Cuspidality criterion for symmetric powers of automorphic representations of GL(2) over function fields

TL;DR

This paper provides a comprehensive criterion for when symmetric powers of cuspidal automorphic representations of GL(2) over function fields remain cuspidal, utilizing the Langlands correspondence and extending number field results.

Contribution

It establishes a new cuspidality criterion for symmetric powers over function fields, leveraging the Langlands correspondence and prior results from number field cases.

Findings

Established a cuspidality criterion for symmetric powers over function fields.

Extended number field results to the setting of function fields.

Utilized the Langlands correspondence for Galois representations.

Abstract

Given a cuspidal automorphic representation of GL(2) over a global function field, we establish a comprehensive cuspidality criterion for symmetric powers. The proof is via passage to the Galois side, possible over function fields thanks to the Langlands correspondence of L. Lafforgue and additional results of G. Henniart and B. Lemaire. Our work is guided by the number fields results of Kim-Shahidi and Ramakrishnan.