Adequate groups of low degree

Guralnick, Robert; Herzig, Florian; Tiep, Pham Huu

doi:10.2140/ant.2015.9.77

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Adequate groups of low degree

Robert Guralnick, Florian Herzig and Pham Huu Tiep

Vol. 9 (2015), No. 1, 77–147

DOI: 10.2140/ant.2015.9.77

Abstract

The notion of adequate subgroups was introduced by Jack Thorne. It is a weakening of the notion of big subgroups used in generalizations of the Taylor–Wiles method for proving the automorphy of certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown by Guralnick, Herzig, Taylor, and Thorne that if the dimension is small compared to the characteristic, then all absolutely irreducible representations are adequate. Here we extend that result by showing that, in almost all cases, absolutely irreducible $k G$ -modules in characteristic $p$ whose irreducible $G^{+}$ -summands have dimension less than $p$ (where $G^{+}$ denotes the subgroup of $G$ generated by all $p$ -elements of $G$ ) are adequate.

Keywords

Artin–Wedderburn theorem, irreducible representations, automorphic representations, Galois representations, adequate representations

Mathematical Subject Classification 2010

Primary: 20C20

Secondary: 11F80

Milestones

Received: 13 April 2014

Accepted: 14 December 2014

Published: 18 February 2015

Authors

Robert Guralnick
Department of Mathematics University of Southern California 3620 South Vermont Ave Los Angeles, CA 90089-2532 United States

Florian Herzig
Department of Mathematics University of Toronto 40 Saint George Street, Room 6290 Toronto, ON M5S 2E4 Canada

Pham Huu Tiep
Department of Mathematics University of Arizona 617 North Santa Rita Avenue Tucson, AZ 85721-0089 United States

Vol. 9, No. 1, 2015