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Problème de Lehmer sur les courbes elliptiques à multiplications complexes

Volume 182 / 2018

Bruno Winckler Acta Arithmetica 182 (2018), 347-396 MSC: Primary 11G50, 14G40; Secondary 11M06. DOI: 10.4064/aa170404-5-11 Published online: 26 February 2018

Abstract

We consider the problem of lower bounds for the canonical height on elliptic curves, aiming for the conjecture of Lehmer. Our main result is an explicit version of a theorem of Laurent (who proved this conjecture for elliptic curves with CM up to an $\varepsilon $ exponent) using arithmetic intersection, emphasizing the dependence on parameters linked to the elliptic curve; if GRH holds, then our lower bound for the canonical height of a non-torsion point only depends on the relative degree of the point, and on the degree of the base field of its elliptic curve. We also provide explicit estimates for the Faltings height of an elliptic curve with CM, thanks to an explicit version of Dirichlet’s theorem on arithmetic progressions, in some sense.

Authors

  • Bruno WincklerÉcole Normale Supérieure (ENS) de Lyon
    Unité de Mathématiques Pures et Appliquées
    46, allée d’Italie
    69364 Lyon Cedex 07, France
    e-mail

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