Local base change via Tate cohomology

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 F / E be a prime degree l l extension of local fields of residue characteristic p ≠ l p \neq l . Let π \pi be an irreducible cuspidal l l -adic representation of G L n ( E ) \mathrm {GL}_n(E) and let ρ \rho be an irreducible cuspidal l l -adic representation of G L n ( F ) \mathrm {GL}_n(F) which is Galois-invariant. Under some minor technical conditions on π \pi and ρ \rho (for instance, we assume that both are level zero) we prove that the mod l \bmod \,l -reductions r l ( π ) r_l(\pi ) and r l ( ρ ) r_l(\rho ) are in base change if and only if the Tate cohomology of ρ \rho with respect to the Galois action is isomorphic, as a modular representation of G L n ( E ) \mathrm {GL}_n(E) , to the Frobenius twist of r l ( π ) 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.