem for eigenvalues of Hecke operators on the space S ( N , k ) of cusp forms of weight k and level N. In this paper, we derive an effective version of Serre's theorem. As a consequence, we estimate, for a given d and prime p coprime to N, the number of eigenvalues of the pth Hecke operator T p acting on S ( N , k ) of degree less than or equal to d. This allows us to determine an effectively computable constant B d such that if J 0 ( N ) is isogenous to a product of Q -simple abelian varieties of dimensions less than or equal to d, then N ⩽ B d . We also study the effective equidistribution of eigenvalues of Frobenius acting on a family of curves over a fixed finite field as well as the eigenvalue distribution of adjacency matrices of families of regular graphs. These results are derived from a general “all-purpose” equidistribution theorem.