Comportement asympotique des hauteurs des points de Heegner
Journal de Théorie des Nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 743-755.

Asymptotic behaviour for the averaged height of Heegner points

The leading order term for the average, over quadratic discriminants satisfying the so-called Heegner condition, of the Néron-Tate height of Heegner points on a rational elliptic curve E has been determined in [13]. In addition, the second order term has been conjectured. In this paper, we prove that this conjectured second order term is the right one; this yields a power saving in the remainder term. Cancellations of Fourier coefficients of GL 2 -cusp forms in arithmetic progressions lie in the core of the proof.

Le terme principal de la moyenne, sur les discriminants quadratiques satisfaisant la condition de Heegner, de la hauteur de Néron-Tate des points de Heegner d’une courbe elliptique rationnelle E a été déterminé dans [13]. Les auteurs ont également conjecturé l’expression du terme suivant. Dans cet article, il est démontré que cette expression est correcte et une asymptotique précise, qui sauve une puissance dans le terme d’erreur, est obtenue. Les annulations des coefficients de Fourier de formes sur GL 2 dans les progressions arithmétiques sont au cœur de la démonstration.

Received:
Published online:
DOI: 10.5802/jtnb.700
Classification: 11G50,  11M41
Guillaume Ricotta 1; Nicolas Templier 2

1 Université Bordeaux 1 Institut de Mathématiques de Bordeaux Laboratoire A2X 351, cours de la libération 33405 Talence Cedex, France
2 Université Montpellier 2 Institut de Mathématiques et de Modélisation de Montpellier Case courrier 051 Place Eugène Bataillon 34095 Montpellier Cedex, France
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Guillaume Ricotta; Nicolas Templier. Comportement asympotique des hauteurs des points de Heegner. Journal de Théorie des Nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 743-755. doi : 10.5802/jtnb.700. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.700/

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