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

Bilu, Yuri; Parent, Pierre; Rebolledo, Marusia

doi:10.5802/aif.2781

Rational points on

X_{0}^{+} (p^{r})

[Points rationnels de

X_{0}^{+} (p^{r})

]

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

Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 957-984.

Résumé
Abstract

En utilisant les récentes bornes d’isogénies obtenues par Gaudron et Rémond, nous prouvons la trivialité de $X_{0}^{+} (p^{r}) (ℚ)$ , pour $r > 1$ et $p$ un nombre premier supérieur à $2 \cdot 10^{11}$ , 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 $11 \leq p \leq 10^{14}$ , $p \neq 13$ . 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 $p^{r} = 13^{2}$ .

Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of $X_{0}^{+} (p^{r}) (ℚ)$ , for $r > 1$ and $p$ a prime number exceeding $2 \cdot 10^{11}$ . 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 $11 \leq p \leq 10^{14}$ , $p \neq 13$ . 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 $p^{r} = 13^{2}$ .

Reçu le : 2011-07-21
Accepté le : 2011-12-01

MR Zbl

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

Affiliations des auteurs :

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

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     title = {Rational points on $X_0^+ (p^r )$},
     journal = {Annales de l'Institut Fourier},
     pages = {957--984},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Bilu, Yuri; Parent, Pierre; Rebolledo, Marusia. Rational points on $X_0^+ (p^r )$. Annales de l'Institut Fourier, Tome 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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