Local Base Change via Tate Cohomology

2015/07/02 by Ronchetti, Niccolò · 1 citation
#FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1507.00745

Abstract

In this paper we propose a new way to realize cyclic base change (a special case of Langlands functoriality) for prime degree extensions of characteristic zero local fields. Let F / E be a prime degree l extension of local fields of residue characteristic p ≠ l. Let π be an irreducible cuspidal l-adic representation of GL_n(E) and ρ be an irreducible cuspidal l-adic representation of GL_n(F) which is Galois-invariant. Under some minor technical conditions on π and ρ (for instance, we assume that both are level zero) we prove that the \bmod l-reductions r_l(π) and r_l(ρ) are in base change if and only if the Tate cohomology of ρ with respect to the Galois action is isomorphic, as a modular representation of GL_n(E), to the Frobenius twist of r_l(π). This proves a special case of a conjecture of Treumann and Venkatesh as they investigate the relationship between linkage and Langlands functoriality.

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• Jacquet modules of Tate cohomology and base change lifting

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