Analytic and Geometric Number Theory Seminar

A result of Kim-Sarnak (2003) gives the best known bounds towards the Ramanujan conjecture for Maass forms. The technique employed has not, until now, been made to apply to general GL2 cusp forms over number fields whose unit group is infinite. In this talk we will show how one can slightly modify the method of Kim-Sarnak, using infinite order Hecke characters, to extend their numerical bound to an arbitrary number field. The proof naturally allows one to obtain new non-vanishing results for L-functions twisted by a wide class of Hecke characters, at points near the edge of the critical strip. These results have been obtained in collaboration with Valentin Blomer.