## Local base change via Tate cohomology

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- by Niccolò Ronchetti
- Represent. Theory
**20**(2016), 263-294 - DOI: https://doi.org/10.1090/ert/486
- Published electronically: September 27, 2016
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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.## References

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## Bibliographic Information

**Niccolò Ronchetti**- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- Email: niccronc@stanford.edu
- 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: https://doi.org/10.1090/ert/486
- MathSciNet review: 3551160