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

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 $G{L}_m(F)$. In particular, we establish that those $L$-functions, via the theory of integral representations, are equal to their corresponding Artin $L$-functions $L(s,{\wedge }^2(\varphi (\pi )))$, $L(s+1/2,\varphi (\pi ))L(s,{\wedge }^2(\varphi (\pi )))$, and $L(s,As(\varphi (\pi )))$ of the associated Langlands parameter $\varphi (\pi )$ 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.

28 pages, incorporated all referee's comments

References in corpus (1)

• The functional equation of the Jacquet-Shalika integral representation of the local exterior-square $L$-function