Related papers: Counting rational points over number fields on a s…

A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space P^{n-1}. In this paper, we achieve Serre's conjecture in the special case of smooth cyclic covers of any degree when n is…

In studying rational points on elliptic K3 surfaces of the form $f(t)y^2=g(x)$, where $f,g$ are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having…

We provide an asymptotic estimate for the number of rational points of bounded height on a non-singular conic over the rationals. The estimate is uniform in the coefficients of the underlying quadratic form.

We construct a (smooth, projective) surface over the field of rational numbers, which is a counterexample to the Hasse principle not accounted for by the Manin obstruction. The construction relies on the classical 4-descent on elliptic…

We prove the "all-the-heights'' version of the Batyrev--Manin--Peyre conjecture for split quintic del Pezzo surfaces, both for counting rational points over global function fields in positive characteristic and for the motivic version over…

We explore log Manin's conjecture for integral points and its connections to $\mathbb A^1$-connectedness. We prove log Manin's conjecture for Campana rational curves and for $\mathbb A^1$-curves on split toric varieties. Our arguments…

The Manin conjecture is established for Ch\^atelet surfaces over Q arising as minimal proper smooth models of the surface Y^2+Z^2=f(X) where f is a totally reducible polynomial of degree 3 without repeated roots. These surfaces do not…

In this paper we give a complete characterization of the intersections between the Norm-Trace curve over $\mathbb{F}_{q^3}$ and the curves of the form $y=ax^3+bx^2+cx+d$, generalizing a previous result by Bonini and Sala, providing more…

Let $X$ be a smooth projective Fano variety over the complex numbers. We study the moduli spaces of rational curves on $X$ using the perspective of Manin's Conjecture. In particular, we bound the dimension and number of components of spaces…

In this paper, we prove an explicit upper bound on the number of rational points on a smooth projective curve of genus at least two over a number field. This gives explicit constants in the uniform Mordell conjecture proposed by Mazur and…

A new, simple method to approach enumerative questions about rational curves on rational surfaces is described. Applications include a short proof of Kontsevich's formula for plane curves and a the solution of the analogous problem for the…

Let $X_4\subset\mathbb{P}^{n+1}$ be a quartic hypersurface of dimension $n\geq 4$ over an infinite field $k$. We show that if either $X_4$ contains a linear subspace $\Lambda$ of dimension $h\geq \max\{2,\dim(\Lambda\cap…

We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer-Manin obstruction to weak approximation. To do so, we exploit the relationship between…

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.

In his book "Cubic forms" Manin discovered that del Pezzo surfaces are related to root systems. To explain the many numerical coincidences Batyrev conjectured that a universal torsor on a del Pezzo surface can be embedded in a certain…

It is shown that rational points over finite fields of moduli spaces of stable quiver representations are counted by polynomials with integer coefficients. These polynomials are constructed recursively using an identity in the Hall algebra…

We prove Manin's conjecture for a split singular quartic del Pezzo surface with singularity type $2\Aone$ and eight lines. This is achieved by equipping the surface with a conic bundle structure. To handle the sum over the family of conics,…

The Tate conjecture for squares of K3 surfaces over finite fields was recently proved by Ito-Ito-Koshikawa. We give a more geometric proof when the characteristic is at least 5. The main idea is to use twisted derived equivalences between…

We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We…

We establish a sharp asymptotic formula for the number of rational points up to a given height and within a given distance from a hypersurface. Our main innovation is a bootstrap method that relies on the synthesis of Poisson summation,…