Automorphic forms, trace formulas and zeta functions

Around 1980, K. Doi and H. Hida found a meaning of the special value of certain degree 3 L-functions, so called adjoint L-functions of cusp forms. They discovered that if a prime divides “algebraic part” of the adjoint L-function of a cusp form, the prime is a congruence prime for the cusp form. E. Ghate and M. Dimitrov proved analogues of Hida’s theorem in Hilbert modular case. E. Urban also proved a similar result in case of cusp forms on GL(2) over imaginary quadratic fields. In this talk, we prove such a result in case of cusp forms on GL(2) over number fields. 13:30-14:30 Gombodorj Bayarmagnai (University of Tokyo) Title: Differential equations satisfied by principal series Whittaker functions on SU(2, 2) Abstract: In this talk, we consider principal series representations of SU(2, 2) with higher dimensional minimal K-type and discuss about a system of differential equations for Whittaker functions associated with these K-types.