Vol. 9, No. 1, 2015

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Adequate groups of low degree

Robert Guralnick, Florian Herzig and Pham Huu Tiep

Vol. 9 (2015), No. 1, 77–147

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 kG-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.

Artin–Wedderburn theorem, irreducible representations, automorphic representations, Galois representations, adequate representations
Mathematical Subject Classification 2010
Primary: 20C20
Secondary: 11F80
Received: 13 April 2014
Accepted: 14 December 2014
Published: 18 February 2015
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
Pham Huu Tiep
Department of Mathematics
University of Arizona
617 North Santa Rita Avenue
Tucson, AZ 85721-0089
United States