Vol. 271, No. 1, 2014

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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
Abstract

Let LK 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(LK), the sharper of which is conditional on the Artin conjecture and the generalized Riemann hypothesis. Our bound is nontrivial when [K : ] 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
Authors
Jeremy Rouse
Department of Mathematics
Wake Forest University
P.O. Box 7388
Winston-Salem, NC 27109
United States
Frank Thorne
Department of Mathematics
University of South Carolina
1523 Greene Street
Columbia, SC 29208
United States