Adequate subgroups II (original) (raw)
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Adequate subgroups and indecomposable modules
Journal of the European Mathematical Society, 2017
The notion of adequate subgroups was introduced by Jack Thorne [60]. It is a weakening of the notion of big subgroups used by Wiles and Taylor in proving automorphy lifting theorems for certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] and [23] that if the dimension is smaller than the characteristic then almost all absolutely irreducible representations are adequate. We extend the results by considering all absolutely irreducible modules in characteristic p of dimension p. This relies on a modified definition of adequacy, provided by Thorne in [61], which allows p to divide the dimension of the module. We prove adequacy for almost all irreducible representations of SL 2 (p a) in the natural characteristic and for finite groups of Lie type as long as the field of definition is sufficiently large. We also essentially classify indecomposable modules in characteristic p of dimension less than 2p − 2 and answer a question of Serre concerning complete reducibility of subgroups in classical groups of low dimension.
Algebra & Number Theory, 2015
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. 1. Introduction 78 2. Linear groups of low degree 81 3. Weak adequacy for SL 2 ކ( p) 85 4. Weak adequacy for Chevalley groups 89 5. Weak adequacy in cross-characteristic 98 6. Weak adequacy for special linear groups 114 7. Extensions and self-extensions, I: Generalities 127 8. Indecomposable representations of SL 2 (q) 131 9. Finite groups with indecomposable modules of small dimension 133 10. Extensions and self-extensions, II 140 References 144
2016
Let l be a prime, and let Γ be a finite subgroup of GL n (F l) = GL(V). With these assumptions we say that Condition (C) holds if for every irreducible Γ-submodule W ⊂ ad 0 V there exists an element g ∈ Γ with an eigenvalue α such that tr e g,α W = 0. Here, e g,α denotes the projection to the generalised α-eigenspace of g. This condition arises in the definition of adequacy in section 2. Let Γ ss denote the subset of Γ consisting of the elements that are semisimple (i.e. of order prime to l).
Appendix : Adequate Subgroups 59 Appendix : Adequate Subgroups
2010
Let l be a prime, and let Γ be a finite subgroup of GLn(Fl) = GL(V ). With these assumptions we say that Condition (C) holds if for every irreducible Γ-submodule W ⊂ ad V there exists an element g ∈ Γ with an eigenvalue α such that tr eg,αW 6= 0. Here, eg,α denotes the projection to the generalised α-eigenspace of g. This condition arises in the definition of adequacy in section 2. Let Γ denote the subset of Γ consisting of the elements that are semisimple (i.e. of order prime to l).
On superfluous subgroups and fully invariant subgroups
Hacettepe University Bulletin of Natural Sciences and Engineering Series B: Mathematics and Statistics
This paper is mainly concerned with Abelian groups having the lifting property with respect to fully invariant and projection invariant subgroups.
A Group Theoretical Characterisation of S-Arithmetic Groups in Higher Rank Semi-Simple Groups
2002
We now state the main theorem proved in the present paper. There are a number of definitions which are used in the statement. They will be explained in sections 1 and 2. Main Theorem. Let Γ be a group satisfying the following conditions. (1) Γ is finitely generated and is virtually torsion free (see the end of the introduction for an explanation of this term). (2) The semi-simple dimension of Γ (denoted s.s. dim (Γ)) is finite and is positive (see sections (1.1)-(1.3) for an explanation of the terms semisimple p−dimension (denoted s.s.p-dim)) and semi-simple dimension (denoted s.s.dim). (3) The semi-simple rank of Γ (denoted s..s. rank (Γ)) is at least two : s.s. rank (Γ) ≥ 2 (see (2.1) for the definition). (4) Γ is hereditarily just infinite (i.e. if Γ ⊂ Γ is a subgroup of finite index and N ⊂ Γ is a normal subgroup of Γ , then either Γ /N is finite or N is finite). (5) Let X def = {l | l is a prime and s.s.l-dim(Γ) < s.s.dim (Γ)} (we prove, under the assumptions (1) through (4), that X is a finite set: see the proof of Theorem (1.17)). Let q be a prime with q not in X. Form the product Ω = ∈X Γ × Γ q. Let Q l (l ∈ X) be the semi-simple quotient of Γ l and Q q that of Γ q (see (1.17) for a proof that under the assumption (2), such quotients exist). Let Ω = l∈X Q l × Q q ; there is a natural map from Γ to Ω which under the assumptions (1)-(4), is a virtual inclusion. Let Γ ⊂ Λ ⊂ Ω and Λ satisfy the properties (1), (2), (3) (but not necessarily (4)) as well as the equalities s.s.p − dim(Λ) = s.s.p − dim(Γ) for every prime p. Then we assume that Λ/Γ is finite. Under these assumptions (1) through (5), Γ is virtually isomorphic to an S−arithmetic subgroup Γ of a group G, which is defined over a number field K and is absolutely simple (here, S is a finite set of places of K containing all the Archimedean ones). Moreover, S−rank (G) def = v∈S K v −rank (G) ≥ 2, Γ ⊂ G(O S) , O S = S-integers on K, and G(O S)/Γ is finite. Remark: We note (see §1 and §2) that all these assumptions (s.s. dimension being finite, semi-simple rank being at least two, and the condition (5)) are properties of a quotient of a certain prop completion of Γ, and as such, are not really dependent on a linear realisation of Γ. However, it turns out (see Theorem (1.17)) that there is a more or less canonical linear realisation of our group Γ under these assumptions. Corollary 1. Suppose Γ is a group satisfying the properties (1) through (5) of the Main Theorem. Suppose that s.s.p−dim(Γ) = s.s.dim(Γ) for every prime p (in other words, suppose that the set X of (5) of the Main Theorem is empty). Then, Γ is isomorphic to an arithmetic subgroup of a higher rank semisimple real Lie group. K such that S−rank (G) := v∈S K v −rank (G) ≥ 2. Let Γ be a subgroup of finite index in G(O s) (O S = S-integers in K). Then Γ satisfies the propertis (1) through (5). It is of interest to note that in fact, we can recover-purely from the abstract properties of Γ-the S-dimension of the semi-simple group in which Γ sits as a lattice , and also its S-rank. However, the field K over which our group G is defined, and the set S of places, are irrecoverable by our methods (our methods use the prop completion of Γ and there are non-isomorphic arithmetic groups with isomorphic profinite completions (see section (3.1)). We prove the Main Theorem in §3. In the course of the proof, we also use Theorem 2. Among the S-arithmetic groups characterised by the Main Theorem, it would be desirable to single out the non-uniform ones. In §3 we also define the notion of a u-element (following [L-M-R1]) and deduce Corollary 3. Let Γ be an abstract group possessing the properties (1) through (5) of the Main Theorem. Suppose Γ has a u−element of infinite order. Then Γ is isomorphic to an S−arithmetic non-uniform lattice.
On some finiteness properties of algebraic groups over finitely generated fields
Comptes Rendus Mathematique, 2016
We present several finiteness results for absolutely almost simple algebraic groups over finitely generated fields that are more general than global fields. We also discuss the relations between the various finiteness properties involved in these results, such as the properness of the global-to-local map in the Galois cohomology of a given K-group G relative to a certain natural set V of discrete valuations of K, and the finiteness of the number of isomorphism classes of K-forms of G having, on the one hand, smooth reduction at V and, on the hand, the same isomorphism classes of maximal K-tori as G. Résumé. Nous présentons plusieurs résultats de finitude pour les groupes algébriques absolument presque simples définis sur des corps de type fini plus généraux que les corps globaux. Nous discutons aussi des liens entre les propriétés de finitude divers qui entrent dans le cadre de notre analyse, tels que la propreté de l'application globale-locale dans la cohomologie galoisienne d'un K-groupe G par rapportà un ensemble convenable V de valuations discrètes de K, et la finitude du nombre de K-formes de G ayant, d'une part, bonne réduction en V , et, d'autre part, possédant les même classes d'isomorphisme de K-tores maximaux que G. même ensemble marcheégalement pour les groupes de type G 2. Dans les sections 2 et 3, il s'agit de l'analyse des groupes algébriques absolument presque simples possédant les mêmes classes d'isomorphisme de tores maximaux sur le corps de définition. Plus précisément, si G est un groupe algébrique simplement connexe absolument presque simple défini sur 1