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

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Abstract

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.

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Authors and Affiliations

  1. Laboratoire Nicolas Oresme, Université de Caen, F14032, Caen cedex, France

    Vincent Bosser

  2. Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051, Basel, Switzerland

    Andrea Surroca

Authors
  1. Vincent Bosser
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  2. Andrea Surroca
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Correspondence to Andrea Surroca.

Additional information

Supported by the contract ANR “HAMOT”, BLAN-0115-01.

Supported by an Ambizione fund PZ00P2_121962of the Swiss National Science Foundation and the Marie Curie IEF 025499 of the European Community.

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Bosser, V., Surroca, A. Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture. Bull Braz Math Soc, New Series 45, 1–23 (2014). https://doi.org/10.1007/s00574-014-0038-x

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  • DOI: https://doi.org/10.1007/s00574-014-0038-x

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