A Mild Tchebotarev Theorem for GL$(n)$

TL;DR

This paper extends the Tchebotarev density theorem to automorphic representations of GL(n) over cyclic extensions, showing they are determined by local data at primes of degree one, with stronger results for quadratic extensions.

Contribution

It demonstrates that cuspidal automorphic representations of GL(n) over cyclic extensions are uniquely determined by local components at primes of degree one, advancing the automorphic analogue of Tchebotarev's theorem.

Findings

Automorphic representations are determined by local data at primes of degree one.

For quadratic extensions, representations are determined even up to isomorphism.

Uses Luo-Rudnick-Sarnak bounds and descent techniques in the proof.

Abstract

It is well known that the Tchebotarev density theorem implies that an irreducible $\ell$-adic representation $\rho$ 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$(n)$ by showing that, given a cyclic extension $K/k$ of number fields of prime degree $p$, a cuspidal automorphic representation $\pi$ of GL$(n,{\mathbb A}_K)$ is determined up to twist equivalence by the knowledge of its local components at the (density one) set $S_{K/k}$ of primes of $K$ of degree $1$ over $k$, and moreover that $\pi$ is determined even up to isomorphism if $p=2$. The proof uses the Luo-Rudnick-Sarnak bound for the Hecke roots of $\pi$, applied to certain Rankin-Selberg $L$-functions of positive type, in conjunction with some Kummer theory and descent along suitable $p$-power extensions arising as nested sequences of cyclic $p^2$-extensions.