Vol. 276, No. 1, 2015

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ISSN: 0030-8730
On the degree of certain local $L$-functions

U. K. Anandavardhanan and Amiya Kumar Mondal

Vol. 276 (2015), No. 1, 1–17
Abstract

Let π be an irreducible supercuspidal representation of GLn(F), where F is a p-adic field. By a result of Bushnell and Kutzko, the group of unramified self-twists of π has cardinality ne, where e is the oF-period of the principal oF-order in Mn(F) attached to π. This is the degree of the local Rankin–Selberg L-function L(s,π × π). In this paper, we compute the degree of the Asai, symmetric square, and exterior square L-functions associated to π. 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
Authors
U. K. Anandavardhanan
Department of Mathematics
Indian Institute of Technology Bombay
Mumbai 400076
India
Amiya Kumar Mondal
Department of Mathematics
Indian Institute of Technology Bombay
Mumbai 400076
India