A p-adic integral for the reciprocal of L-functions

TL;DR

This paper develops a p-adic analog of the Langlands-Shahidi method to construct reciprocals of p-adic L-functions using Eisenstein series, providing explicit measures for SL(2) related to Dirichlet L-functions.

Contribution

It introduces a novel p-adic approach to invert certain L-functions, extending the Langlands-Shahidi method to the p-adic setting for the group SL(2).

Findings

Constructed explicit p-adic measures for reciprocals of Dirichlet L-functions.

Extended the Langlands-Shahidi method to the p-adic context.

Demonstrated the method for the group SL(2).

Abstract

We introduce an analog of part of the Langlands-Shahidi method to the p-adic setting, constructing reciprocals of certain p-adic L-functions using the nonconstant terms of the Fourier expansions of Eisenstein series. We carry out the method for the group SL(2), and give explicit p-adic measures whose Mellin transforms are reciprocals of Dirichlet L-functions.