Related papers: Congruences for rational points on varieties over …

We show that every finite abelian group $G$ occurs as the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$. We produce partial results for abelian varieties over a general finite…

For any affine hypersurface defined by a complete symmetric polynomial in $k\geq 3$ variables of degree $m$ over the finite field $\mathbb{F}_{q}$ of $q$ elements, a special case of our theorem says that this hypersurface has at least…

Under suitable hypotheses, we prove that a form of a projective homogeneous variety $G/P$ defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of…

A well-known conjecture of Orlov asks whether the existence of a full exceptional collection implies rationality of the underlying variety. We prove this conjecture for arithmetic toric varieties over general fields. We also investigate a…

We prove that a smooth proper universally CH_0-trivial variety X over a field k has universally trivial Brauer group. This fills a gap in the literature concerning the p-torsion of the Brauer group when k has characteristic p.

Chevalley's theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a pseudo-abelian…

We give examples of sequences of smooth non-isotrivial curves for every genus at least two, defined over a rational function field of positive characteristic, such that the (finite) number of rational points of the curves in the sequence…

Let $k$ be an algebraically closed field. Chambert-Loir proved that the \'etale fundamental group of a normal rationally chain connected variety over $k$ is finite. We prove that the fundamental group scheme of a normal rationally chain…

A result of Graber, Harris, and Starr shows that a rationally connected variety defined over the function field of a curve over the complex numbers always has a rational point. Similarly, a separably rationally connected variety over a…

Wan conjectures that if $X$ and $Y$ form a strong mirror pair of Calabi-Yau varieties over a finite field $F_q$ with $q$ elements, then X and Y have the same number of $F_{q^k}$-rational points modulo $q^k$. We prove this conjecture under…

A smooth, proper, retract rational variety over a field $k$ is known to be $\mathbb{A}^1$-connected. We improve on this result, in the case when $k$ is infinite, showing that such varieties are naively $\mathbb{A}^1$-connected.

Fix a non-negative integer g and a positive integer I dividing 2g-2. For any Henselian, discretely valued field K whose residue field is perfect and admits a degree I cyclic extension, we construct a curve C over K of genus g and index I.…

We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a…

We study the set of common F_q-rational zeros of systems of multivariate symmetric polynomials with coefficients in a finite field F_q. We establish certain properties on these polynomials which imply that the corresponding set of zeros…

(On the fundamental group of rationnally connected varieties.) I show that the fundamental group of a normal variety which is rationally chain connected is finite. The proof holds in non-zero characteristic. Je d\'emontre que le groupe…

Let $k$ be a field of characteristic different from 2. Let $G$ be a simply connected or adjoint semisimple algebraic $k$-group which does not contain a simple factor of type $E_8$ and such that every exceptional simple factor of type other…

Let $K$ denote a number field and a finite set $S$ of places of $K$ and $\phi:\PP^n\rightarrow\PP^n$ be rational morphism defined over $K$. The main result of this paper proves that there are only finitely many twists of $\phi$ defined over…

Let P^n denote the n-dimensional projective space defined over the algebraic closure of a finite field F_q, let V contained P^n be a complete intersection defined over F_q of dimension r and singular locus of dimension at most s, and let…

Given a smooth projective variety $X$ over a number field $k$ and $P\in X(k)$, the first author conjectured that in a precise sense, any sequence that approximates $P$ sufficiently well must lie on a rational curve. We prove this conjecture…

By a theorem of Chevalley the image of a morphism of varieties is a constructible set. The algebraic version of this fact is usually stated as a result on "extension of specializations" or "lifting of prime ideals". We present a difference…