Related papers: The multi-height distribution implies the Batyrev-…

Let $L$ be a simply-connected simple connected algebraic group over a number field $F$, and $H$ be a semisimple absolutely maximal connected $F$-subgroup of $L$. Under a cohomological condition, we prove an asymptotic formula for the number…

We study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin's…

Let $\pi : E\to B$ be an elliptic surface defined over a number field $K$, where $B$ is a smooth projective curve, and let $P: B \to E$ be a section defined over $K$ with canonical height $\hat{h}_E(P)\not=0$. In this article, we show that…

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 employ the photography method to obtain a lower bound for the number of solutions to a nonlinear elliptic problem on a Riemannian orbifold in function of the Lusternik--Schnirelmann category of its submanifold of points with largest…

The proof by Ullmo and Zhang of Bogomolov's conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for ``small points'' in the associated complex abelian variety. We study the…

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…

Let $f(x) = x^{2g+1} + c_1 x^{2g} + \dots + c_{2g+1} \in k[x]$ be a polynomial of nonzero discriminant, and let $J$ denote the Jacobian of the odd hyperelliptic curve $C : y^2 = f(x)$. We show that the morphism $J \to \mathbb{P}^{2^g-1}$…

Empirical analysis is often the first step towards the birth of a conjecture. This is the case of the Birch-Swinnerton-Dyer (BSD) Conjecture describing the rational points on an elliptic curve, one of the most celebrated unsolved problems…

We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.

We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the N\'eron-Tate height of generators of its Mordell-Weil group. The…

For a single cusped hyperbolic 3-manifold, Hodgson proved that there are only finitely many Dehn fillings of it whose trace fields have bounded degree. In this paper, we conjecture the same for manifolds with more cusps, and give the first…

Following ideas by Beardon, Minda and Baribeau, Rivard, Wegert in the context of the complex Schwarz-Pick Lemma, we use iterated hyperbolic difference quotients to prove a quaternionic multipoint Schwarz-Pick Lemma, in the context of the…

We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field. This bound depends only on the dimensions and degrees of the irreducible components and holds for very…

In this manuscript, we initiate the study of the number of rational points with bounded denominators, contained in a non-isotropic $\delta_1\times\ldots\times \delta_R$ neighborhood of a compact submanifold $\mathcal{M}$ of codimension $R$…

Let $k$ be a number field and $K$ a finite extension of $k$. We count points of bounded height in projective space over the field $K$ generating the extension $K/k$. As the height gets large we derive asymptotic estimates with a…

Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are…

Standard Markov chain Monte Carlo methods struggle to explore distributions that are concentrated in the neighbourhood of low-dimensional structures. These pathologies naturally occur in a number of situations. For example, they are common…

Caro and Pasten gave an explicit upper bound on the number of rational points on a hyperbolic surface that is embedded in an abelian variety of rank at most one. We show how to use their method to produce a refined bound on the number of…

We develop a rigorous theoretical framework for principal manifold estimation that recovers a latent low-dimensional manifold from a point cloud observed in a high-dimensional ambient space. Our framework accommodates manifolds with…