We compute the Bessel models of every irreducible representation of the finite group \operatorname{GSp}(4,q) .

In this paper, we introduce the notion of a bi- \overline{\mathbb{Q}} -structure on the tangent space at a CM point on a locally Hermitian symmetric domain. We prove that this bi- \overline{\mathbb{Q}} -structure decomposes into the direct sum of 1 -dimensional bi- \overline{\mathbb{Q}} -subspaces, and make this decomposition explicit for the moduli space of abelian varieties \mathbb{A}_{g} .We propose an analytic subspace conjecture , which is the analogue of the Wüstholz’s analytic subgroup theorem in this context. We show that this conjecture, applied to \mathbb{A}_{g} , implies that all quadratic \overline{\mathbb{Q}} -relations among the holomorphic periods of CM abelian varieties arise from elementary ones.

We study four types of (co)cartesian fibrations of \infty -bicategories over a given base \mathcal{B} , and prove that they encode the four variance flavors of \mathcal{B} -indexed diagrams of \infty -categories. We then use this machinery to setup a general theory of marked (co)limits for diagrams valued in an \infty -bicategory, capable of expressing lax, weighted and pseudo limits. When the \infty -bicategory at hand arises from a model category tensored over marked simplicial sets, we show that this notion of marked (co)limit can be calculated as a suitable form of a weighted homotopy limit on the model categorical level, thus showing in particular the existence of these marked (co)limits in a wide range of examples. We finish by discussing a notion of cofinality appropriate to this setting and use it to deduce the unicity of marked (co)limits, provided they exist.

Let R be a local Artin ring with residue field k of positive characteristic. We prove that every finite flat group scheme over R whose special fiber belongs to a certain explicit class of non-commutative k -group schemes is killed by its order. This is achieved via a classification result which relies on the study of the infinitesimal deformation theory for such non-commutative k -group schemes. The main result answers positively in a new case a question of Grothendieck in SGA 3 on whether all finite flat group schemes are killed by their order, and improves the currently best known result due to Schoof.

Loop torsors over Laurent polynomial rings in characteristic 0 were originally introduced in relation to infinite dimensional Lie theory. Applications to other areas require a theory that can yields results in positive characteristic, and for group schemes that are not of finite type. The relation between loop and so-called toral torsors, is one of the central questions in the area. The present paper addresses this question in full generality.

We introduce notions of lax semiadditive and lax additive (\infty,2) -categories, categorifying the classical notions of semiadditive and additive 1 -categories. To establish a well-behaved axiomatic framework, we develop a calculus of lax matrices and use it to prove that in locally cocomplete (\infty,2) -categories lax limits and lax colimits agree and are absolute. In the lax additive setting, we categorify fundamental constructions from homological algebra such as mapping complexes and mapping cones and establish their basic properties.

We prove conservativity results for weak Kőnig’s lemma that extend the celebrated result of Harrington (for \Pi^{1}_{1} -statements) and are somewhat orthogonal to the extension by Simpson, Tanaka and Yamazaki (for statements of the form \forall X\exists!Y\psi with arithmetical \psi ). In particular, we show that \mathsf{WKL}_{0} is conservative over \mathsf{RCA}_{0} for well-ordering principles. We also show that compactness (which characterizes weak Kőnig’s lemma) is dispensable for certain results about continuous functions with isolated singularities.

We show that for a braided Hopf algebra in the category of comodules over a cosemisimple coquasitriangular Hopf algebra, the Hochschild cohomological dimension, the left and right global dimensions and the projective dimensions of the trivial left and right module all coincide. We also provide convenient criteria for smoothness and the twisted Calabi–Yau property for such braided Hopf algebras (without the cosemisimplicity assumption on H ), in terms of properties of the trivial module. These generalize well-known results in the case of ordinary Hopf algebras. As an illustration, we study the case of the coordinate algebra on the two-parameter braided quantum group \mathrm{SL}_{2} .

For a prime p and a smooth proper p -adic formal scheme \mathscr{X} over \mathcal{O}_K where K is a p -adic field, we study a series of conditions ( \mathrm{Cr}_s ), s\geq 0 that partially control the G_K ‑action on the image of the associated Breuil–Kisin prismatic cohomology \def\prisma{\vartriangle\hspace*{-3.9pt}{\tiny\vartriangle}} \mathsf{R}\Gamma_{\prisma}(\mathscr{X}/\mathfrak{S})\ inside the A_{\mathrm{inf}} -prismatic cohomology \def\prisma{\vartriangle\hspace*{-3.9pt}{\tiny\vartriangle}} \mathsf{R}{\Gamma_{\prisma}}(\mathscr{X}_{A_{\mathrm{inf}}}/A_{\mathrm{inf}}).\ The condition ( \mathrm{Cr}_{0} ) is a crystallinity criterion for a Breuil–Kisin–Fargues G_K -module of Gee and Liu, and leads to a proof of crystallinity of \mathrm{H}^{i}_{\text{\'{e}t}}(\mathscr{X}_{\overline{\eta}},\mathbb{Q}_{p}) that avoids the crystalline comparison. Using the higher conditions ( \mathrm{Cr}_{s} ), we are able to adapt the strategy of Caruso and Liu to establish ramification bounds for the mod p representations \mathrm{H}^{i}_{\text{\'{e}t}}(\mathscr{X}_{\overline{\eta}},\mathbb{Z}/p\mathbb{Z}) , for arbitrarily large e and i . This extends and/or improves existing bounds in various situations.

We compute the wrapped Fukaya category \mathcal{W}(T^{*}S^{1}, D) of a cylinder relative to a divisor D= \{p_{0},\dots,p_{n}\} of n+1 points, proving a mirror equivalence with the category of perfect complexes on a crepant resolution (over k\llbracket t_{0},\dots,t_{n}\rrbracket ) of the singularity uv=t_{0}t_{1}\cdots t_{n} . Upon making the base-change t_{i}= f_{i}(x,y) , we obtain the derived category of any crepant resolution of the cA_{n} singularity given by the equation uv= f_{0}\cdots f_{n} . These categories inherit braid group actions via the action on \mathcal{W}(T^{*}S^{1},D) of the mapping class group of T^{*}S^{1} fixing D . We also give geometric models for the derived contraction algebras associated to a cA_{n} singularity in terms of the relative Fukaya category of the disc.