Appendix A: Adequacy of representations of finite groups of Lie type

TL;DR

This paper investigates the adequacy of certain representations of finite groups of Lie type, particularly SL(2,q), and demonstrates their application in automorphy and number theory, extending the concept of adequate representations.

Contribution

It proves that specific representations of SL(2,q) are adequate and generalizes adequacy results for finite groups of Lie type in natural characteristic.

Findings

Certain SL(2,q) representations are adequate

Adequacy results are extended to broader classes of finite groups of Lie type

Applications to automorphy of Galois representations and modular forms

Abstract

Thorne introduced the notion of adequate representations as a weakening of the big representations used by Wiles and Taylor and others. In this appendix to Dieulefait's paper, Automorphy of Symm5(GL(2)) and base change, we show that certain representations of SL(2,q) are adequate. This is used by Dieulefait to prove results about Hecke eigenforms of level 1 and newforms. We also prove some general results about adequacy for representations of finite groups of Lie type in the natural characteristic.