Local newforms and formal exterior square L-functions

TL;DR

This paper proves that Jacquet-Shalika integrals for newforms in generic representations of GL_n(F) realize the formal exterior square L-functions, linking local zeta integrals with Galois-theoretic L-functions.

Contribution

It establishes that Jacquet-Shalika integrals attain the formal exterior square L-functions for newforms, connecting integral representations with Galois side L-functions.

Findings

Jacquet-Shalika integrals match formal exterior square L-functions for newforms.

Formal exterior square L-functions coincide with exterior square L-functions for certain principal series.

The results bridge local zeta integrals and Galois representations in the context of GL_n(F).

Abstract

Let F be a non-archimedean local field of characteristic zero. Jacquet and Shalika attached a family of zeta integrals to unitary irreducible generic representations $\pi$ of GL_n(F). In this paper, we show that Jacquet-Shalika integral attains a certain L-function, so called the formal exterior square L-function, when the Whittaker function is associated to a newform for $\pi$. By consideration on the Galois side, formal exterior square L-functions are equal to exterior square L-functions for some principal series representations.