A mild Tchebotarev theorem for GL(n)GL(n)

It is well known that the Tchebotarev density theorem implies that an irreducible ℓ-adic representation ρ of the absolute Galois group of a number field K   is determined (up to isomorphism) by the characteristic polynomials of Frobenius elements at any set of primes of density 1. In this Note we make some progress on the automorphic side for GL(n)GL(n) by showing that, for any cyclic extension K/kK/k of number fields of prime degree p, a cuspidal automorphic representation π   of GL(n,AK)GL(n,AK) is determined up to twist equivalence, even up to isomorphism if p=2p=2, by the knowledge of its local components at the (density one) set SK/kSK/k of primes of K of degree 1 over k. The proof uses the Luo–Rudnick–Sarnak bound, certain L  -functions of positive type, Kummer theory, and automorphic descent along suitable nested sequences of cyclic p2p2-extensions.