Local exterior square and Asai $L$-functions for $\mathrm{GL}(n)$ in odd characteristic

Yeongseong Jo

Vol. 322 (2023), No. 2, 301–340

DOI: 10.2140/pjm.2023.322.301

Abstract

Let $F$ be a nonarchimedean local field of odd characteristic $p>0$. We consider local exterior square $L$-functions $L(s,\pi ,{\wedge }^2)$, Bump–Friedberg $L$-functions $L(s,\pi ,\mathrm{BF}<!--FUNCTION\; APPLICATION-->)$, and Asai $L$-functions $L(s,\pi ,\mathrm{As}<!--FUNCTION\; APPLICATION-->)$ of an irreducible admissible representation $\pi$ of ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_m(F)$. In particular, we establish that those $L$-functions, via the theory of integral representations, are equal to their corresponding Artin $L$-functions $L\left(s,{\wedge }^2(\varphi (\pi ))\right)$, $L\left(s+\frac{1}{2},\varphi (\pi )\right)L\left(2s,{\wedge }^2(\varphi (\pi ))\right)$, and $L\left(s,\mathrm{As}<!--FUNCTION\; APPLICATION-->(\varphi (\pi ))\right)$ 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 Lomelí.

Keywords

Bernstein–Zelevinsky derivatives, local exterior square and Asai $L$-functions in positive characteristic, Rankin–Selberg methods

Mathematical Subject Classification

Primary: 11F70

Secondary: 11F85, 22E50

Milestones

Received: 20 April 2022

Revised: 22 December 2022

Accepted: 11 February 2023

Published: 23 May 2023