Related papers: On Bruhat-Tits theory over a higher dimensional ba…

Let $k$ be a perfect field of characteristic $p>0$, $\mathcal{V}$ a complete discrete valuation ring with residue field $k$ and field of fractions $K$ of characteristic 0, and $S$ a separated $k$-scheme of finite type. When $S$ is smooth…

In \cite{FGLNP}, Fox, Gromov, Lafforgue, Naor and Pach, in a respond to a question of Gromov \cite{G}, constructed bounded degree geometric expanders, namely, simplical complexes having the affine overlapping property. Their explicit…

Among connected linear algebraic groups, quasi-reductive groups generalize pseudo-reductive groups, which in turn form a useful relaxation of the notion of reductivity. We study quasi-reductive groups over non-archimedean local fields,…

In our previous paper "Bruhat-Tits theory from Berkovich's point of view. I ? Realizations and compactifications of buildings", we investigated realizations of the Bruhat-Tits building B(G,k) of a connected and reductive linear algebraic…

We extend the work of Pappas-Rapoport-Zhu on twisted affine Grassmannians to wildly ramified, quasi-split, and residually split groups, assuming the maximal torus is induced. This relies on the construction, inspired by Tits, of certain…

In this article we consider sheaf quotients of affine superschemes by affine supergroups that act on them freely. The necessary and sufficient conditions for such quotients to be affine are given. If $G$ is an affine supergroup and $H$ is…

We provide a new condition for an absolutely almost simple algebraic group to have good reduction with respect to a discrete valuation of the base field which is formulated in terms of the existence of maximal tori with special properties.…

This is a companion paper to our previous work, where we proved the finiteness of the Tate-Shafarevich group for an arbitrary torus $T$ over a finitely generated field $K$ with respect to any divisorial set $V$ of places of $K$. Here, we…

It is well-known that the coset spaces G(k((z)))/G(k[[z]]), for a reductive group G over a field k, carry the geometric structure of an inductive limit of projective k-schemes. This k-ind-scheme is known as the affine Grassmannian for G.…

In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicate to describe the position of decomposition groups…

For a split Kac-Moody group G over an ultrametric field K, S. Gaussent and the author defined an ordered affine hovel on which the group acts; it generalizes the Bruhat-Tits building which corresponds to the case when G is reductive. This…

Given a Chevalley group $\mathcal{G}$ of classical type and a Borel subgroup $\mathcal{B} \subseteq \mathcal{G}$, we compute the $\Sigma$-invariants of the $S$-arithmetic groups $\mathcal{B}(\mathbb{Z}[1/N])$, where $N$ is a product of…

Let $G$ be the universal Chevalley-Demazure group scheme corresponding to a reduced irreducible root system of rank $\geq 2$, and let $R$ be a commutative ring. We analyze the linear representations $\rho \colon G(R)^+ \to GL_n (K)$ over an…

Given a Henselian and Japanese discrete valuation ring $A$ and a flat and projective $A$-scheme $X$, we follow the approach of Biswas-dos Santos to introduce a full subcategory of coherent modules on $X$ which is then shown to be Tannakian.…

Let $k$ be a number field and $X$ a smooth integral affine variety equipped with a morphism $f : X \to A^1_k$ to the affine line. Assume that all fibres of $f$ are split, for instance that they are geometrically integral. Assume that the…

The aim of this paper is to lay the foundations for the cohomological study of Bruhat-Tits group schemes over a semi-local Dedekind ring. In particular, we obtain a simplified proof of the Grothendieck-Serre conjecture in this case and also…

We provide an equivalence between the category of affine, smooth group schemes over the ring of generalized dual numbers $k[I]$, and the category of extensions of the form $1 \rightarrow \text{Lie}(G, I) \rightarrow E \rightarrow G…

In this paper, we classify irreducible representations of affine group superschemes over fields $F$ of characteristic not two in terms of those over a separable closure $F^{\mathrm{sep}}$ and their Galois twists. We also compute the…

We complete the proof of the Nisnevich conjecture in equal characteristic: for a smooth algebraic variety $X$ over a field $k$, a $k$-smooth divisor $D \subset X$, and a reductive $X$-group $G$ whose base change $G_D$ is totally isotropic,…

Let $G$ be a reductive algebraic group---possibly non-connected---over a field $k$ and let $H$ be a subgroup of $G$. If $G= GL_n$ then there is a degeneration process for obtaining from $H$ a completely reducible subgroup $H'$ of $G$; one…