  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 $p \neq l$. 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 $\bmod
l$-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.
