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

Bosser, Vincent ; Surroca, Andrea

In: Bulletin of the Brazilian Mathematical Society, New Series, 2014, vol. 45, no. 1, p. 1-23

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    Titre
    • Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture
    Auteur
    • Bosser, Vincent. Laboratoire Nicolas Oresme, Université de Caen, F14032, Caen cedex, France
    • Surroca, Andrea. Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051, Basel, Switzerland
    Type de document
    • Postprint
    Langue
    • Anglais
    Publié dans
    • Bulletin of the Brazilian Mathematical Society, New Series, 2014, vol. 45, no. 1, p. 1-23. Springer Berlin Heidelberg
    Autre version électronique
    Identifiant OAI-PMH
    • oai:doc.rero.ch:326282
    Summary
    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.