Comportement asymptotique des hauteurs des points de Heegner

TL;DR

This paper proves the conjectured second order term for the average Neron-Tate height of Heegner points on elliptic curves, refining previous asymptotic results and demonstrating power savings in the error term.

Contribution

It confirms the conjectured second order term for the average height of Heegner points, advancing understanding of their asymptotic behavior on elliptic curves.

Findings

Confirmed the conjectured second order term in the asymptotic expansion.

Achieved a power saving in the remainder term of the average height.

Utilized cancellations of Fourier coefficients of GL(2)-cusp forms in the proof.

Abstract

The leading order term for the average, over quadratic discriminants satisfying the so-called Heegner condition, of the Neron-Tate height of Heegner points on a rational elliptic curve E has been determined in [12]. 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.