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 p...

The purpose of this paper is to exhibit an explicit formula which describes the projection operator from the space of C°° automorphic forms to the subspace of holomorphic cusp forms, and to apply it to the zeta functions of Rankin type. Fix a number k > 0 such that 2k G Z. Let TV be a positive integer such that TV = 0 mod(4) if k fi Z, and let x(Z/NZ) —+ C be a Dirichlet character modulo TV. De...

We study the detailed structure of the distribution of Eichler-Shimura periods of an automorphic form on a compact hyperbolic surface. We show that these periods do not cluster around the asymptotic period over a homology class discovered by Zelditch. 0. Introduction and Results Let M = H/Γ be a compact hyperbolic surface, where Γ is a discrete subgroup of PSL(2,R) = SL(2,R)/{±I}. Such a surfac...

1. The trace formula of Selberg reduces the problem of calculating the dimension of a space of automorphic forms, at least when there is a compact fundamental domain, to the evaluation of certain integrals. Some of these integrals have been evaluated by Selberg. An apparently different class of definite integrals has occurred in Harish-Chandra’s investigations of the representations of semi-sim...