Adequate groups of low degree

Abstract

Abstract. The notion of adequate subgroups was introduced by Jack Thorne [42]. 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 in [22] that if the dimension is small compared to the characteristic then all absolutely irreducible representations are adequate. Here we extend the result by showing that, in almost all cases, absolutely irreducible kG-modules in characteristic p, whose irreducible G

Serre weights for $GSp_4$ over totally real fields

“…For small p ≤ 7, if the image of ρ contains Sp 4 (F p ) for p > 2 or ρ Sym 3 τ for some τ : G F −→ GL 2 (F p ) so that τ contains SL 2 (F p ) for p > 5 , then ρ(G F (ζp) ) is adequate. This follows immediately from Corollary 1.5 of [45] and Proposition 2.1.1 of [7]. Since all subgroups in GSp 4 (F p ) which act irreducibly on F ⊕4 p via the natural inclusion GSp 4 (F p ) ⊂ GL 4 (F p ) are classified , for example, in [21], we would be able to prove a similar result of Proposition 2.1.1 of [7].…”

Adequate subgroups and indecomposable modules

“…We have shown that A i ∼ = B i for all i; in particular W j ∼ = W ǫ 1 . Now we have that m = 2, and dim k Ext 1 L 2 (A 2 , B 2 ) equals 0 if p a > 5 and 1 if p a = 5, see [23,Lemma 8.1]. Again by Lemma 3.3, B 2 ).…”

“…In fact, as shown in [23,Corollary 1.5], if dim V < p − 3, then the (p ± 1)/2-dimensional representations of SL 2 (p) are the only two counterexamples. More precisely, in [23] we extend Theorem 1.1 to the more general situation that dim W < p and show that almost always (G, V ) is adequate: Theorem 1.2. Let k be a field of characteristic p and G a finite group.…”

“…Throughout this section, we assume that k is an algebraically closed field of characteristic p > 3. First we recall several intermediate results proved in [23]:…”

“…Given the assumption of Theorem 1.5, suppose that H is as in the extraspecial case (e) of Theorem 2.1. Then [23,Theorem 2.4], H 1 (G + , k) = 0, and so we are done.…”