Rational points on $X_0^+ (p^r )$

[Points rationnels de $X_0^+ (p^r )$]

Bilu, Yuri ; Parent, Pierre ; Rebolledo, Marusia

Annales de l'Institut Fourier, Tome 63 (2013), p. 957-984 / Harvested from Numdam

Résumé

En utilisant les récentes bornes d’isogénies obtenues par Gaudron et Rémond, nous prouvons la trivialité de $X_0^+ (p^r )(\mathbb{Q} )$, pour ${r>1}$ et $p$ un nombre premier supérieur à $2\cdot 10^{11}$, ce qui inclut le cas des courbes $X_{\mathrm{split}} (p)$. Nous montrons ensuite, avec l’aide de calculs sur machine, la même propriété pour $p$ dans l’intervalle $11\le p\le 10^{14}$, $p\ne 13$. La combinaison de ces résultats complète l’étude qualitative des points de $X_0^+ (p^r )$ entreprise dans nos travaux précédents, à la seule exception du cas $p^r=13^2$.

Abstract

Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of $X_0^+ (p^r )(\mathbb{Q})$, for ${r>1}$ and $p$ a prime number exceeding $2\cdot 10^{11}$. This includes the case of the curves $X_{\mathrm{split}} (p)$. We then prove, with the help of computer calculations, that the same holds true for $p$ in the range $11\le p\le 10^{14}$, $p\ne 13$. The combination of those results completes the qualitative study of rational points on $X_0^+ (p^r )$ undertook in our previous work, with the only exception of $p^r=13^2$.

Détail

Publié le : 2013-01-01

MR 3137477 | Zbl 06227477

DOI : https://doi.org/10.5802/aif.2781
Classification:  11G18,  11G05,  11G16
Mots clés: courbes elliptiques, courbes modulaires, points rationnels, méthode de Runge, bornes d’isogénies, points de Gross-Heegner

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