A Theorem on GL(n) a la Tchebotarev

Abstract

Let $K/F$ be a finite Galois extension of number fields. It is well known that the Tchebotarev density theorem implies that an irreducible, finitely ramified $p$-adic representation $\rho$ of the absolute Galois group of $K$ is determined (up to equivalence) by the characteristic polynomials of Frobenius elements Fr$_v$ at any set of primes $v$ of $K$ of degree $1$ over $F$. Here we prove an analogue for GL$(n)$, namely that a cuspidal automorphic representation $\pi$ of GL$(n, {\mathbb A}_K)$ is determined up by the knowledge of its local components at the primes of degree one over $F$. We prove in fact a stronger theorem, stimulated by a question of Michael Rapoport and Wei Zhang, relaxing to an extent the Galois hypothesis. The method uses, besides the Rankin-Selberg theory of L-functions and the Luo-Rudnick-Sarnak bound for the Hecke roots of $\pi$, certain consequences of class field theory via Galois cohomology. In an earlier paper (\cite{Ra2}) we obtained such a result up to twist equivalence for $K/F$ cyclic of prime degree by using basic Kummer theory. We make use of suitable solvable base changes $\pi_M$, relative to certain auxiliary succession of abelian extensions $E/F$, with $M$ being an abelian extension of the compositum $EK$, and deduce that $\pi_M \simeq \pi'_M$, and then descend this isomorphism to one over $K$. A key ingredient for progress here is the use of global Tate duality and a local-global result arising from class field theory. In fact we prove the main result for {\it isobaric} automorphic representations, which are analogues of {\it semisimple} Galois representations. In the last section we introduce a notion of {\it semi-temperedness}, which is much weaker than temperedness, but allows for the deduction of the main result without any hypothesis whatsoever on $K/F$.