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

DOI : https://doi.org/10.5802/aif.2781
Classification:  11G18,  11G05,  11G16
Keywords: Elliptic curves, modular curves, rational points, Runge’s method, isogeny bounds, Gross-Heegner points
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {3},
     year = {2013},
     pages = {957-984},
     doi = {10.5802/aif.2781},
     mrnumber = {3137477},
     zbl = {06227477},
     language = {en},
     url = {http://www.numdam.org/item/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. http://www.numdam.org/item/AIF_2013__63_3_957_0/

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

[2] (http://www.numbertheory.org/classnos/) | MR 2820153 | Zbl 1225.11088

[3] Baran, B. An exceptional isomorphism between modular curves of level 13 (preprint (available on the author’s webpage)) | MR 2806555 | Zbl pre05900985

[4] Bilu, Yuri; Illengo, Marco Effective Siegel’s theorem for modular curves, Bull. Lond. Math. Soc., Tome 43 (2011) no. 4, pp. 673-688 (http://arXiv.org/pdf/0905.0418) | Article | MR 2753610 | Zbl pre05960666

[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 1975445 | Zbl 1041.11045

[6] Bilu, Yuri; Parent, Pierre Serre’s uniformity problem in the split Cartan case, Ann. of Math. (2), Tome 173 (2011) no. 1, pp. 569-584 (http://arXiv.org/pdf/0807.4954) | Article | MR 2685127 | Zbl pre05947723

[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 2078072 | Zbl 1106.11020

[8] Bruin, Nils; Stoll, Michael The Mordell-Weil sieve: proving non-existence of rational points on curves, LMS J. Comput. Math., Tome 13 (2010), pp. 272-306 | Article | MR 2685127 | Zbl pre05947723

[9] Chen, Imin Jacobians of modular curves associated to normalizers of Cartan subgroups of level p n , C. R. Math. Acad. Sci. Paris, Tome 339 (2004) no. 3, pp. 187-192 | Article | MR 330050 | Zbl 0281.14010

[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 2058644 | Zbl 0281.14010 | Zbl 1166.11335

[11] Elkies, Noam D. On elliptic K-curves, Modular curves and abelian varieties, Birkhäuser, Basel (Progr. Math.) Tome 224 (2004), pp. 81-91 | MR 718935 | Zbl 1166.11335 | Zbl 0588.14026

[12] Faltings, G. Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., Tome 73 (1983) no. 3, pp. 349-366 | Article | MR 1737228 | Zbl 0588.14026 | Zbl 0960.14010

[13] Galbraith, Steven D. Rational points on X 0 + (p), Experiment. Math., Tome 8 (1999) no. 4, pp. 311-318 | Article | Numdam | MR 1925998 | Zbl 0960.14010 | Zbl 1035.14008

[14] Galbraith, Steven D. Rational points on X 0 + (N) and quadratic -curves, J. Théor. Nombres Bordeaux, Tome 14 (2002) no. 1, pp. 205-219 | Article | Numdam | MR 1925998 | Zbl 1035.14008

[15] Gaudron, É.; Rémond, G. Théorème des périodes et degrés minimaux d’isogénies (2011) (submitted http://arXiv.org/pdf/1105.1230) | MR 1801716 | Zbl 1297.11058

[16] González, Josep On the j-invariants of the quadratic Q-curves, J. London Math. Soc. (2), Tome 63 (2001) no. 1, pp. 52-68 | Article | MR 563921 | Zbl 1010.11028 | Zbl 0433.14032

[17] Gross, Benedict H. Arithmetic on elliptic curves with complex multiplication, Springer, Berlin, Lecture Notes in Mathematics, Tome 776 (1980) (With an appendix by B. Mazur) | MR 894322 | Zbl 0433.14032 | Zbl 0623.10019

[18] Gross, Benedict H. Heights and the special values of L-series, Number theory (Montreal, Que., 1985), Amer. Math. Soc., Providence, RI (CMS Conf. Proc.) Tome 7 (1987), pp. 115-187 | MR 1482891 | Zbl 0623.10019 | Zbl 0894.11019

[19] Hibino, Takeshi; Murabayashi, Naoki Modular equations of hyperelliptic X 0 (N) and an application, Acta Arith., Tome 82 (1997) no. 3, pp. 279-291 | MR 648603 | Zbl 0894.11019 | Zbl 0492.12002

[20] Kubert, Daniel S.; Lang, Serge Modular units, Springer-Verlag, New York, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], Tome 244 (1981) | MR 1037140 | Zbl 0492.12002 | Zbl 0722.14027

[21] Masser, D. W.; Wüstholz, G. Estimating isogenies on elliptic curves, Invent. Math., Tome 100 (1990) no. 1, pp. 1-24 | Article | Numdam | MR 488287 | Zbl 0722.14027 | Zbl 0394.14008

[22] Mazur, B. Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977) no. 47, p. 33-186 (1978) | Article | Numdam | MR 482230 | Zbl 0394.14008 | Zbl 0386.14009

[23] Mazur, B. Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math., Tome 44 (1978) no. 2, pp. 129-162 | Article | MR 1861089 | Zbl 0386.14009 | Zbl 1020.11041

[24] Merel, Loïc Sur la nature non-cyclotomique des points d’ordre fini des courbes elliptiques, Duke Math. J., Tome 110 (2001) no. 1, pp. 81-119 (With an appendix by E. Kowalski and P. Michel) | Article | MR 2349658 | Zbl 1020.11041 | Zbl 1220.11074

[25] Merel, Loïc Normalizers of split Cartan subgroups and supersingular elliptic curves, Diophantine geometry, Ed. Norm., Pisa (CRM Series) Tome 4 (2007), pp. 237-255 | Numdam | MR 742701 | Zbl 1220.11074 | Zbl 0574.14023

[26] Momose, Fumiyuki Rational points on the modular curves X split (p), Compositio Math., Tome 52 (1984) no. 1, pp. 115-137 | Numdam | MR 866046 | Zbl 0574.14023 | Zbl 0621.14018

[27] Momose, Fumiyuki Rational points on the modular curves X 0 + (p r ), J. Fac. Sci. Univ. Tokyo Sect. IA Math., Tome 33 (1986) no. 3, pp. 441-466 | MR 1892103 | Zbl 0621.14018 | Zbl 1032.11024

[28] Momose, Fumiyuki; Shimura, Mahoro Lifting of supersingular points on X 0 (p r ) and lower bound of ramification index, Nagoya Math. J., Tome 165 (2002), pp. 159-178 | MR 2135276 | Zbl 1032.11024 | Zbl 1167.11310

[29] Parent, Pierre J. R. Towards the triviality of X 0 + (p r )() for r>1, Compos. Math., Tome 141 (2005) no. 3, pp. 561-572 | Article | MR 1865384 | Zbl 1167.11310 | Zbl 0986.11046

[30] Pellarin, Federico Sur une majoration explicite pour un degré d’isogénie liant deux courbes elliptiques, Acta Arith., Tome 100 (2001) no. 3, pp. 203-243 | Article | MR 2375318 | Zbl 0986.11046 | Zbl 1167.11023

[31] Rebolledo, Marusia Module supersingulier, formule de Gross-Kudla et points rationnels de courbes modulaires, Pacific J. Math., Tome 234 (2008) no. 1, pp. 167-184 | Article | MR 2375318 | Zbl 1167.11023

[32] Serre, Jean-Pierre Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math., Tome 15 (1972) no. 4, pp. 259-331 | Article | MR 387283 | Zbl 0235.14012

[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 314766 | Zbl 0221.10029

[34] Silverman, Joseph H. Heights and elliptic curves, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York (1986), pp. 253-265 | MR 861979 | Zbl 0603.14020