Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of , for and a prime number exceeding . This includes the case of the curves . We then prove, with the help of computer calculations, that the same holds true for in the range , . The combination of those results completes the qualitative study of rational points on undertook in our previous work, with the only exception of .
En utilisant les récentes bornes d’isogénies obtenues par Gaudron et Rémond, nous prouvons la trivialité de , pour et un nombre premier supérieur à , ce qui inclut le cas des courbes . Nous montrons ensuite, avec l’aide de calculs sur machine, la même propriété pour dans l’intervalle , . La combinaison de ces résultats complète l’étude qualitative des points de entreprise dans nos travaux précédents, à la seule exception du cas .
Accepted:
DOI: 10.5802/aif.2781
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
@article{AIF_2013__63_3_957_0, author = {Bilu, Yuri and Parent, Pierre and Rebolledo, Marusia}, 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}, volume = {63}, number = {3}, year = {2013}, doi = {10.5802/aif.2781}, mrnumber = {3137477}, zbl = {06227477}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2781/} }
TY - JOUR AU - Bilu, Yuri AU - Parent, Pierre AU - Rebolledo, Marusia TI - Rational points on $X_0^+ (p^r )$ JO - Annales de l'Institut Fourier PY - 2013 SP - 957 EP - 984 VL - 63 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2781/ DO - 10.5802/aif.2781 LA - en ID - AIF_2013__63_3_957_0 ER -
%0 Journal Article %A Bilu, Yuri %A Parent, Pierre %A Rebolledo, Marusia %T Rational points on $X_0^+ (p^r )$ %J Annales de l'Institut Fourier %D 2013 %P 957-984 %V 63 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2781/ %R 10.5802/aif.2781 %G en %F AIF_2013__63_3_957_0
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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