Algebraicity and the $p$-adic Interpolation of Special $L$-values for certain Classical Groups

TL;DR

This paper develops explicit integral representations for standard L-functions of classical groups, proving algebraicity of special L-values and constructing p-adic L-functions, advancing understanding in automorphic forms and number theory.

Contribution

It explicitly computes ramified local integrals, constructs local sections of Eisenstein series, and establishes algebraicity and p-adic interpolation of special L-values for various classical groups.

Findings

Explicit formulas for ramified local integrals

Construction of p-adic L-functions for classical groups

Proof of algebraicity of special L-values

Abstract

In this paper, we calculate the ramified local integrals in the doubling method and present an integral representation of standard $L$-functions for classical groups. We explicitly construct local sections of Eisenstein series such that the local ramified integrals represent certain ramified $L$-factors. As an application, we prove algebraicity of special $L$-values and construct $p$-adic $L$-functions for symplectic, unitary, quaternionic unitary and quaternionic orthogonal groups.