Arithmetic of automorphic representations

Abstract

There are two types of automorphic representations, algebraic automorphic representations and non-algebraic automorphic representations. Modular forms are examples of algebraic automorphic representations, and Maass forms are examples of non-algebraic automorphic representations. In this thesis, we investigate the arithmetic of automorphic representations for each type. First, we provide the result of the number of proportions of Hecke eigenvalues in two distinct newforms. Second, we investigate the sufficient conditions for having infinitely many quadratic character such that the central values of L-function of twists of modular forms by the quadratic character is not divided by a fixed prime. Finally, we introduce the equidistribution of Hecke eigenvalues of non-algebraic automorphic representations such as Maass forms.