Ian Petrow

Bounds for traces of Hecke operators and points on curves over finite fields

Consider the space of classical holomorphic cusp forms with respect to a congruence subgroup, and the algebra of Hecke operators acting on this space. In this talk I will discuss estimates for the traces of these Hecke operators, as well as their applications to the analytic theory of elliptic curves over a finite field. For example, such bounds on traces of Hecke operators allow one to estimate the number of F_q-points on modular curves, going beyond what the Riemann hypothesis for curves over finite fields can tell us directly. I will also discuss a Petersson trace formula for newforms for general nebentype characters, which is one of the main tools used in the proof of the results in this talk.