A Mild Tchebotarev Theorem for GL(n)

Details A Mild Tchebotarev Theorem for GL(n) A Mild Tchebotarev Theorem for GL(n)

2010-03-23

arxiv.org/abs/1003.4498v1

Mathematics

Number Theory

15 pages

Scientific paper

It is well known that the Tchebotarev density theorem implies that an irreducible ell-adic representation R 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. The analogue for cusp forms of GL(n) is open for n>2. In this Note we make some progress by showing that, given a cyclic extension K/k of number fields of prime degree p, a cuspidal automorphic representation Pi of GL(n,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.

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