Related papers: A Proof of the Barsotti-Chevalley Theorem on Algeb…

We develop theorems which produce a multitude of hyperbolic triples for the finite classical groups. We apply these theorems to prove that every quasisimple group except Alt(5) and SL_2(5) is a Beauville group. In particular, we settle a…

In the present paper we introduce the property AA of a subsemigroup of the endomorphism semigroup of an abelian variety, which holds for semigroup of endomorphisms of an abelian variety defined over a number field, and show that the orbit…

We establish a natural connection of the $q$-Virasoro algebra $D_{q}$ introduced by Belov and Chaltikian with affine Kac-Moody Lie algebras. More specifically, for each abelian group $S$ together with a one-to-one linear character $\chi$,…

We give a completely algebraic proof of the Bogomolov-Tian-Todorov theorem. More precisely, we shall prove that if X is a smooth projective variety with trivial canonical bundle defined over an algebraically closed field of characteristic…

Consider the real free Lie algebra $\mathfrak{fr}_n$ with generators $\omega_1$, \dots, $\omega_n$. Since it is positively graded, it has a completion $\overline{\mathfrak{fr}}_n$ consisting of formal series. By the Campbell--Hausdorff…

We extend in several directions invariant theory results of Chevalley, Shephard and Todd, Mitchell and Springer. Their results compare the group algebra for a finite reflection group with its coinvariant algebra, and compare a group…

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because…

In this paper we give a geometric proof of the Karpelevich's theorem that asserts that a semisimple Lie subgroup of isometries, of a symmetric space of non compact type, has a totally geodesic orbit. In fact, this is equivalent to a…

This paper contains two results concerning the equivariant K-theory of toric varieties. The first is a formula for the equivariant K-groups of an arbitrary affine toric variety, generalizing the known formula for smooth ones. In fact, this…

We finish the proof of the conjecture of F. Bogomolov and F. Pop: Let $F_{1}$ and $F_{2}$ be fields finitely-generated and of transcendence degree $\geq 2$ over $k_{1}$ and $k_{2}$, respectively, where $k_{1}$ is either $\bar{\mathbb{Q}}$…

We propose two definitions of configuration Lie groupoids and in both the cases we prove a Fadell-Neuwirth type fibration theorem for a class of Lie groupoids. We show that this is the best possible extension, in the sense that, for the…

In this article we study the algebraic structure of fine Mordell--Weil groups, plus/minus Mordell--Weil groups, Selmer groups, and plus/minus Selmer groups in the cyclotomic $\mathbb{Z}_p$-extensions of abelian number fields. As a first, we…

We give a criterion for a group homomorphism on a valued abelian group to be surjective and to preserve spherical completeness. We apply this to give a criterion for the existence of integration on a valued differential field. Further, we…

We prove that the Borel space of torsion-free Abelian groups with domain $\omega$ is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing…

Given two hyperbolic curves over p-adic local fields, the absolute anabelian conjecture claims that any isomorphism between their \'etale fundamental group comes from an isomorphism of schemes. This conjecture was proven by S. Mochizuki for…

This paper describes a formal theory of smooth vector fields, Lie groups and the Lie algebra of a Lie group in the theorem prover Isabelle. Lie groups are abstract structures that are composable, invertible and differentiable. They are…

We show that the number of rational points on the fibres of a proper morphism of smooth varieties over a finite field k whose generic fibre has a ``trival'' Chow group of zero cycles is congruent to 1 mod |k|. As a consequence we prove that…

In this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic $p>0$ is isogenous to another one defined over a finite field. We also show that the category of…

We study the geometry of algebraic monoids. We prove that the group of invertible elements of an irreducible algebraic monoid is an algebraic group, open in the monoid. Moreover, if this group is reductive, then the monoid is affine. We…

Freiman's theorem asserts, roughly speaking, if that a finite set in a torsion-free abelian group has small doubling, then it can be efficiently contained in (or controlled by) a generalised arithmetic progression. This was generalised by…