Corrigendum - Points entiers et théorèmes de Bertini arithmétiques
Counting points of fixed degree and bounded height
Counting rational points on a certain exponential-algebraic surface
We study the distribution of rational points on a certain exponential-algebraic surface and we prove, for this surface, a conjecture of A. J. Wilkie.
Counting rational points on del Pezzo surfaces with a conic bundle structure
For any number field k, upper bounds are established for the number of k-rational points of bounded height on non-singular del Pezzo surfaces defined over k, which are equipped with suitable conic bundle structures over k.
Cycles on Siegel threefolds and derivatives of Eisenstein series
Descente infinie et hauteur p-adique sur les courbes elliptiques à multiplication complexe.
Diophantine approximation and deformation
Dynamical systems arising from elliptic curves
We exhibit a family of dynamical systems arising from rational points on elliptic curves in an attempt to mimic the familiar toral automorphisms. At the non-archimedean primes, a continuous map is constructed on the local elliptic curve whose topological entropy is given by the local canonical height. Also, a precise formula for the periodic points is given. There follows a discussion of how these local results may be glued together to give a map on the adelic curve. We are able to give a map whose...
Elliptic curves associated with simplest quartic fields
We are studying the infinite family of elliptic curves associated with simplest cubic fields. If the rank of such curves is 1, we determine the whole structure of the Mordell-Weil group and find all integral points on the original model of the curve. Note however, that we are not able to find them on the Weierstrass model if the parameter is even. We have also obtained similar results for an infinite subfamily of curves of rank 2. To our knowledge, this is the first time that so much information...
Equidistribution and the heights of totally real and totally p-adic numbers
C. J. Smyth was among the first to study the spectrum of the Weil height in the field of all totally real numbers, establishing both lower and upper bounds for the limit infimum of the height of all totally real integers, and determining isolated values of the height. Later, Bombieri and Zannier established similar results for totally p-adic numbers and, inspired by work of Ullmo and Zhang, termed this the Bogomolov property. In this paper, we use results on equidistribution of points of low height...
Equidistribution of dynamically small subvarieties over the function field of a curve
Equidistribution of Small Points, Rational Dynamics, and Potential Theory
Given a rational function on of degree at least 2 with coefficients in a number field , we show that for each place of , there is a unique probability measure on the Berkovich space such that if is a sequence of points in whose -canonical heights tend to zero, then the ’s and their -conjugates are equidistributed with respect to .The proof uses a polynomial lift of to construct a two-variable Arakelov-Green’s function for each . The measure is obtained by taking the...
Erratum: Rational points of bounded height on Fano varieties. Invent. Math. 95, 421-435 (1989).
Estimating heights using auxiliary functions
Explicit local heights.
Families of hypersurfaces of large degree
Grauert and Manin showed that a non-isotrivial family of compact complex hyperbolic curves has finitely many sections. We consider a generic moving enough family of high enough degree hypersurfaces in a complex projective space. We show the existence of a strict closed subset of its total space that contains the image of all its sections.
Fonctions zêta des hauteurs
Ce papier présente les récents progrès concernant les fonctions zêta des hauteurs associées à la conjecture de Manin. En particulier, des exemples où on peut prouver un prolongement méromorphe de ces fonctions sont détaillés.
Generalised Hermite constants, Voronoi theory and heights on flag varieties
This paper explores the study of the general Hermite constant associated with the general linear group and its irreducible representations, as defined by T. Watanabe. To that end, a height, which naturally applies to flag varieties, is built and notions of perfection and eutaxy characterising extremality are introduced. Finally we acquaint some relations (e.g., with Korkine–Zolotareff reduction), upper bounds and computation relative to these constants.
Generators and integer points on the elliptic curve y² = x³ - nx
Let E be an elliptic curve over the rationals ℚ given by y² = x³ - nx with a positive integer n. We consider first the case where n = N² for a square-free integer N. Then we show that if the Mordell-Weil group E(ℚ ) has rank one, there exist at most 17 integer points on E. Moreover, we show that for some parameterized N a certain point P can be in a system of generators for E(ℚ ), and we determine the integer points in the group generated by the point P and the torsion points. Secondly, we consider...
Generators and integral points on twists of the Fermat cubic
We study integral points and generators on cubic twists of the Fermat cubic curve. The main results assert that integral points can be in a system of generators in the case where the Mordell-Weil rank is at most two. As a corollary, we explicitly describe the integral points on the curve.