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Representation Theory

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Local base change via Tate cohomology

Author: Niccolò Ronchetti
Journal: Represent. Theory 20 (2016), 263-294
MSC (2010): Primary 11F70, 11S37, 22E50
Published electronically: September 27, 2016
MathSciNet review: 3551160
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Abstract: 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 $p \neq l$. Let $\pi$ be an irreducible cuspidal $l$-adic representation of $\mathrm {GL}_n(E)$ and let $\rho$ be an irreducible cuspidal $l$-adic representation of $\mathrm {GL}_n(F)$ which is Galois-invariant. Under some minor technical conditions on $\pi$ and $\rho$ (for instance, we assume that both are level zero) we prove that the $\bmod l$-reductions $r_l(\pi )$ and $r_l(\rho )$ are in base change if and only if the Tate cohomology of $\rho$ with respect to the Galois action is isomorphic, as a modular representation of $\mathrm {GL}_n(E)$, to the Frobenius twist of $r_l(\pi )$. 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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Additional Information

Niccolò Ronchetti
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305

Received by editor(s): July 2, 2015
Received by editor(s) in revised form: April 21, 2016, and July 18, 2016
Published electronically: September 27, 2016
Article copyright: © Copyright 2016 American Mathematical Society