Cohomology of GL(2)

Abstract

Let $F$ be a number field. Let $G=GL(2)$ over $F$. Let $\mathbb{A}$ and $\mathbb{A}_f$ denote the ring of adeles and finite adeles of $F$ respectively. Let $K_{\infty}$ denote the maximal compact subgroup of $G_{\infty}=\prod_{\nu} G(F_{\nu})$ thickened by the center, where the product runs over all archimedean places $\nu$ of $F$ and $F_{\nu}$ denotes the completion of $F$ at $\nu$. For a fixed open compact subgroup $K_f \subset G(\mathbb{A}_f)$, let $S^G_{K_f} = G(F) \backslash G(\mathbb{A})/K_fK_{\infty}$ be a locally symmetric space attached to $G$. Let $r[d]$ be an irreducible representation of $G$ of highest weight $d$, and let $\mathcal{F}_d$ denote the corresponding sheaf on $S^G_{K_f}$. The goal of this project is to understand the inner cohomology denoted $H_!^{\bullet}(S^G_{K_f}, \mathcal{F}_d)$, which by definition is the image of compactly supported cohomology in full cohomology, via its relation to automorphic forms of $G$. It is known that inner cohomology contains cuspidal cohomology, which is genered by cusp forms on $G$. The problem is to classify inner cohomology classes that are not cuspidal. In this thesis, we deal with $F=\mathbb{Q}$ and $F=\mathbb{Q}(\sqrt{-n})$ where $n$ is a square free positive integer. We give a description of the inner cohomology mentioned above in these two cases.