Related papers: Local description of phylogenetic group-based mode…

A finitely generated group $G$ is said to be condensed if its isomorphism class in the space of finitely generated marked groups has no isolated points. We prove that every product variety $\mathcal{UV}$, where $\mathcal{U}$ (respectively,…

Let K be a field not of characteristic 2 such that every finite separable extension of K is cyclic. Let A be an abelian variety over K. If K is infinite, then A(K) is Zariski-dense in A. If K is not locally finite, the rank of A over K is…

We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose…

The characteristic polynomials of abelian varieties over the finite field $\mathbb{F}_q$ with $q=p^n$ elements have a lot of arithmetic and geometric information. They have been explicitly described for abelian varieties up to dimension 4,…

Let $Z$ be an affine algebraic variety and $X$ be a smooth flexible variety. We develop some criteria under which $Z$ admits a closed embedding into $X$. In particular, we show that if $X$ is isomorphic (as an algebraic variety) to a…

Modelling the substitution of nucleotides along a phylogenetic tree is usually done by a hidden Markov process. This allows to define a distribution of characters at the leaves of the trees and one might be able to obtain polynomial…

A classical theorem by K. Ribet asserts that an abelian variety defined over the maximal cyclotomic extension $K$ of a number field has only finitely many torsion points. We show that this statement can be viewed as a particular case of a…

Recently there have been several attempts to provide a whole set of generators of the ideal of the algebraic variety associated to a phylogenetic tree evolving under an algebraic model. These algebraic varieties have been proven to be…

In 2012, Zilber used model-theoretic techniques to show that a curve of high genus over an algebraically closed field is determined by its Jacobian (viewed only as an abstract group with a distinguished subset for an image of the curve). In…

If X is a CW complex, one can assign to each point of X an ordered abelian group of finite rank whose subset of positive elements depends continuously on the points of X. A locally trivial bundle which arises in this way we denote by E(X).…

Associated to an abelian variety $A$ of dimension $g$ over a number field $K$ is a Galois representation $\rho_A\colon Gal(\bar{K}/K)\to GL_{2g}(\hat{\mathbb{Z}})$. The representation $\rho_A$ encodes the Galois action on the torsion points…

We study two families of $g$-dimensional abelian varieties, induced by distinct rational maps defined on a common variety $\overline{\mathcal A}$ and mapping to two bases $\overline{S}_1$ and $\overline{S}_2$. Two non-torsion sections…

Let $G$ be a split semisimple linear algebraic group over a field and let $X$ be a generic twisted flag variety of $G$. Extending the Hilbert basis techniques to Laurent polynomials over integers we give an explicit presentation of the…

We give a classification of the degrees of the points with rational $j$-invariant on the modular curves $X_{0}(n)$ and $X_{1}(n)$. The degrees which occur infinitely often are computed unconditionally, while those which occur finitely often…

Let $X$ be a smooth projective geometrically connected variety defined over a number field $K$. We prove that the geometric \'etale cohomology of $X$ with $\mathbb{Q}/\mathbb{Z}$-coefficients has finitely many classes invariant under the…

A simple abelian variety $A$ defined over a number field $k$ is called of $\mathrm{GL}_n$-type if there exists a number field of degree $2\dim(A)/n$ which is a subalgebra of $\mathrm{End}^0(A)$. We say that $A$ is genuinely of…

We describe the set of characteristic polynomials of abelian varieties of dimension 3 over finite fields.

This paper deals with two families of algebraic varieties arising from applications. First, the k-factor model in statistics, consisting of n-times-n covariance matrices of n observed Gaussian variables that are pairwise independent given k…

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers, the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over…

A Gelfand model for a finite group $G$ is a complex linear representation of $G$ that contains each of its irreducible representations with multiplicity one. For a finite group $G$ with a faithful representation $V$, one constructs a…