Minoration du rang des courbes elliptiques sur les corps de classes de Hilbert

TL;DR

The paper establishes a lower bound on the rank of elliptic curves over the rationals generated by Heegner points of large discriminant, supporting the Birch and Swinnerton-Dyer conjecture.

Contribution

It provides a new lower bound on the rank of elliptic curves generated by Heegner points with large discriminant, linking to BSD conjecture.

Findings

Rank exceeds |D|^{0.0009} for large discriminant D

Supports the Birch and Swinnerton-Dyer conjecture

Connects Heegner points to rank growth

Abstract

Soit $E/\BmQ$ une courbe elliptique. Soit $D<0$ un discriminant fondamental suffisamment grand. Si $E(\bar{\BmQ})$ contient des points de Heegner de discriminant $D$, ces points engendrent un sous-groupe dont le rang est sup\'erieur \`a $\pabs{D}^{0.0009}$. Ce r\'esultat est en accord avec la conjecture de Birch et Swinnerton-Dyer. --- Let $E/\BmQ$ be an elliptic curve. Let $D<0$ be a sufficiently large fundamental discriminant. If $E(\bar{\BmQ})$ contains Heegner points of discriminant $D$, these points generate a subgroup of rank greater than $\pabs{D}^{0.0009}$. This result is in agreement with the conjecture of Birch and Swinnerton-Dyer.