On the degree of certain local $L$-functions

U. K. Anandavardhanan and Amiya Kumar Mondal

Vol. 276 (2015), No. 1, 1–17

DOI: 10.2140/pjm.2015.276.1

Abstract

Let $\pi$ be an irreducible supercuspidal representation of ${GL}_n\left(F\right)$, where $F$ is a $p$-adic field. By a result of Bushnell and Kutzko, the group of unramified self-twists of $\pi$ has cardinality $n∕e$, where $e$ is the ${\mathfrak{o}}_F$-period of the principal ${\mathfrak{o}}_F$-order in $M_n\left(F\right)$ attached to $\pi$. This is the degree of the local Rankin–Selberg $L$-function $L\left(s,\pi \times {\pi }^{\vee }\right)$. In this paper, we compute the degree of the Asai, symmetric square, and exterior square $L$-functions associated to $\pi$. As an application, assuming $p$ is odd, we compute the conductor of the Asai lift of a supercuspidal representation, where we also make use of the conductor formula for pairs of supercuspidal representations due to Bushnell, Henniart, and Kutzko (1998).

Keywords

Asai $L$-function, symmetric square $L$-function, exterior square $L$-function, degree of a local $L$-function

Mathematical Subject Classification 2010

Primary: 22E50

Secondary: 11F33, 11F70, 11F85

Milestones

Received: 31 March 2014

Revised: 9 November 2014

Accepted: 9 December 2014

Published: 1 July 2015