Finite flat commutative group schemes over complete discrete valuation fields: classification, structural results; application to reduction of Abelian varieties

This article is on the inverse Galois problem in Galois theory of linear iterative differential equations in positive characteristic. We show that it has an affirmative answer for reduced algebraic group schemes over any iterative differential field which is finitely generated over its algebraically closed field of constants. We also introduce the notion of equivalence of iterative derivations on a given field - a condition which implies that the inverse Galois problem over e...

In this paper, we classify the possible group structures on the set of $R$-valued points of an abelian variety, where $R$ is any real closed field. We make use of a family of abelian varieties that, in effect, allows one to quantify over all abelian varieties of a fixed dimension and degree of polarization in a first-order fashion.

The finite monodromy groups of abelian varieties over number fields have been introduced by Grothendieck. They represent the local obstruction to semi-stable reduction. In this paper we prove a criteria for finite groups to be realized as finite monodromy groups in given dimension. An application to the degree of semi-stability gives an effective version of Grothendieck's semi-stable reduction theorem in terms of the degree of the extension with regards to the dimension of th...

Given an abelian variety over a field with a discrete valuation, Grothendieck defined a certain open normal subgroup of the absolute inertia group. This subgroup encodes information on the extensions over which the abelian variety acquires semistable reduction. We study this subgroup, and use it to obtain information on the extensions over which the abelian variety acquires semistable reduction.

Given a relative faithfully flat pointed scheme over the spectrum of a discrete valuation ring $X \to S$ this paper is motivated by the study of the natural morphism from the fundamental group scheme of the generic fiber $X_\eta $ to the generic fiber of the fundamental group scheme of $X$. Given a torsor $T \to X_\eta $ under an affine group scheme $G$ over the generic fiber of $X$, we address the question to find a model of this torsor over $X$, focusing in particular on th...

Let $\mathcal{O}_K$ be a discrete valuation ring with fraction field $K$ of characteristic $0$ and algebraically closed residue field $k$ of characteristic $p > 0$. Let $A/K$ be an abelian variety of dimension $g$ with a $K$-rational point of order $p$. In this article, we are interested in the reduction properties that $A/K$ can have. After discussing the general case, we specialize to $g=1$, and we study the possible Kodaira types that can occur.

Let $G$ be a reductive group over a field $k$ which is algebraically closed of characteristic $p \neq 0$. We prove a structure theorem for a class of subgroup schemes of $G$, for $p$ bounded below by the Coxeter number of $G$. As applications, we derive semi-simplicity results, generalizing earlier results of Serre proven in 1998, and also obtain an analogue of Luna's \'etale slice theorem for suitable bounds on $p$.

Let $k$ be an algebraically closed field of characteristic $p \geq 0$ and $V$ be a faithful $k$-rational representation of a finite $\ell$-group $G$, where $\ell$ is a prime number. The Noether problem asks whether $V/G$ is a stably rational variety. While if $\ell=p$ it is well-known that $V/G$ is always rational, when $\ell\neq p$, Saltman and then Bogomolov constructed $\ell$-groups for which $V/G$ is not stably rational. Hence, the geometry of $V/G$ depends heavily on the...

In this paper we generalize the Deuring theorem on a reduction of elliptic curve with complex multiplication. More precisely, for an Abelian variety $A$, arising after reduction of an Abelian variety with complex multiplication by a CM field $K$ over a number field at a pace of good reduction. We establish a connection between a decomposition of the first truncated Barsotti-Tate group scheme $A[p]$ and a decomposition of $p\cO_{K}$ into prime ideals. In particular, we produce...

Given an elliptic curve $E$ over a perfect defectless henselian valued field $(F,\mathrm{val})$ with perfect residue field $\textbf{k}_F$ and valuation ring $\mathcal{O}_F$, there exists an integral separated smooth group scheme $\mathcal{E}$ over $\mathcal{O}_F$ with $\mathcal{E}\times_{\text{Spec } \mathcal{O}_F}\text{Spec } F\cong E$. If $\text{char}(\textbf{k}_F)\neq 2,3$ then one can be found over $\mathcal{O}_{F^{alg}}$ such that the definable group $\mathcal{E}(\mathca...