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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Local base change via Tate cohomology
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by Niccolò Ronchetti PDF
Represent. Theory 20 (2016), 263-294 Request permission


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
  • Email:
  • 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
  • © Copyright 2016 American Mathematical Society
  • Journal: Represent. Theory 20 (2016), 263-294
  • MSC (2010): Primary 11F70, 11S37, 22E50
  • DOI:
  • MathSciNet review: 3551160