Rational points on X 0 + (p r )
[Points rationnels de X 0 + (p r )]
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 + (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 .

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 .

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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     title = {Rational points on $X_0^+ (p^r )$},
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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. http://www.numdam.org/articles/10.5802/aif.2781/

[1] (http://www.sagemath.org/)

[2] (http://www.numbertheory.org/classnos/) | MR | Zbl

[3] Baran, B. An exceptional isomorphism between modular curves of level 13 preprint (available on the author’s webpage) | MR

[4] Bilu, Yuri; Illengo, Marco Effective Siegel’s theorem for modular curves, Bull. Lond. Math. Soc., Volume 43 (2011) no. 4, pp. 673-688 (http://arXiv.org/pdf/0905.0418) | DOI | MR

[5] Bilu, Yuri; Parent, Pierre Runge’s method and modular curves, Int. Math. Res. Not. IMRN (2011) no. 9, pp. 1997-2027 (http://arXiv.org/pdf/0907.3306) | MR | Zbl

[6] Bilu, Yuri; Parent, Pierre Serre’s uniformity problem in the split Cartan case, Ann. of Math. (2), Volume 173 (2011) no. 1, pp. 569-584 (http://arXiv.org/pdf/0807.4954) | DOI | MR

[7] Bilu, Yuri F. Baker’s method and modular curves, A panorama of number theory or the view from Baker’s garden (Zürich, 1999), Cambridge Univ. Press, Cambridge, 2002, pp. 73-88 | MR | Zbl

[8] Bruin, Nils; Stoll, Michael The Mordell-Weil sieve: proving non-existence of rational points on curves, LMS J. Comput. Math., Volume 13 (2010), pp. 272-306 | DOI | MR

[9] Chen, Imin Jacobians of modular curves associated to normalizers of Cartan subgroups of level p n , C. R. Math. Acad. Sci. Paris, Volume 339 (2004) no. 3, pp. 187-192 | DOI | MR | Zbl

[10] Deligne, P.; Rapoport, M. Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, p. 143-316. Lecture Notes in Math., Vol. 349 | MR | Zbl

[11] Elkies, Noam D. On elliptic K-curves, Modular curves and abelian varieties (Progr. Math.), Volume 224, Birkhäuser, Basel, 2004, pp. 81-91 | MR | Zbl

[12] Faltings, G. Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., Volume 73 (1983) no. 3, pp. 349-366 | DOI | EuDML | MR | Zbl

[13] Galbraith, Steven D. Rational points on X 0 + (p), Experiment. Math., Volume 8 (1999) no. 4, pp. 311-318 | DOI | EuDML | Numdam | MR | Zbl

[14] Galbraith, Steven D. Rational points on X 0 + (N) and quadratic -curves, J. Théor. Nombres Bordeaux, Volume 14 (2002) no. 1, pp. 205-219 | DOI | EuDML | Numdam | MR | Zbl

[15] Gaudron, É.; Rémond, G. Théorème des périodes et degrés minimaux d’isogénies (2011) (submitted http://arXiv.org/pdf/1105.1230) | MR | Zbl

[16] González, Josep On the j-invariants of the quadratic Q-curves, J. London Math. Soc. (2), Volume 63 (2001) no. 1, pp. 52-68 | DOI | MR | Zbl

[17] Gross, Benedict H. Arithmetic on elliptic curves with complex multiplication, Lecture Notes in Mathematics, 776, Springer, Berlin, 1980 (With an appendix by B. Mazur) | MR | Zbl

[18] Gross, Benedict H. Heights and the special values of L-series, Number theory (Montreal, Que., 1985) (CMS Conf. Proc.), Volume 7, Amer. Math. Soc., Providence, RI, 1987, pp. 115-187 | MR | Zbl

[19] Hibino, Takeshi; Murabayashi, Naoki Modular equations of hyperelliptic X 0 (N) and an application, Acta Arith., Volume 82 (1997) no. 3, pp. 279-291 | MR | Zbl

[20] Kubert, Daniel S.; Lang, Serge Modular units, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 244, Springer-Verlag, New York, 1981 | MR | Zbl

[21] Masser, D. W.; Wüstholz, G. Estimating isogenies on elliptic curves, Invent. Math., Volume 100 (1990) no. 1, pp. 1-24 | DOI | EuDML | Numdam | MR | Zbl

[22] Mazur, B. Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977) no. 47, p. 33-186 (1978) | DOI | EuDML | Numdam | MR | Zbl

[23] Mazur, B. Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math., Volume 44 (1978) no. 2, pp. 129-162 | DOI | EuDML | MR | Zbl

[24] Merel, Loïc Sur la nature non-cyclotomique des points d’ordre fini des courbes elliptiques, Duke Math. J., Volume 110 (2001) no. 1, pp. 81-119 (With an appendix by E. Kowalski and P. Michel) | DOI | MR | Zbl

[25] Merel, Loïc Normalizers of split Cartan subgroups and supersingular elliptic curves, Diophantine geometry (CRM Series), Volume 4, Ed. Norm., Pisa, 2007, pp. 237-255 | Numdam | MR | Zbl

[26] Momose, Fumiyuki Rational points on the modular curves X split (p), Compositio Math., Volume 52 (1984) no. 1, pp. 115-137 | EuDML | Numdam | MR | Zbl

[27] Momose, Fumiyuki Rational points on the modular curves X 0 + (p r ), J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 33 (1986) no. 3, pp. 441-466 | MR | Zbl

[28] Momose, Fumiyuki; Shimura, Mahoro Lifting of supersingular points on X 0 (p r ) and lower bound of ramification index, Nagoya Math. J., Volume 165 (2002), pp. 159-178 | MR | Zbl

[29] Parent, Pierre J. R. Towards the triviality of X 0 + (p r )() for r>1, Compos. Math., Volume 141 (2005) no. 3, pp. 561-572 | DOI | MR | Zbl

[30] Pellarin, Federico Sur une majoration explicite pour un degré d’isogénie liant deux courbes elliptiques, Acta Arith., Volume 100 (2001) no. 3, pp. 203-243 | DOI | MR | Zbl

[31] Rebolledo, Marusia Module supersingulier, formule de Gross-Kudla et points rationnels de courbes modulaires, Pacific J. Math., Volume 234 (2008) no. 1, pp. 167-184 | DOI | MR | Zbl

[32] Serre, Jean-Pierre Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math., Volume 15 (1972) no. 4, pp. 259-331 | DOI | EuDML | MR | Zbl

[33] Shimura, Goro Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971 (Kanô Memorial Lectures, No. 1) | MR | Zbl

[34] Silverman, Joseph H. Heights and elliptic curves, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 253-265 | MR | Zbl

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