The difference between the ordinary height and the canonical height on elliptic curves

Abstract

We estimate the bounds for the difference between the ordinary height and the canonical height on elliptic curves over number fields. Our result is an improvement of the recent result of Cremona, Prickett, and Siksek [J.E. Cremona, M. Prickett, S. Siksek, Height difference bounds for elliptic curves over number fields, J. Number Theory 116 (2006) 42-68]. Our bounds are usually sharper than the other known bounds.

Valuations of Somos 4 sequences and canonical local heights on elliptic curves

“…In fact, the constants C v and C v can be explicitly estimated. See [3,18] and the literature cited there. We will use a result in [3] later.…”

Bornes optimales pour la différence entre la hauteur de Weil et la hauteur de Néron–Tate sur les courbes elliptiques sur Q

“…Il est donc naturel de se demander si, étant donnée une équation (1.1), on peut calculer le supremum et l'infimum de cette différence. Des résultats dans cette direction ont été obtenus par Dem ′ janenko [3] et Zimmer [11], Silverman [6], Siksek [5], Cremona, Prickett et Siksek [2], et Uchida [10]. Le lecteur est renvoyé à l'introduction de [2] pour plus de détails.…”

An Arakelov-Theoretic Approach to Naïve Heights on Hyperelliptic Jacobians

“…The problems of computing the height of a given point on the Jacobian of a curve and computing the (finite) sets of points of bounded height on the Jacobian have been studied since the work of Tate in the 1960s, who gave a simpler formula for Néron's height. Using this formula, Tate (unpublished), Dem'janenko [Dem68], Zimmer [Zim76], Silverman [Sil90] and more recently Cremona, Prickett and Siksek [CPS06], Uchida [Uch08] and Bruin [Bru13] have given increasingly refined algorithms for the case of elliptic curves. Meanwhile, in the direction of increasing genus, Flynn and Smart [FS97] gave an algorithm for the above problems for genus 2 curves building on work of Flynn [Fly93], which was later modified by Stoll ([Sto99] and [Sto02]).…”