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Sur le rang des courbes elliptiques sur les corps de classes de Hilbert

Part of:Arithmetic algebraic geometryArithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  10 February 2011

Nicolas Templier
Nicolas Templier*
Affiliation:
Institute for Advanced Study, Princeton, NJ 08540, USA (email: nicolas.templier@normalesup.org)
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Abstract

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Let E/ℚ be an elliptic curve and let D<0 be a sufficiently large fundamental discriminant. If $E(\overline {\BmQ })$ contains Heegner points of discriminant D, those points generate a subgroup of rank at least |D|δ, where δ>0 is an absolute constant. This result is compatible with the Birch and Swinnerton-Dyer conjecture.

Résumé

Soit E/ℚ une courbe elliptique. Soit D<0 un discriminant fondamental suffisamment grand. Si $E(\overline {\BmQ })$ contient des points de Heegner de discriminant D, ces points engendrent un sous-groupe dont le rang est supérieur à |D|δ, où δ>0 est une constante absolue. Ce résultat est en accord avec la conjecture de Birch et Swinnerton-Dyer.

Keywords

automorphic formsequidistributionL-functionsHeegner pointselliptic curves

MSC classification

Secondary:11G05: Elliptic curves over global fields14G40: Arithmetic varieties and schemes; Arakelov theory; heights11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture11G15: Complex multiplication and moduli of abelian varieties

Information

Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 4 , July 2011, pp. 1087 - 1104
Copyright
Copyright © Foundation Compositio Mathematica 2011

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