A Banach space determined by the Weil height
A Bogomolov property for curves modulo algebraic subgroups
Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least . The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.
A dynamical interpretation of the global canonical height on an elliptic curve.
A “Hardy-Littlewood” approach to the -unit equation
A natural series for the natural logarithm.
A new record for the canonical height on an elliptic curve over .
A Note on heights in certain infinite extensions of
We study the behaviour of the absolute Weil height of algebraic numbers in certain infinite extensions of . In particular, we obtain a Northcott type property for infinite abelian extensions of finite exponent and also a Bogomolov type property for certain fields which are a -adic analog of totally real fields. Moreover, we obtain a non-archimedean analog of a uniform distribution theorem of Bilu in the archimedean case.
A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves.
A relative Dobrowolski lower bound over abelian extensions
Algebraic S-integers of fixed degree and bounded height
Let k be a number field and S a finite set of places of k containing the archimedean ones. We count the number of algebraic points of bounded height whose coordinates lie in the ring of S-integers of k. Moreover, we give an asymptotic formula for the number of S̅-integers of bounded height and fixed degree over k, where S̅ is the set of places of k̅ lying above the ones in S.
An equivalent of Kronecker's theorem for powers of an algebraic number and structure of linear recurrences of fixed length
Approximation diophantienne sur les variétés semi-abéliennes
Arithmetic Distance Functions and Height Fuctions in Diophantine Geometry.
Bicyclotomic polynomials and impossible intersections
In a recent paper we proved that there are at most finitely many complex numbers such that the points and are both torsion on the Legendre elliptic curve defined by . In a sequel we gave a generalization to any two points with coordinates algebraic over the field and even over . Here we reconsider the special case and with complex numbers and .
Borne polynomiale pour le nombre de points rationnels des courbes
Soit un polynôme en deux variables, de degré et à coefficients entiers dans pour . Alors le nombre de zéros rationnels de est soit infini soit plus petit que . Nous montrons aussi une version plus générale sur les corps de nombres.
Canonical local heights and multiplication formulas for the Jacobians of curves of genus 2
Comportement asympotique des hauteurs des points de Heegner
Le terme principal de la moyenne, sur les discriminants quadratiques satisfaisant la condition de Heegner, de la hauteur de Néron-Tate des points de Heegner d’une courbe elliptique rationnelle a été déterminé dans [13]. Les auteurs ont également conjecturé l’expression du terme suivant. Dans cet article, il est démontré que cette expression est correcte et une asymptotique précise, qui sauve une puissance dans le terme d’erreur, est obtenue. Les annulations des coefficients de Fourier de formes...
Comptage de courbes sur le plan projectif éclaté en trois points alignés
Nous établissons une version de la conjecture de Manin pour le plan projectif éclaté en trois points alignés, le corps de base étant un corps global de caractéristique positive.
Computation of -adic heights and log convergence.
Constructions of plane curves with many points