L-functions for Classical Groups: The Integral Representations, Algebraicity and the p-adic Interpolations

Abstract

The main theme of this thesis is the study of special values of L-functions through integral representations. We present an integral representation of the standard L-functions for classical groups via the doubling method. Our computations, comparing with the well-known result for partial L-functions by Piatetski-Shapiro and Rallis, include all ramified local integrals with the explicit choice of local sections for Eisenstein series. When the classical group admits a Shimura variety, we have a well-defined notion of algebraic modular forms. In this case, we calculate the Fourier expansion of Eisenstein series from which the properties of their special values can be easily read off. Utilizing our integral representations, we then prove the algebraicity of certain special L-values for modular forms on some classical groups. Furthermore, by our proper choice of the local sections for Eisenstein series, we construct the p-adic L-functions interpolating these special L-values.

Generalizing the classical doubling method, Cai, Friedberg, Ginzburg and Kaplan presents an integral representation for Sp(2n)×GL(k) by the twisted doubling method. In the final chapter of the thesis, we present another integral representation for the L-functions of Sp(2n)×GL(k) via a non-unique model and obtain some analytic results.