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On the rank of elliptic curves on the field of Hilbert classes. (Sur le rang des courbes elliptiques sur les corps de classes de Hilbert.) (French. English summary) Zbl 1276.11096

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

MSC:

11G05 Elliptic curves over global fields
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G15 Complex multiplication and moduli of abelian varieties
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