Recent zbMATH articles in MSC 20C33https://www.zbmath.org/atom/cc/20C332021-06-15T18:09:00+00:00WerkzeugAdequate subgroups and indecomposable modules.https://www.zbmath.org/1460.200032021-06-15T18:09:00+00:00"Guralnick, Robert"https://www.zbmath.org/authors/?q=ai:guralnick.robert-m"Herzig, Florian"https://www.zbmath.org/authors/?q=ai:herzig.florian"Tiep, Pham Huu"https://www.zbmath.org/authors/?q=ai:tiep.pham-huuSummary: ``The notion of adequate subgroups was introduced by \textit{J. Thorne} [J. Inst. Math. Jussieu 11, No. 4, 855--920 (2012; Zbl 1269.11054)]. 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 [\textit{R. Guralnick} et al., J. Eur. Math. Soc. (JEMS) 19, No. 4, 1231--1291 (2017; Zbl 06705852); Algebra Number Theory 9, No. 1, 77--147 (2015; Zbl 1365.20008)] 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 \textit{J. A. Thorne} [Math. Z. 285, No. 1--2, 1--38 (2017; Zbl 1427.11050)], which allows
\(p\) to divide the dimension of the module. We prove adequacy for almost all irreducible
representations of \(\mathrm{SL}_2(p^a)\) 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.''
Let \(G\) be a finite group.
From the introduction:
``more generally, we say that an absolutely irreducible
representation \(\rho: G \to \mathrm{GL}(V)\) is adequate if:
(i) \(H^1(G, k) = 0\);
(ii) \(H^1(G, (V^* \otimes V)/k) = 0\);
(iii) \(\mathrm{End}(V)\) is spanned by the elements \(\rho(g)\) with \(g\) semisimple.''
The authors collect a lot of information. There are many cases to consider.
Reviewer: Wilberd van der Kallen (Utrecht)Weight elimination in Serre-type conjectures.https://www.zbmath.org/1460.110772021-06-15T18:09:00+00:00"Le, Daniel"https://www.zbmath.org/authors/?q=ai:le.daniel"Hung, Bao V. Le"https://www.zbmath.org/authors/?q=ai:le-hung.bao-v"Levin, Brandon"https://www.zbmath.org/authors/?q=ai:levin.brandonSummary: We prove the weight elimination direction of the Serre weight conjectures as formulated by Herzig for forms of \(U(n)\) which are compact at infinity and split at places dividing \(p\) in generic situations. That is, we show that all modular weights for a mod \(p\) Galois representation are contained in the set predicted by Herzig. Under some additional hypotheses, we also show modularity of all the ``obvious'' weights.