Hauteur asymptotique des points de Heegner

(Asymptotic height of Heegner points)





Accepted for publication 13 June 2006.

Published in Canadian Journal of Mathematics, Vol. 60, No. 6, 1406—1436 (2008).





E is a fixed elliptic curve over the rational numbers.




     To study the Néron-Tate height of Heegner points on E.




We get asymptotic formulae for

·       the Néron-Tate height of Heegner points on E on average over a subset of discriminants: it is governed by the degree of the modular parametrisation of E as geometry suggests,

·       the Néron-Tate height of traces of Heegner points on average over a subset of discriminants: we find a difference according to the rank of the elliptic curve.

By Gross-Zagier formulae, it consists in proving asymptotic formulae for the first moments of


·       the derivative of the Rankin-Selberg convolution of E with a certain weight one theta series attached to the principal ideal class of an imaginary quadratic field,

·       the twisted L-function of E by a quadratic Dirichlet character.


Many experimental results are discussed.




These results give some insight to the problem of the discretisation of odd quadratic twists of elliptic curves.


 heegner.ps     heegner.pdf    heegnerview.pdf


The previous texts are not the published one. For they should not be quoted. Reprints of published versions may be asked by e-mails.