Rational points on X 0 + (p r )
Annales de l'Institut Fourier, Volume 63 (2013) no. 3, pp. 957-984.

Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of X 0 + (p r )(), for r>1 and p a prime number exceeding 2·10 11 . This includes the case of the curves X split (p). We then prove, with the help of computer calculations, that the same holds true for p in the range 11p10 14 , p13. 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 .

En utilisant les récentes bornes d’isogénies obtenues par Gaudron et Rémond, nous prouvons la trivialité de X 0 + (p r )(), pour r>1 et p un nombre premier supérieur à 2·10 11 , ce qui inclut le cas des courbes X split (p). Nous montrons ensuite, avec l’aide de calculs sur machine, la même propriété pour p dans l’intervalle 11p10 14 , p13. 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 .

Received:
Accepted:
DOI: 10.5802/aif.2781
Classification: 11G18, 11G05, 11G16
Keywords: Elliptic curves, modular curves, rational points, Runge’s method, isogeny bounds, Gross-Heegner points
Mot clés : courbes elliptiques, courbes modulaires, points rationnels, méthode de Runge, bornes d’isogénies, points de Gross-Heegner

Bilu, Yuri 1; Parent, Pierre 2; Rebolledo, Marusia 3

1 IMB, Université Bordeaux 1 351 cours de la Libération 33405 Talence CEDEX, FRANCE
2 IMB, Université Bordeaux 1 351 cours de la Libération 33405 Talence CEDEX FRANCE
3 Université Blaise Pascal Clermont-Ferrand 2 Laboratoire de Mathématiques Campus universitaire des Cézeaux 63177 Aubière FRANCE
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Bilu, Yuri; Parent, Pierre; Rebolledo, Marusia. Rational points on $X_0^+ (p^r )$. Annales de l'Institut Fourier, Volume 63 (2013) no. 3, pp. 957-984. doi : 10.5802/aif.2781. https://aif.centre-mersenne.org/articles/10.5802/aif.2781/

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