The local exterior square and Asai $L$-functions for $GL(n)$ in odd characteristic

TL;DR

This paper proves that certain local $L$-functions for $GL(n)$ over odd characteristic fields match their Langlands parameter counterparts, confirming key aspects of the local Langlands correspondence for these functions.

Contribution

It establishes the equality of local exterior square, Bump-Friedberg, and Asai $L$-functions with their Artin $L$-functions via integral representations in odd characteristic.

Findings

Proved the equality of local $L$-functions and Artin $L$-functions for supercuspidal representations.

Extended the local Langlands correspondence to include these specific $L$-functions.

Utilized local-global techniques to establish the identities.

Abstract

Let $F$ be a non-archimedean local field of odd characteristic $p > 0$. In this paper, we consider local exterior square $L$-functions $L(s,\pi,\wedge^2)$, Bump-Friedberg $L$-functions $L(s,\pi,BF)$, and Asai $L$-functions $L(s,\pi,As)$ of an irreducible admissible representation $\pi$ of $GL_m(F)$. In particular, we establish that those $L$-functions, via the theory of integral representations, are equal to their corresponding Artin $L$-functions $L(s,\wedge^2(\phi(\pi)))$, $L(s+1/2,\phi(\pi))L(s,\wedge^2(\phi(\pi)))$, and $L(s,As(\phi(\pi)))$ of the associated Langlands parameter $\phi(\pi)$ under the local Langlands correspondence. These are achieved by proving the identity for irreducible supercuspidal representations, exploiting the local to global argument due to Henniart and Lomeli.