Zeros of Poincaré series
Noam Kimmel (MPIM Bonn)

We explore the zeros of certain Poincaré series P(k,m) of weight k and index m for the full modular group. These are distinguished modular forms, which have played a key role in the analytic theory of modular forms. We study the zeros of P(k,m) when the weight k tends to infinity. The case where the index m is constant was considered by Rankin who showed that in this case almost all of the zeros lie on the unit arc |z|=1. In this talk we will explore the location of the zeros when the index m grows with the weight k, finding a range of different limit laws. Along the way, we also establish a version of Quantum Unique Ergodicity for some ranges.

Zoom (680 4828 0736, The password is the first Fourier coefficient of the modular j-function (as digits)).