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.

__Purpose__:

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

__Results__:

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.

__Application__:

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.