Related papers: The groups of points on abelian varieties over fin…

Suppose $F$ is either a global field or a finitely generated extension of ${\mathbf Q}$, $A$ is an abelian variety over $F$, and $\ell$ is a prime not equal to the characteristic of $F$. Let $Z$ denote the center of the endomorphism algebra…

We classify finite groups acting by birational transformations of a non-trivial Severi--Brauer surface over a field of characteristc zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism…

Let U:=L\G be a homogeneous variety defined over a number field K, where G is a connected semisimple K-group and L is a connected maximal semisimple K-subgroup of G with finite index in its normalizer. Assuming that G(K_v) acts transitively…

Given a grading $\Gamma: A=\oplus_{g\in G}A_g$ on a nonassociative algebra $A$ by an abelian group $G$, we have two subgroups of the group of automorphisms of $A$: the automorphisms that stabilize each homogeneous component $A_g$ (as a…

Faltings's isogeny theorem states that two abelian varieties are isogenous over a number field precisely when the characteristic polynomials of the reductions at almost all prime ideals of the number field agree. This implies that two…

Let K be a CM-field, i.e., a totally complex quadratic extension of a totally real field F. Let X be a g-dimensional abelian variety admitting an algebra embedding of F into the rational endomorphisms of X. Let A be the product of X and…

We study the Brauer group of an abelian variety A over an algebraically closed field of characteristic p focusing on the p-primary torsion, the key part of which is a certain quasi-algebraic unipotent group U_A. We determine its dimension…

Let $\operatorname{K}_0(\operatorname{Var}_k)$ denote the Grothendieck ring of $k$-varieties over an algebraically closed field $k$. Larsen and Lunts asked if two $k$-varieties having the same class in $\operatorname{K}_0…

We study degree bounds on rational but not necessarily polynomial generators for the field $\mathbf{k}(V)^G$ of rational invariants of a linear action of a finite abelian group. We show that lattice-theoretic methods used recently by the…

It is proved that for any finite dimensional representation of a prime order group over the field of rational numbers, polynomial invariants of degree at most $3$ separate the orbits. A result providing an upper degree bound for separating…

This work presents the conjugacy classes of finite abelian subgroups of the Cremona group of the plane. Using a well-known theory, this problem amounts to the study of automorphism groups of some Del Pezzo surfaces and conic bundles. We…

We study quotients of principally polarized abelian varieties with real multiplication by Galois-stable finite subgroups and describe when these quotients are principally polarizable. We use this characterization to provide an algorithm to…

An abelian variety over a field K is said to have big monodromy, if the image of the Galois representation on l-torsion points, for almost all primes l contains the full symplectic group. We prove that all abelian varieties over a finitely…

The main result of the paper is the following theorem. Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^2$. Suppose that $A$ acts coprimely on a finite group $G$ and assume that for each $a\in…

Let $\mathbb{F}_q$ stand for the finite field of odd characteristic $p$ with $q$ elements ($q=p^{n},n\in \mathbb{N} $) and $\mathbb{F}_q^*$ denote the set of all the nonzero elements of $\mathbb{F}_{q}$. Let $m$ and $t$ be positive…

We introduce the notion of a variety (or more generally a motive) of CM-type which generalises the well known notion of abelian variety of CM-type. Just as in that particular case it will turn out that the cohomology of the variety is…

We define a rank variety for a module of a noncocommutative Hopf algebra $A = \Lambda \rtimes G$ where $\Lambda = k[X_1, ..., X_m]/(X_1^{\ell}, ..., X_m^{\ell})$, $G = ({\mathbb Z}/\ell{\mathbb Z})^m$, and $\text{char} k$ does not divide…

We investigate properties of the Albanese map and the fundamental group of a complex projective variety with many rational points over some function field, and prove that every linear quotient of the fundamental group of such a variety is…

We initiate the study of finite abelian groups that faithfully act on 3-dimensional rationally connected varieties. We show that these groups can be naturally divided into three types: the groups of product type are finite abelian groups…

Let $A$ be an abelian variety defined over a number field $K$. The number of torsion points that are rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$ of $L$ over $K$. Under the following three…