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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
DOI: https://doi.org/10.1090/ert/486
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
Email: niccronc@stanford.edu

DOI: https://doi.org/10.1090/ert/486
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