Problème de Lehmer sur les courbes elliptiques à multiplications complexes

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 a epsilon exponent) using arithmetic intersection, enlightening the dependence with parameters linked to the elliptic curve. If GRH holds, then this dependence is reduced to the degree of the base field of the elliptic curve and the relative degree of the algebraic non-torsion point we consider. We also provide an explicit estimate 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.
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Contributor : Bruno Winckler <>
Submitted on : Tuesday, March 21, 2017 - 5:58:25 PM
Last modification on : Thursday, January 11, 2018 - 6:12:31 AM
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  • HAL Id : hal-01493577, version 1



Bruno Winckler. Problème de Lehmer sur les courbes elliptiques à multiplications complexes. 2017. ⟨hal-01493577⟩



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