Effective equidistribution of Hecke eigenvalues

For a fixed prime p, we consider the space S(N,k) of cusp forms of weight k and level N, with N coprime to p. In 1995, J.-P. Serre proved the existence of a measure up with respect to which the eigenvalues of the pth Hecke operator acting on S(N,k) are equidistributed as k+N tends to infinity. We will derive an effective version of Serre's theorem and apply it to study the factorization of J0(N) into simple abelian varieties. Our methods can also be applied to study the variation of eigenvalues of the Frobenius automorphism acting on a family of curves mod p and the variation of eigenvalues of adjacency matrices of regular graphs. (This is joint work with Kaneenika Sinha.)