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

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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https://hal-normandie-univ.archives-ouvertes.fr/hal-02151923
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Submitted on : Monday, June 10, 2019 - 5:12:45 PM
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Vincent Bosser, Andrea Surroca. Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture. Boletim da Sociedade Brasileira de Matemática / Bulletin of the Brazilian Mathematical Society, Springer Verlag, 2014, 45 (1), ⟨10.1007/s00574-014-0038-x⟩. ⟨hal-02151923⟩

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