### 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^{+}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.

Original language | English (US) |
---|---|

Pages (from-to) | 77-148 |

Number of pages | 72 |

Journal | Algebra and Number Theory |

Volume | 9 |

Issue number | 1 |

DOIs | |

State | Published - 2015 |

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### Keywords

- Adequate representations
- Artin–Wedderburn theorem
- Automorphic representations
- Galois representations
- Irreducible representations

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Algebra and Number Theory*,

*9*(1), 77-148. https://doi.org/10.2140/ant.2015.9.77

**Adequate groups of low degree.** / Guralnick, Robert; Herzig, Florian; Tiep, Pham Huu.

Research output: Contribution to journal › Article

*Algebra and Number Theory*, vol. 9, no. 1, pp. 77-148. https://doi.org/10.2140/ant.2015.9.77

}

TY - JOUR

T1 - Adequate groups of low degree

AU - Guralnick, Robert

AU - Herzig, Florian

AU - Tiep, Pham Huu

PY - 2015

Y1 - 2015

N2 - 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+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.

AB - 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+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.

KW - Adequate representations

KW - Artin–Wedderburn theorem

KW - Automorphic representations

KW - Galois representations

KW - Irreducible representations

UR - http://www.scopus.com/inward/record.url?scp=84924180070&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84924180070&partnerID=8YFLogxK

U2 - 10.2140/ant.2015.9.77

DO - 10.2140/ant.2015.9.77

M3 - Article

AN - SCOPUS:84924180070

VL - 9

SP - 77

EP - 148

JO - Algebra and Number Theory

JF - Algebra and Number Theory

SN - 1937-0652

IS - 1

ER -