Local Base Change via Tate Cohomology

Abstract

In this paper 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 $p\ne l$. Let $\pi$ be an irreducible cuspidal $l$-adic representation of ${\mathrm{GL}}_n(E)$ and $\rho$ 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 $\mathrm{mod}l$-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.

arXiv comment31 pages. Typos corrected, referee report addressed, all results unchanged. To appear in the AMS Journal of Representation Theory