Related papers: Frobenian multiplicative functions and rational po…

We study rational points on conic bundles over elliptic curves with positive rank over a number field. We show that the etale Brauer-Manin obstruction is insufficient to explain failures of the Hasse principle for such varieties. We then…

We prove a fibration property for varieties with Hilbert-type properties and give applications to rational points on varieties with nef tangent bundle.

A powerful method pioneered by Swinnerton-Dyer allows one to study rational points on pencils of curves of genus 1 by combining the fibration method with a sophisticated form of descent. A variant of this method, first used by Skorobogatov…

Corvaja and Zannier conjectured that an abelian variety over a number field satisfies a modified version of the Hilbert property. We investigate their conjecture for products of elliptic curves using Kawamata's structure result for ramified…

The number of rational points of a plane non-singular algebraic curve X defined over a finite field is computed, provided that the generic point of X is not an inflexion and that X is Frobenius non-classical with respect to conics.

Let $X$ be a projective variety with log terminal singularities and vanishing augmented irregularity. In this paper we prove that if $X$ admits a relatively minimal genus one fibration then it does contain a subvariety of codimension one…

Using Weil descent, we give bounds for the number of rational points on two families of curves over finite fields with a large abelian group of automorphisms: Artin-Schreier curves of the form $y^q-y=f(x)$ with $f\in\Fqr[x]$, on which the…

We tensorize the Faber spline system from [14] to prove sequence space isomorphisms for multivariate function spaces with higher mixed regularity. The respective basis coefficients are local linear combinations of discrete function values…

Given an extension of number fields $E \subset F$ and a projective variety $X$ over $F$, we compare the problem of counting the number of rational points of bounded height on $X$ with that of its Weil restriction over $E$. In particular, we…

We continue to study birational geometry of Fano fibrations $\pi\colon V\to {\mathbb P}^1$ the fibers of which are Fano double hypersurfaces of index 1. For a majority of families of this type, which do not satisfy the condition of…

Let $[X,\lambda]$ be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either $X$ is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce…

We give a complete conjectural formula for the number $e_r(d,m)$ of maximum possible ${\mathbb{F}}q$-rational points on a projective algebraic variety defined by $r$ linearly independent homogeneous polynomial equations of degree $d$ in…

Given systems of two (inhomogeneous) quadratic equations in four variables, it is known that the Hasse principle for integral points may fail. Sometimes this failure can be explained by some integral Brauer-Manin obstruction. We study the…

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus $g$ over ${\mathbb F}_q$. As a consequence of Katz-Sarnak theory, we first get for any…

We consider the problem of counting the number of rational points of bounded height in the zero-loci of Brauer group elements on semi-simple algebraic groups over number fields. We obtain asymptotic formulae for the counting problem for…

We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite…

We study the existence problem and the enumeration problem for sections of Serre fibrations over compact orientable surfaces. When the fundamental group of the fiber is finite, a complete solution is given in terms of 2-dimensional…

Let $X$ be an algebraic variety, defined over the rationals. This paper gives upper bounds for the number of rational points on $X$, with height at most $B$, for the case in which $X$ is a curve or a surface. In the latter case one excludes…

We study rational points on a smooth variety X over a complete local field K with algebraically closed residue field, and models of X with tame quotient singularities. If a model of X is the quotient of a Galois action on a weak N\'eron…

This is meant to be a survey article for the Cubo Journal. We discuss the existence and number of rational points over a finite field, the Hodge type over the complex numbers, and the motivic conjectures which are controlling those…