Arithmetic properties of L-functions attached to Hilbert modular forms

Abstract

Scope and Method of Study: The goal of this thesis is to study some arithmetic properties of L-functions attached to Hilbert modular forms. We mainly use a representation theoretical point of view for the study, which can be done by associating Hilbert modular forms of our interests with automorphic representations of GL(2). Furthermore, their L-functions are deeply related. We use this realization to analyze the critical L-values for Hilbert modular forms, which reduces some technical difficulties.

Findings and Conclusions: The thesis focuses on three main theorems which concern: Algebraicity theorem; Congruence property; and Non-vanishing property. The first theorem is completed by interpreting the Mellin transform cohomologically, and the second follows from analyzing it integrally. The third theorem is obtained by studying the completed L-functions of Hilbert modular forms.