Let R be a discrete valuation ring of field of fractions K and of residue field k of characteristic $p> 0$. In an earlier work, we studied the question of extending torsors over K-curves into torsors over R-regular models of the curves in the case when the structural K-group scheme of the torsor admits a finite flat model over R. In this paper, we first give a simpler description of the problem in the case where the curve is semistable using recent work in Holmes, Molcho, Orecchia, and Poiret (2023, Journal für die Reine und Angewandte Mathematik [Crelle’s Journal] 230, 115–159) and Molcho and Wise (2022, Compositio Mathematica 158, 1477–1562). Second, if R is assumed to be Henselian and Japanese, we solve the problem of extending torsors by combining our previous work together with results in Antei and Emsalem (2018, Nagoya Mathematical Journal 230, 18–34) and Phung and Dos Santos (2023, Algebraic Geometry 230, 1–40), including the case where the structural group does not admit a finite flat R-model.

Let ${\mathcal {R}} \subset \mathbb {P}^1_{\mathbb {C}}$ be a finite subset of markings. Let G be an almost simple simply-connected algebraic group over $\mathbb {C}$. Let $K_G$ denote the compact real form of G. Suppose for each lasso l around the marked point, a conjugacy class $C_l$ in $K_G$ is prescribed. The aim of this paper is to give verifiable criteria for the existence of an irreducible homomorphism of $\pi _{1}(\mathbb P^1_{\mathbb {C}} \,{\backslash}\, {\mathcal {R}})$ into $K_G$ such that the image of l lies in $C_l$.

We prove that after inverting the Planck constant $h$, the Bezrukavnikov–Kaledin quantization $(X, {\mathcal {O}}_h)$ of symplectic variety $X$ in characteristic $p$ with $H^2(X, {\mathcal {O}}_X) =0$ is Morita equivalent to a certain central reduction of the algebra of differential operators on $X$.

In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme $G$ over a valuation ring $V$ with fraction field $K$, a $G$-torsor over $V$ is trivial if it is trivial over $K$. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture.

Let $G$ be a reductive group over an algebraically closed field $k$ of separably good characteristic $p>0$ for $G$. Under these assumptions, a Springer isomorphism $\phi : \mathcal {N}_{\mathrm {red}}(\mathfrak {g}) \rightarrow \mathcal {V}_{\mathrm {red}}(G)$ from the nilpotent scheme of $\mathfrak {g}$ to the unipotent scheme of $G$ always exists and allows one to integrate any $p$-nilpotent element of $\mathfrak {g}$ into a unipotent element of $G$. One should wonder whether such a punctual integration can lead to an integration of restricted $p$-nil $p$-subalgebras of $\mathfrak {g}= \operatorname {Lie}(G)$. We provide a counter-example of the existence of such an integration in general, as well as criteria to integrate some restricted $p$-nil $p$-subalgebras of $\mathfrak {g}$ (that are maximal in a certain sense). This requires the generalisation of the notion of infinitesimal saturation first introduced by Deligne and the extension of one of his theorems on infinitesimally saturated subgroups of $G$ to the previously mentioned framework.

We generalize the works of Pappas–Rapoport–Zhu on twisted affine Grassmannians to the wildly ramified case under mild assumptions. This rests on a construction of certain smooth affine $\mathbb {Z}[t]$ -groups with connected fibers of parahoric type, motivated by previous work of Tits. The resulting $\mathbb {F}_p(t)$ -groups are pseudo-reductive and sometimes non-standard in the sense of Conrad–Gabber–Prasad, and their $\mathbb {F}_p [\hspace {-0,5mm}[ {t} ]\hspace {-0,5mm}] $ -models are parahoric in a generalized sense. We study their affine Grassmannians, proving normality of Schubert varieties and Zhu’s coherence theorem.

In characteristic $0$ , symplectic automorphisms of K3 surfaces (i.e., automorphisms preserving the global $2$ -form) and non-symplectic ones behave differently. In this paper, we consider the actions of the group schemes $\mu _{n}$ on K3 surfaces (possibly with rational double point [RDP] singularities) in characteristic p, where n may be divisible by p. We introduce the notion of symplecticness of such actions, and we show that symplectic $\mu _{n}$ -actions have similar properties, such as possible orders, fixed loci, and quotients, to symplectic automorphisms of order n in characteristic $0$ . We also study local $\mu _n$ -actions on RDPs.

The Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete valuation rings, as well as a suitable presentation lemma for smooth relative curves in mixed characteristic that facilitates passage to the relative affine line via excision and patching.

We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] and about Galois actions [14]. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.

We prove that if G is a finite flat group scheme of p-power rank over a perfect field of characteristic p, then the second crystalline cohomology of its classifying stack $H^2_{\text {crys}}(BG)$ recovers the Dieudonné module of G. We also provide a calculation of the crystalline cohomology of the classifying stack of an abelian variety. We use this to prove that the crystalline cohomology of the classifying stack of a p-divisible group is a symmetric algebra (in degree $2$) on its Dieudonné module. We also prove mixed-characteristic analogues of some of these results using prismatic cohomology.

Soient K un corps discrètement valué et hensélien, ${\mathcal {O}}$ son anneau d’entiers supposé excellent, $\kappa $ son corps résiduel supposé parfait et G un K-groupe quasi-réductif, c’est-à-dire lisse, affine, connexe et à radical unipotent déployé trivial. On construit l’immeuble de Bruhat-Tits ${\mathcal {I}}(G, K)$ pour $G(K)$ de façon canonique, améliorant les constructions moins canoniques de M. Solleveld sur les corps locaux, et l’on associe un ${\mathcal {O}}$ -modèle en groupes ${\mathcal {G}}_{\Omega }$ de G à chaque partie non vide et bornée $\Omega $ contenue dans un appartement de ${\mathcal {I}}(G,K)$ . On montre que les groupes parahoriques ${\mathcal {G}}_{\textbf {f}}$ attachés aux facettes peuvent être caractérisés en fonction de la géométrie de leurs grassmanniennes affines, ainsi que dans la thèse de T. Richarz. Ces résultats sont appliqués ailleurs à l’étude des grassmanniennes affines tordues entières.

According to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti–Tate group. We extend this result to 1-motives.

This paper is dedicated to a problem raised by Jacquet Tits in 1956: the Weyl group of a Chevalley group should find an interpretation as a group over what is nowadays called $\mathbb{F}_{1}$, the field with one element. Based on Part I of The geometry of blueprints, we introduce the class of Tits morphisms between blue schemes. The resulting Tits category$\text{Sch}_{{\mathcal{T}}}$ comes together with a base extension to (semiring) schemes and the so-called Weyl extension to sets. We prove for ${\mathcal{G}}$ in a wide class of Chevalley groups—which includes the special and general linear groups, symplectic and special orthogonal groups, and all types of adjoint groups—that a linear representation of ${\mathcal{G}}$ defines a model $G$ in $\text{Sch}_{{\mathcal{T}}}$ whose Weyl extension is the Weyl group $W$ of ${\mathcal{G}}$. We call such models Tits–Weyl models. The potential of Tits–Weyl models lies in (a) their intrinsic definition that is given by a linear representation; (b) the (yet to be formulated) unified approach towards thick and thin geometries; and (c) the extension of a Chevalley group to a functor on blueprints, which makes it, in particular, possible to consider Chevalley groups over semirings. This opens applications to idempotent analysis and tropical geometry.

Order three elements in the exceptional groups of type ${{G}_{2}}$ are classified up to conjugation over arbitrary fields. Their centralizers are computed, and the associated classification of idempotents in symmetric composition algebras is obtained. Idempotents have played a key role in the study and classification of these algebras.

Over an algebraically closed field, there are two conjugacy classes of order three elements in ${{G}_{2}}$ in characteristic not 3 and four of them in characteristic 3. The centralizers in characteristic 3 fail to be smooth for one of these classes.

Let $U$ be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck’s étale fundamental group $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(U)$. In this paper, we consider a purely inseparable analogue of this problem, formulated in terms of Nori’s profinite fundamental group scheme $\unicode[STIX]{x1D70B}^{N}(U)$, and give a partial answer to it.

We describe a general method for expanding a truncated $G$-iterative Hasse–Schmidt derivation, where $G$ is an algebraic group. We give examples of algebraic groups for which our method works.

This paper concerns the classification of isogeny classes of $p$-divisible groups with saturated Newton polygons. Let $S$ be a normal Noetherian scheme in positive characteristic $p$ with a prime Weil divisor $D$. Let ${\mathcal{X}}$ be a $p$-divisible group over $S$ whose geometric fibers over $S\setminus D$ (resp. over $D$) have the same Newton polygon. Assume that the Newton polygon of ${\mathcal{X}}_{D}$ is saturated in that of ${\mathcal{X}}_{S\setminus D}$. Our main result (Corollary 1.1) says that ${\mathcal{X}}$ is isogenous to a $p$-divisible group over $S$ whose geometric fibers are all minimal. As an application, we give a geometric proof of the unpolarized analogue of Oort’s conjecture (Oort, J. Amer. Math. Soc. 17(2) (2004), 267–296; 6.9).

We consider the distribution of $p$-power group schemes among the torsion of abelian varieties over finite fields of characteristic $p$, as follows. Fix natural numbers $g$ and $n$, and let ${\it\xi}$ be a non-supersingular principally quasipolarized Barsotti–Tate group of level $n$. We classify the $\mathbb{F}_{q}$-rational forms ${\it\xi}^{{\it\alpha}}$ of ${\it\xi}$. Among all principally polarized abelian varieties $X/\mathbb{F}_{q}$ of dimension $g$ with $X[p^{n}]_{\bar{\mathbb{F}}_{q}}\cong {\it\xi}_{\bar{\mathbb{F}}_{q}}$, we compute the frequency with which $X[p^{n}]\cong {\it\xi}^{{\it\alpha}}$. The error in our estimate is bounded by $D/\sqrt{q}$, where $D$ depends on $g$, $n$, and $p$, but not on ${\it\xi}$.

The celebrated Smith–Minkowski–Siegel mass formula expresses the mass of a quadratic lattice $(L,Q)$ as a product of local factors, called the local densities of $(L,Q)$. This mass formula is an essential tool for the classification of integral quadratic lattices. In this paper, we will describe the local density formula explicitly by observing the existence of a smooth affine group scheme $\underline{G}$ over $\mathbb{Z}_{2}$ with generic fiber $\text{Aut}_{\mathbb{Q}_{2}}(L,Q)$, which satisfies $\underline{G}(\mathbb{Z}_{2})=\text{Aut}_{\mathbb{Z}_{2}}(L,Q)$. Our method works for any unramified finite extension of $\mathbb{Q}_{2}$. Therefore, we give a long awaited proof for the local density formula of Conway and Sloane and discover its generalization to unramified finite extensions of $\mathbb{Q}_{2}$. As an example, we give the mass formula for the integral quadratic form $Q_{n}(x_{1},\dots ,x_{n})=x_{1}^{2}+\cdots +x_{n}^{2}$ associated to a number field $k$ which is totally real and such that the ideal $(2)$ is unramified over $k$.

Let $k$ be an infinite field. Let $R$ be the semi-local ring of a finite family of closed points on a $k$-smooth affine irreducible variety, let $K$ be the fraction field of $R$, and let $G$ be a reductive simple simply connected $R$-group scheme isotropic over $R$. Our Theorem 1.1 states that for any Noetherian $k$-algebra $A$ the kernel of the map

$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{1}(R\otimes _{k}A,G)\rightarrow H_{\acute{\text{e}}\text{t}}^{1}(K\otimes _{k}A,G)\end{eqnarray}$$

$R$

$K$

$A=k$

$R$