On the existence of large degree Galois representations for fields of small discriminant

Jeremy Rouse and Frank Thorne

Vol. 271 (2014), No. 1, 243–256

DOI: 10.2140/pjm.2014.271.243

Abstract

Let $L∕K$ be a Galois extension of number fields. We prove two lower bounds on the maximum of the degrees of the irreducible complex representations of $Gal\left(L∕K\right)$, the sharper of which is conditional on the Artin conjecture and the generalized Riemann hypothesis. Our bound is nontrivial when $\left[K:\mathbb{Q}\right]$ is small and $L$ has small root discriminant, and might be summarized as saying that such fields can’t be “too abelian”.

Keywords

Artin $L$-function, Galois representation, Rankin–Selberg $L$-function

Mathematical Subject Classification 2010

Primary: 11R29

Secondary: 11R42

Milestones

Received: 4 March 2013

Revised: 17 June 2014

Accepted: 22 July 2014

Published: 10 September 2014