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Andrea Surroca

Publications and source records attributed to Andrea Surroca.

7 recordsLinked to original sources
arXiv2013

Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker's method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be derived from this elliptic approach.

math.NT↗
arXiv2012

Quantitative Chevalley-Weil theorem for curves

The classical Chevalley-Weil theorem asserts that for an étale covering of projective varieties over a number field K, the discriminant of the field of definition of the fiber over a K-rational point is uniformly bounded. We obtain a fully explicit version of this theorem in dimension 1.

math.NT↗
arXiv2012

Upper bound for the height of S-integral points on elliptic curves

We establish new upper bounds for the height of the S-integral points of an elliptic curve. This bound is explicitly given in terms of the set S of places of the number field K involved, but also in terms of the degree of K, as well as the rank, the regulator and the height of a basis of the Mordell-Weil group of the curve. The proof uses the elliptic analogue of Baker's method, based on lower bounds for linear forms in elliptic logarithms.

math.NT↗
arXiv2006

Sur l'effectivite du theoreme de Siegel et la conjecture abc

Nous montrons qu'un raffinement du théorème de Siegel sur les points entiers de courbes algébriques impliquerait la conjecture abc de Masser-Oesterlé. Nous formulons une hypothèse "Siegel uniforme" qui est une majoration de la hauteur des points S-entiers de la courbe, en termes du corps de rationalité et de l'ensemble de places S. La validité de l'hypothèse pour une quelconque courbe algébrique de caractéristique d'Euler-Poincaré strictement négative, impliquerait une version de la conjecture abc. Ceci étend aux points S-entiers des résultats précédents de L. Moret-Bailly, et est en quelque sorte, un énoncé réciproque de ce que nous avons montré dans math.NT/0408168, en suivant les idées proposées par N. Elkies. Le principal outil géométrique employé est un théorème de G.V. Belyi. Nous montrons aussi quelques versions inconditionnelles de ces énoncés : un résultat allant dans le sens de la conjecture abc, valable sur tout corps de nombres, ainsi que des bornes pour la hauteur des solutions en S-entiers de certaines équations diophantiennes classiques.

math.NT↗
arXiv2004

Siegel's theorem and the abc conjecture

Following N. Elkies ("ABC implies Mordell") we show that the abc conjecture of Masser-Oesterle implies an effective version of Siegel's theorem about integral points on algebraic curves, i.e. an upper bound for the S-integral points where the dependence on S is explicit. The converse statement is also announced in this note. For both results, the main geometric tool is a theorem of G.V. Belyi.

math.NT↗