Recent zbMATH articles in MSC 11F75https://www.zbmath.org/atom/cc/11F752021-04-16T16:22:00+00:00WerkzeugGeneral Serre weight conjectures.https://www.zbmath.org/1456.110932021-04-16T16:22:00+00:00"Gee, Toby"https://www.zbmath.org/authors/?q=ai:gee.toby"Herzig, Florian"https://www.zbmath.org/authors/?q=ai:herzig.florian"Savitt, David"https://www.zbmath.org/authors/?q=ai:savitt.davidSummary: We formulate a number of related generalisations of the weight part of Serre's conjecture to the case of \(\mathrm{GL}_n\) over an arbitrary number field, motivated by the formalism of the Breuil-Mézard conjecture. We give evidence for these conjectures, and discuss their relationship to previous work. We generalise one of these conjectures to the case of connected reductive groups which are unramified over \(\mathbb{Q}_p\), and we also generalise the second author's previous conjecture for \(\mathrm{GL}_n/\mathbb{Q}\) to this setting, and show that the two conjectures are generically in agreement.Supercuspidal ramifications and traces of adjoint lifts.https://www.zbmath.org/1456.110522021-04-16T16:22:00+00:00"Banerjee, Debargha"https://www.zbmath.org/authors/?q=ai:banerjee.debargha"Mandal, Tathagata"https://www.zbmath.org/authors/?q=ai:mandal.tathagataSummary: In this paper, we write down the local Brauer classes of the endomorphism algebras of motives attached to non-CM primitive Hecke eigenforms for the supercuspidal prime \(p = 2\). The same for odd supercuspidal primes are determined by \textit{S. Bhattacharya} and \textit{E. Ghate} [Proc. Am. Math. Soc. 143, No. 11, 4669--4684 (2015; Zbl 1378.11053)]. We also treat the case of odd unramified supercuspidal primes of level zero also removing a mild hypothesis of them. As an intermediate step, we write down a description of the inertial Galois representation even for \(p = 2\) generalizing the construction of \textit{E. Ghate} and \textit{A. Mézard} [Trans. Am. Math. Soc. 361, No. 5, 2243--2261 (2009; Zbl 1251.11044)]. Some numerical examples using age and MFDB are provided supporting some of our theorems.Profinite invariants of arithmetic groups.https://www.zbmath.org/1456.200232021-04-16T16:22:00+00:00"Kammeyer, Holger"https://www.zbmath.org/authors/?q=ai:kammeyer.holger"Kionke, Steffen"https://www.zbmath.org/authors/?q=ai:kionke.steffen"Raimbault, Jean"https://www.zbmath.org/authors/?q=ai:raimbault.jean"Sauer, Roman"https://www.zbmath.org/authors/?q=ai:sauer.romanOne says that an arithmetic group has the conguence subgroup property if the conguence kernel of it is finite.
The main result of the paper states that the sign of the Euler characteristic of an arithmetic group \(\Gamma\) with the congruence subgroup property is determined by its profinite completion (or equivalently by the family of its finite quotients). More precisely, Theorem 1.1 states that two arithmetic groups \(\Gamma_1,\Gamma_2\) with the conguence subgroup property and commensurable profinite completions have the same sign of their Euler characteristic.
Note that by the result of \textit{M. Aka} [J. Algebra 352, No. 1, 322--340 (2012; Zbl 1254.20026)] an arithmetic group with the conguence subgroup property is determined by its profinite completion up to finitely many isomorphism calsses among arithmetic groups.
It is also shown in the paper that two natural generalizations of Theorem 1.1 do not hold. Namely, the Euler charateristic of \(\Gamma\) is not determined by the profinite completion. Also the profinite completion does not determine the sign of the Euler characteristic of finitely generated residually finite group of type \(F\).
Reviewer: Pavel Zalesskij (Brasília)