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Published January 2015 | public
Journal Article

A mild Tchebotarev theorem for GL(n)

Abstract

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) by showing that, for any cyclic extension K/k of number fields of prime degree p, a cuspidal automorphic representation π of GL(n,A_K) is determined up to twist equivalence, even up to isomorphism if p=2, by the knowledge of its local components at the (density one) set S_(K/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 p^2-extensions.

Additional Information

© 2014 Elsevier Inc. Received 31 March 2014; Received in revised form 27 August 2014; Accepted 27 August 2014; Available online 16 September 2014. We thank the many people who have shown interest in this work over the past few years, especially to those who have used it and have encouraged, like K. Martin, to have it published. Thanks are also due to the NSF for partial support through the grants DMS-0701089 and DMS-1001916. This article is dedicated to the memory of Steve Rallis from whom this author learnt a lot in conversations over the years.

Additional details

Created:
August 22, 2023
Modified:
March 5, 2024