Théorie des nombres
Minoration de rangs de courbes elliptiques
[Lower bounds for ranks of elliptic curves]
Comptes Rendus. Mathématique, Volume 346 (2008) no. 23-24, pp. 1225-1230.

We state a quantitative result concerning the group of rational points on an elliptic curve that are defined over certain Hilbert class fields. We provide a lower bound for the rank. We present also an analytic approach based on the proof of an estimate for sums of Fourier coefficients of a modular form along values taken by a quadratic polynomial. This estimate is a non-split version of the shifted convolution problem.

Nous énonçons dans cette Note un résultat quantitatif concernant le groupe des points rationnels d'une courbe elliptique qui sont définis sur certains corps de classes de Hilbert. Il s'agit d'établir une borne inférieure pour le rang. Nous présentons également une approche analytique qui se fonde sur l'estimation de sommes de coefficients d'une forme modulaire aux valeurs d'un polynôme quadratique. Cette estimée est une version non scindée d'un problème de convolution décalée.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.10.016
Templier, Nicolas 1

1 Institute for Advanced Study, Princeton, NJ 08540, États-Unis
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Templier, Nicolas. Minoration de rangs de courbes elliptiques. Comptes Rendus. Mathématique, Volume 346 (2008) no. 23-24, pp. 1225-1230. doi : 10.1016/j.crma.2008.10.016. http://www.numdam.org/articles/10.1016/j.crma.2008.10.016/

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