Rational points on $X_0^{+}\left(p^r\right)$

Bilu, Yuri; Parent, Pierre; Rebolledo, Marusia

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

En utilisant les récentes bornes d’isogénies obtenues par Gaudron et Rémond, nous prouvons la trivialité de $X_0^{+}\left(p^r\right)\left(\mathbb{Q}\right)$, pour $r>1$ et $p$ un nombre premier supérieur à $2·{10}^{11}$, ce qui inclut le cas des courbes $X_{\mathrm{split}}\left(p\right)$. 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 $X_0^{+}\left(p^r\right)$ 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 $X_0^{+}\left(p^r\right)\left(\mathbb{Q}\right)$, for $r>1$ and $p$ a prime number exceeding $2·{10}^{11}$. This includes the case of the curves $X_{\mathrm{split}}\left(p\right)$. 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 $X_0^{+}\left(p^r\right)$ undertook in our previous work, with the only exception of $p^r={13}^2$.

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

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