Local base change via Tate cohomology

Ronchetti, Niccolò

doi:10.1090/ert/486

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.

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