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 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 dans les progressions arithmétiques sont au cœur de la démonstration.
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 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 -cusp forms in arithmetic progressions lie in the core of the proof.
@article{JTNB_2009__21_3_743_0,
author = {Ricotta, Guillaume and Templier, Nicolas},
title = {Comportement asympotique des hauteurs des points de {Heegner}},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {743--755},
publisher = {Universit\'e Bordeaux 1},
volume = {21},
number = {3},
year = {2009},
doi = {10.5802/jtnb.700},
mrnumber = {2605545},
language = {fr},
url = {http://archive.numdam.org/articles/10.5802/jtnb.700/}
}
TY - JOUR AU - Ricotta, Guillaume AU - Templier, Nicolas TI - Comportement asympotique des hauteurs des points de Heegner JO - Journal de théorie des nombres de Bordeaux PY - 2009 SP - 743 EP - 755 VL - 21 IS - 3 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.700/ DO - 10.5802/jtnb.700 LA - fr ID - JTNB_2009__21_3_743_0 ER -
%0 Journal Article %A Ricotta, Guillaume %A Templier, Nicolas %T Comportement asympotique des hauteurs des points de Heegner %J Journal de théorie des nombres de Bordeaux %D 2009 %P 743-755 %V 21 %N 3 %I Université Bordeaux 1 %U http://archive.numdam.org/articles/10.5802/jtnb.700/ %R 10.5802/jtnb.700 %G fr %F JTNB_2009__21_3_743_0
Ricotta, Guillaume; Templier, Nicolas. Comportement asympotique des hauteurs des points de Heegner. Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 743-755. doi : 10.5802/jtnb.700. http://archive.numdam.org/articles/10.5802/jtnb.700/
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