Non-split Extensions and Hecke L-functions of Imaginary Quadratic Fields

Abstract

In this thesis, we present an adelic construction of some non-split extension of rational mixed Hodge structures arising from the cohomology of modular curves, which witnesses the order 1 vanishing of Hecke L-functions of imaginary quadratic fields at s=0, as predicted by Beilinson's Conjecture. This recasts the construction of Skinner in the adelic language. The main calculation is based on an adelic description of the cohomology of modular curves in terms of the (s,K)-cohomology of automorphic forms, an adelic description of Hecke L-functions in terms of Tate's Zeta integrals, and an adelic description of Eisenstein series in terms of Godements's flat sections.