# Asymptotic distribution of Hecke eigensystems

-
Marco Sangiovanni, Princeton University

The aim of this talk is to explain a result due to Serre that proves that, as the level and weight varies, the Hecke eigensystems associated to classical cusp forms are equidistributed with respect to some explicit continuous measure. The main tool of the proof is the Eichler-Selberg trace formula which computes in an explicit way the traces of Hecke operators. I will also explain some interesting applications of this result; for example, the fact that the Ramanujan conjecture provides, at least asymptotically, an optimal bound for Hecke eigenvalues, and the fact that algebraic extensions associated to eigenforms have asymptotically "big degree".