Serre’s uniformity problem in the split Cartan case

Abstract

We prove that there exists an integer $p_{0}$ such that $X_{\mathrm{split}} (p)(\Bbb{Q} )$ is made of cusps and CM-points for any prime ${p>p_0}$. Equivalently, for any non-\rm CM elliptic curve $E$ over $\Bbb{Q}$ and any prime ${p>p_0}$ the image of $\mathrm{Gal} (\overline{\Bbb{Q}} /\Bbb{Q} )$ by the representation induced by the Galois action on the $p$-division points of $E$ is not contained in the normalizer of a split Cartan subgroup. This gives a partial answer to an old question of Serre.

  • [Be90] D. Bertrand, "Hauteurs et isogénies," Astérisque, vol. 183, pp. 107-125, 1990.
    @article {Be90, MRKEY = {1065157},
      AUTHOR = {Bertrand, D.},
      TITLE = {Hauteurs et isogénies},
      JOURNAL = {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {183},
      YEAR = {1990},
      PAGES = {107--125},
      ISSN = {0303-1179},
      MRCLASS = {11G05 (11J86 14K02)},
      MRNUMBER = {1065157},
      MRREVIEWER = {Marc Hindry},
      ZBLNUMBER = {0729.14025},
      }
  • [BP08] Go to document Y. Bilu and P. Parent, "Integral $j$-invariants and Cartan structures for elliptic curves," C. R. Math. Acad. Sci. Paris, vol. 346, iss. 11-12, pp. 599-602, 2008.
    @article {BP08, MRKEY = {2423260},
      AUTHOR = {Bilu, Yu. and Parent, Pierre},
      TITLE = {Integral {$j$}-invariants and {C}artan structures for elliptic curves},
      JOURNAL = {C. R. Math. Acad. Sci. Paris},
      FJOURNAL = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
      VOLUME = {346},
      YEAR = {2008},
      NUMBER = {11-12},
      PAGES = {599--602},
      ISSN = {1631-073X},
      MRCLASS = {11G05},
      MRNUMBER = {2423260},
      MRREVIEWER = {Matthew H. Baker},
      DOI = {10.1016/j.crma.2008.04.002},
      ZBLNUMBER = {1165.11053},
      }
  • [BP10] Go to document Y. Bilu and P. Parent, "Runge’s method and modular curves," Internat. Math. Res. Notes, p. 31, 2010.
    @article{BP10,
      author = {Bilu, Yu. and Parent, Pierre},
      TITLE ={Runge's method and modular curves},
      JOURNAL={ Internat. Math. Res. Notes},
      PAGES={31 pages},
      NOTE={article ID rnq141},
      DOI={10.1093/imrn/rnq141},
      YEAR={2010},
      }
  • [Bo83] Go to document E. Bombieri, "On Weil’s “théorème de décomposition”," Amer. J. Math., vol. 105, iss. 2, pp. 295-308, 1983.
    @article {Bo83, MRKEY = {701562},
      AUTHOR = {Bombieri, Enrico},
      TITLE = {On {W}eil's ``théorème de décomposition''},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {105},
      YEAR = {1983},
      NUMBER = {2},
      PAGES = {295--308},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {14H25 (14G25)},
      MRNUMBER = {701562},
      MRREVIEWER = {David Goss},
      DOI = {10.2307/2374261},
      }
  • [Ch04] Go to document I. Chen, "Jacobians of modular curves associated to normalizers of Cartan subgroups of level $p^n$," C. R. Math. Acad. Sci. Paris, vol. 339, iss. 3, pp. 187-192, 2004.
    @article {Ch04, MRKEY = {2078072},
      AUTHOR = {Chen, Imin},
      TITLE = {Jacobians of modular curves associated to normalizers of {C}artan subgroups of level {$p\sp n$}},
      JOURNAL = {C. R. Math. Acad. Sci. Paris},
      FJOURNAL = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
      VOLUME = {339},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {187--192},
      ISSN = {1631-073X},
      MRCLASS = {11G18 (11G40)},
      MRNUMBER = {2078072},
      MRREVIEWER = {Matthew H. Baker},
      DOI = {10.1016/j.crma.2004.04.027},
      ZBLNUMBER = {1106.11020},
      }
  • [Co05] Go to document A. C. Cojocaru, "On the surjectivity of the Galois representations associated to non-CM elliptic curves," Canad. Math. Bull., vol. 48, iss. 1, pp. 16-31, 2005.
    @article {Co05, MRKEY = {2118760},
      AUTHOR = {Cojocaru, Alina Carmen},
      TITLE = {On the surjectivity of the {G}alois representations associated to non-{CM} elliptic curves},
      NOTE = {with an appendix by Ernst Kani},
      JOURNAL = {Canad. Math. Bull.},
      FJOURNAL = {Canadian Mathematical Bulletin. Bulletin Canadien de Mathématiques},
      VOLUME = {48},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {16--31},
      ISSN = {0008-4395},
      CODEN = {CMBUA3},
      MRCLASS = {11G05 (11F80 11N36 11R45)},
      MRNUMBER = {2118760},
      MRREVIEWER = {Ravi K. Ramakrishna},
      URL = {http://journals.cms.math.ca/ams/ams-redirect.php?Journal=CMB&Volume=48&FirstPage=16},
      ZBLNUMBER = {1062.11031},
      }
  • [CH05] Go to document A. C. Cojocaru and C. Hall, "Uniform results for Serre’s theorem for elliptic curves," Int. Math. Res. Not., vol. 2005, iss. 50, pp. 3065-3080, 2005.
    @article {CH05, MRKEY = {2189500},
      AUTHOR = {Cojocaru, Alina Carmen and Hall, Chris},
      TITLE = {Uniform results for {S}erre's theorem for elliptic curves},
      JOURNAL = {Int. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {2005},
      NUMBER = {50},
      PAGES = {3065--3080},
      ISSN = {1073-7928},
      MRCLASS = {11G05},
      MRNUMBER = {189500},
      MRREVIEWER = {Robert Juricevic},
      DOI = {10.1155/IMRN.2005.3065},
      VOLUME = {2005},
      ZBLNUMBER = {1178.11045},
      }
  • [Fa84] Go to document G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern," Invent. Math., vol. 73, iss. 3, pp. 349-366, 1983.
    @article {Fa84, MRKEY = {718935},
      AUTHOR = {Faltings, G.},
      TITLE = {Endlichkeitssätze für abelsche {V}arietäten über {Z}ahlkörpern},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {73},
      YEAR = {1983},
      NUMBER = {3},
      PAGES = {349--366},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11D41 (11G30 14G25)},
      MRNUMBER = {718935},
      MRREVIEWER = {James Milne},
      DOI = {10.1007/BF01388432},
      }
  • [Ka04] K. Kato, "$p$-adic Hodge theory and values of zeta functions of modular forms," Astérisque, vol. 295, p. ix, 117-290, 2004.
    @article {Ka04, MRKEY = {2104361},
      AUTHOR = {Kato, Kazuya},
      TITLE = {{$p$}-adic {H}odge theory and values of zeta functions of modular forms},
      JOURNAL = {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {295},
      YEAR = {2004},
      PAGES = {ix, 117--290},
      ISSN = {0303-1179},
      MRCLASS = {11F85 (11F67 11G40 11R33 11S80 14G10 14G35)},
      MRNUMBER = {2104361},
      MRREVIEWER = {Fabrizio Andreatta},
      }
  • [Kr97] A. Kraus, "Une remarque sur les points de torsion des courbes elliptiques," C. R. Acad. Sci. Paris Sér. I Math., vol. 321, iss. 9, pp. 1143-1146, 1995.
    @article {Kr97, MRKEY = {1360773},
      AUTHOR = {Kraus, Alain},
      TITLE = {Une remarque sur les points de torsion des courbes elliptiques},
      JOURNAL = {C. R. Acad. Sci. Paris Sér. I Math.},
      FJOURNAL = {Comptes Rendus de l'Académie des Sciences. Série I. Mathématique},
      VOLUME = {321},
      YEAR = {1995},
      NUMBER = {9},
      PAGES = {1143--1146},
      ISSN = {0764-4442},
      CODEN = {CASMEI},
      MRCLASS = {11G05},
      MRNUMBER = {1360773},
      MRREVIEWER = {Toshihiro Hadano},
      ZBLNUMBER = {0862.11037},
      }
  • [KL81] D. S. Kubert and S. Lang, Modular Units, New York: Springer-Verlag, 1981, vol. 244.
    @book {KL81, MRKEY = {648603},
      AUTHOR = {Kubert, Daniel S. and Lang, Serge},
      TITLE = {Modular {{U}}nits},
      SERIES = {Grundl. Math. Wissen.},
      VOLUME = {244},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1981},
      PAGES = {xiii+358},
      ISBN = {0-387-90517-0},
      MRCLASS = {12A45 (10D05 14G25)},
      MRNUMBER = {648603},
      MRREVIEWER = {Roland Gillard},
      ZBLNUMBER = {0492.12002},
      }
  • [Le08] Go to document A. Levin, "Variations on a theme of Runge: effective determination of integral points on certain varieties," J. Théor. Nombres Bordeaux, vol. 20, iss. 2, pp. 385-417, 2008.
    @article {Le08, MRKEY = {2477511},
      AUTHOR = {Levin, Aaron},
      TITLE = {Variations on a theme of {R}unge: effective determination of integral points on certain varieties},
      JOURNAL = {J. Théor. Nombres Bordeaux},
      FJOURNAL = {Journal de Théorie des Nombres de Bordeaux},
      VOLUME = {20},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {385--417},
      ISSN = {1246-7405},
      MRCLASS = {11G35 (14G05)},
      MRNUMBER = {2477511},
      MRREVIEWER = {Abderrahmane Nitaj},
      URL = {http://jtnb.cedram.org/item?id=JTNB_2008__20_2_385_0},
      ZBLNUMBER = {1179.11018},
      }
  • [MW90] Go to document D. W. Masser and G. Wüstholz, "Estimating isogenies on elliptic curves," Invent. Math., vol. 100, iss. 1, pp. 1-24, 1990.
    @article {MW90, MRKEY = {1037140},
      AUTHOR = {Masser, D. W. and W{ü}stholz, G.},
      TITLE = {Estimating isogenies on elliptic curves},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {100},
      YEAR = {1990},
      NUMBER = {1},
      PAGES = {1--24},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11G05 (11J89 14G25 14K02)},
      MRNUMBER = {1037140},
      MRREVIEWER = {Marc Hindry},
      DOI = {10.1007/BF01231178},
      }
  • [MW93] Go to document D. W. Masser and G. Wüstholz, "Galois properties of division fields of elliptic curves," Bull. London Math. Soc., vol. 25, iss. 3, pp. 247-254, 1993.
    @article {MW93, MRKEY = {1209248},
      AUTHOR = {Masser, D. W. and W{ü}stholz, G.},
      TITLE = {Galois properties of division fields of elliptic curves},
      JOURNAL = {Bull. London Math. Soc.},
      FJOURNAL = {The Bulletin of the London Mathematical Society},
      VOLUME = {25},
      YEAR = {1993},
      NUMBER = {3},
      PAGES = {247--254},
      ISSN = {0024-6093},
      CODEN = {LMSBBT},
      MRCLASS = {11G05},
      MRNUMBER = {1209248},
      MRREVIEWER = {Kenneth A. Ribet},
      DOI = {10.1112/blms/25.3.247},
      }
  • [Ma76] B. Mazur, "Rational points on modular curves," in Modular Functions of One Variable, V, New York: Springer-Verlag, 1977, vol. 601, pp. 107-148.
    @incollection {Ma76, MRKEY = {0450283},
      AUTHOR = {Mazur, B.},
      TITLE = {Rational points on modular curves},
      BOOKTITLE = {Modular {{F}}unctions of {{O}}ne {{V}}ariable, {{{\rm V}}}},
      VENUE={{P}roc. {S}econd {I}nternat. {C}onf., {U}niv. {B}onn, {B}onn, 1976},
      PAGES = {107--148},
      SERIES={Lecture Notes in Math.},
      VOLUME={601},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      MRCLASS = {14G25 (10D15 14H25 14K15)},
      MRNUMBER = {56 \#8579},
      MRREVIEWER = {M. A. Kenku},
      ZBLNUMBER = {0357.14005},
      }
  • [Ma77] Go to document B. Mazur, "Modular curves and the Eisenstein ideal," Inst. Hautes Études Sci. Publ. Math., iss. 47, pp. 33-186 (1978), 1977.
    @article {Ma77, MRKEY = {488287},
      AUTHOR = {Mazur, B.},
      TITLE = {Modular curves and the {E}isenstein ideal},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {47},
      YEAR = {1977},
      PAGES = {33--186 (1978)},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {14G25 (10D05)},
      MRNUMBER = {488287},
      MRREVIEWER = {M. Ohta},
      URL = {http://www.numdam.org/item?id=PMIHES_1977__47__33_0},
      ZBLNUMBER = {0394.14008},
      }
  • [Ma78] Go to document B. Mazur, "Rational isogenies of prime degree (with an appendix by D. Goldfeld)," Invent. Math., vol. 44, iss. 2, pp. 129-162, 1978.
    @article {Ma78, MRKEY = {482230},
      AUTHOR = {Mazur, B.},
      TITLE = {Rational isogenies of prime degree (with an appendix by {D}. {G}oldfeld)},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {44},
      YEAR = {1978},
      NUMBER = {2},
      PAGES = {129--162},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14K07 (10D35 14G25)},
      MRNUMBER = {482230},
      MRREVIEWER = {V. V. Shokurov},
      DOI = {10.1007/BF01390348},
      }
  • [Me05] L. Merel, "Normalizers of split Cartan subgroups and supersingular elliptic curves," in Diophantine Geometry, Ed. Norm., Pisa, 2007, vol. 4, pp. 237-255.
    @incollection {Me05, MRKEY = {2349658},
      AUTHOR = {Merel, Lo{ï}c},
      TITLE = {Normalizers of split {C}artan subgroups and supersingular elliptic curves},
      BOOKTITLE = {Diophantine {{G}}eometry},
      SERIES = {CRM Series},
      VOLUME = {4},
      PAGES = {237--255},
      PUBLISHER = {Ed. Norm., Pisa},
      YEAR = {2007},
      MRCLASS = {11G05 (11G18 11G35)},
      MRNUMBER = {2349658},
      MRREVIEWER = {Andrew Bremner},
      ZBLNUMBER = {05263288},
      }
  • [Mo84] Go to document F. Momose, "Rational points on the modular curves $X_{{ split}}(p)$," Compositio Math., vol. 52, iss. 1, pp. 115-137, 1984.
    @article {Mo84, MRKEY = {742701},
      AUTHOR = {Momose, Fumiyuki},
      TITLE = {Rational points on the modular curves {$X\sb {{\rm split}}(p)$}},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {52},
      YEAR = {1984},
      NUMBER = {1},
      PAGES = {115--137},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {11G18 (11F11 11G30 14G05 14G25)},
      MRNUMBER = {742701},
      MRREVIEWER = {Ernst Kani},
      URL = {http://www.numdam.org/item?id=CM_1984__52_1_115_0},
      ZBLNUMBER = {0574.14023},
      }
  • [Pa05] Go to document P. J. R. Parent, "Towards the triviality of $X^+_0(p^r)(\Bbb Q)$ for $r>1$," Compositio Math., vol. 141, iss. 3, pp. 561-572, 2005.
    @article {Pa05, MRKEY = {2135276},
      AUTHOR = {Parent, Pierre J. R.},
      TITLE = {Towards the triviality of {$X\sp +\sb 0(p\sp r)(\Bbb Q)$} for {$r>1$}},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {141},
      YEAR = {2005},
      NUMBER = {3},
      PAGES = {561--572},
      ISSN = {0010-437X},
      MRCLASS = {11G18 (11G05)},
      MRNUMBER = {2135276},
      MRREVIEWER = {Henri Darmon},
      DOI = {10.1112/S0010437X04001022},
      }
  • [Pe01] Go to document F. Pellarin, "Sur une majoration explicite pour un degré d’isogénie liant deux courbes elliptiques," Acta Arith., vol. 100, iss. 3, pp. 203-243, 2001.
    @article {Pe01, MRKEY = {1865384},
      AUTHOR = {Pellarin, Federico},
      TITLE = {Sur une majoration explicite pour un degré d'isogénie liant deux courbes elliptiques},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Acta Arithmetica},
      VOLUME = {100},
      YEAR = {2001},
      NUMBER = {3},
      PAGES = {203--243},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {11G05 (11G50 11J89)},
      MRNUMBER = {1865384},
      MRREVIEWER = {Antoine Chambert-Loir},
      DOI = {10.4064/aa100-3-1},
      ZBLNUMBER = {0986.11046},
      }
  • [Re08] Go to document M. Rebolledo, "Module supersingulier, formule de Gross-Kudla et points rationnels de courbes modulaires," Pacific J. Math., vol. 234, iss. 1, pp. 167-184, 2008.
    @article {Re08, MRKEY = {2375318},
      AUTHOR = {Rebolledo, Marusia},
      TITLE = {Module supersingulier, formule de {G}ross-{K}udla et points rationnels de courbes modulaires},
      JOURNAL = {Pacific J. Math.},
      FJOURNAL = {Pacific Journal of Mathematics},
      VOLUME = {234},
      YEAR = {2008},
      NUMBER = {1},
      PAGES = {167--184},
      ISSN = {0030-8730},
      CODEN = {PJMAAI},
      MRCLASS = {11G05 (11G18 11R52 14G05 14G10)},
      MRNUMBER = {2375318},
      MRREVIEWER = {Am{\'ı}lcar Pacheco},
      DOI = {10.2140/pjm.2008.234.167},
      ZBLNUMBER = {1167.11023},
      }
  • [Se72] Go to document J. Serre, "Propriétés galoisiennes des points d’ordre fini des courbes elliptiques," Invent. Math., vol. 15, iss. 4, pp. 259-331, 1972.
    @article {Se72, MRKEY = {0387283},
      AUTHOR = {Serre, Jean-Pierre},
      TITLE = {Propriétés galoisiennes des points d'ordre fini des courbes elliptiques},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {15},
      YEAR = {1972},
      NUMBER = {4},
      PAGES = {259--331},
      ISSN = {0020-9910},
      MRCLASS = {14G25 (14K15)},
      MRNUMBER = {52 \#8126},
      MRREVIEWER = {J. W. S. Cassels},
      DOI = {10.1007/BF01405086},
      ZBLNUMBER = {0235.14012},
      }
  • [Si84] J. H. Silverman, "Heights and elliptic curves," in Arithmetic Geometry, New York: Springer-Verlag, 1986, pp. 253-265.
    @incollection {Si84, MRKEY = {861979},
      AUTHOR = {Silverman, Joseph H.},
      TITLE = {Heights and elliptic curves},
      BOOKTITLE = {Arithmetic {{G}}eometry},
      VENUE={{S}torrs, {C}onn., 1984},
      PAGES = {253--265},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1986},
      MRCLASS = {11G05 (11D41)},
      MRNUMBER = {861979},
      ZBLNUMBER = {0603.14020},
      }
  • [Vi10] E. Viada, Minimal elliptic isogenies.
    @misc{Vi10,
      author={Viada, E.},
      TITLE={Minimal elliptic isogenies},
      NOTE={preprint},
      }

Authors

Yuri Bilu

Institut de Mathématiques de Bordeaux
Université Bordeaux I
33 405 Talence
France

Pierre Parent

Institut de Mathématiques de Bordeaux
Université Bordeaux I
33 405 Talence
France