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

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.