Explicit Chabauty–Kim for the split Cartan modular curve of level 13

Abstract

We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in a method to compute a finite set of $p$-adic points, containing the rational points, on a curve of genus $g \ge 2$ over the rationals whose Jacobian has Mordell–Weil rank $g$ and Picard number greater than one, and which satisfies some additional conditions. This is then applied to determine the rational points of the modular curve $X_{\mathrm { s}}(13)$, completing the classification of non-CM elliptic curves over $\mathbf {Q} $ with split Cartan level structure due to Bilu–Parent and Bilu–Parent–Rebolledo.

  • [AS05] Go to document A. Agashe and W. Stein, "Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero," Math. Comp., vol. 74, iss. 249, pp. 455-484, 2005.
    @ARTICLE{AS05,
      author = {Agashe, Amod and Stein, William},
      title = {Visible evidence for the {B}irch and {S}winnerton-{D}yer conjecture for modular abelian varieties of analytic rank zero},
      note = {With an appendix by J. Cremona and B. Mazur},
      journal = {Math. Comp.},
      fjournal = {Mathematics of Computation},
      volume = {74},
      year = {2005},
      number = {249},
      pages = {455--484},
      issn = {0025-5718},
      mrclass = {11G40 (11G10)},
      mrnumber = {2085902},
      mrreviewer = {Chandan Singh Dalawat},
      doi = {10.1090/S0025-5718-04-01644-8},
      url = {https://doi.org/10.1090/S0025-5718-04-01644-8},
      zblnumber = {1084.11033},
      }
  • [Bar14a] Go to document B. Baran, "An exceptional isomorphism between modular curves of level 13," J. Number Theory, vol. 145, pp. 273-300, 2014.
    @ARTICLE{Bar14a,
      author = {Baran, Burcu},
      title = {An exceptional isomorphism between modular curves of level 13},
      journal = {J. Number Theory},
      fjournal = {Journal of Number Theory},
      volume = {145},
      year = {2014},
      pages = {273--300},
      issn = {0022-314X},
      mrclass = {11G05 (11F70)},
      mrnumber = {3253304},
      mrreviewer = {Rupam Barman},
      doi = {10.1016/j.jnt.2014.05.017},
      url = {https://doi.org/10.1016/j.jnt.2014.05.017},
      zblnumber = {1300.11055},
      }
  • [Bar14b] Go to document B. Baran, "An exceptional isomorphism between level 13 modular curves via Torelli’s theorem," Math. Res. Lett., vol. 21, iss. 5, pp. 919-936, 2014.
    @ARTICLE{Bar14b,
      author = {Baran, Burcu},
      title = {An exceptional isomorphism between level 13 modular curves via {T}orelli's theorem},
      journal = {Math. Res. Lett.},
      fjournal = {Mathematical Research Letters},
      volume = {21},
      year = {2014},
      number = {5},
      pages = {919--936},
      issn = {1073-2780},
      mrclass = {14G35 (11G18 14G25 14H40)},
      mrnumber = {3294556},
      mrreviewer = {José Mar\'{i}a Tornero},
      doi = {10.4310/MRL.2014.v21.n5.a1},
      url = {https://doi.org/10.4310/MRL.2014.v21.n5.a1},
      zblnumber = {1327.14120},
      }
  • [BB15] Go to document J. S. Balakrishnan and A. Besser, "Coleman-Gross height pairings and the $p$-adic sigma function," J. Reine Angew. Math., vol. 698, pp. 89-104, 2015.
    @ARTICLE{BB15,
      author = {Balakrishnan, Jennifer S. and Besser, Amnon},
      title = {Coleman-{G}ross height pairings and the {$p$}-adic sigma function},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {698},
      year = {2015},
      pages = {89--104},
      issn = {0075-4102},
      mrclass = {11G50 (11F23)},
      mrnumber = {3294652},
      mrreviewer = {Jörg Jahnel},
      doi = {10.1515/crelle-2012-0095},
      url = {https://doi.org/10.1515/crelle-2012-0095},
      zblnumber = {1348.11091},
      }
  • [BBM16] Go to document J. S. Balakrishnan, A. Besser, and S. J. Müller, "Quadratic Chabauty: $p$-adic heights and integral points on hyperelliptic curves," J. Reine Angew. Math., vol. 720, pp. 51-79, 2016.
    @ARTICLE{BBM16,
      author = {Balakrishnan, Jennifer S. and Besser, Amnon and Müller, J. Steffen},
      title = {Quadratic {C}habauty: {$p$}-adic heights and integral points on hyperelliptic curves},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {720},
      year = {2016},
      pages = {51--79},
      issn = {0075-4102},
      mrclass = {11G30 (11S80)},
      mrnumber = {3565969},
      mrreviewer = {Werner Bley},
      doi = {10.1515/crelle-2014-0048},
      url = {https://doi.org/10.1515/crelle-2014-0048},
      zblnumber = {1350.11067},
      }
  • [BBM17] Go to document J. S. Balakrishnan, A. Besser, and S. J. Müller, "Computing integral points on hyperelliptic curves using quadratic Chabauty," Math. Comp., vol. 86, iss. 305, pp. 1403-1434, 2017.
    @ARTICLE{BBM17,
      author = {Balakrishnan, Jennifer S. and Besser, Amnon and Müller, J. Steffen},
      title = {Computing integral points on hyperelliptic curves using quadratic {C}habauty},
      journal = {Math. Comp.},
      fjournal = {Mathematics of Computation},
      volume = {86},
      year = {2017},
      number = {305},
      pages = {1403--1434},
      issn = {0025-5718},
      mrclass = {11G30 (11S80 11Y50 14G05 14G40 14H50)},
      mrnumber = {3614022},
      mrreviewer = {Mohamed Talbi},
      doi = {10.1090/mcom/3130},
      url = {https://doi.org/10.1090/mcom/3130},
      zblnumber = {1376.11053},
      }
  • [BCP97] Go to document W. Bosma, J. Cannon, and C. Playoust, "The Magma algebra system. I. The user language," J. Symbolic Comput., vol. 24, iss. 3-4, pp. 235-265, 1997.
    @ARTICLE{BCP97,
      author = {Bosma, Wieb and Cannon, John and Playoust, Catherine},
      title = {The {M}agma algebra system. {I}. {T}he user language},
      note = {Computational algebra and number theory (London, 1993)},
      journal = {J. Symbolic Comput.},
      fjournal = {Journal of Symbolic Computation},
      volume = {24},
      year = {1997},
      number = {3-4},
      pages = {235--265},
      issn = {0747-7171},
      mrclass = {68Q40},
      mrnumber = {1484478},
      doi = {10.1006/jsco.1996.0125},
      url = {https://doi.org/10.1006/jsco.1996.0125},
      zblnumber = {0898.68039},
      }
  • [BD16] Go to document J. S. Balakrishnan and N. Dogra, "Quadratic Chabauty and rational points, I: $p$-adic heights," Duke Math. J., vol. 167, iss. 11, pp. 1981-2038, 2018.
    @ARTICLE{BD16,
      author = {Balakrishnan, Jennifer S. and Dogra, Netan},
      title = {Quadratic {C}habauty and rational points, {I}: {$p$}-adic heights},
      note = {With an appendix by J. Steffen Müller},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {167},
      year = {2018},
      number = {11},
      pages = {1981--2038},
      issn = {0012-7094},
      mrclass = {14G05 (11G50 14G40)},
      mrnumber = {3843370},
      doi = {10.1215/00127094-2018-0013},
      url = {https://doi.org/10.1215/00127094-2018-0013},
      zblnumber = {1401.14123},
      }
  • [BD17] J. S. Balakrishnan and N. Dogra, "Quadratic Chabauty and rational points, II: Generalised height functions on Selmer varieties," , 2017.
    @ARTICLE{BD17,
      author = {Balakrishnan, Jennifer S. and Dogra, Netan},
      title = {Quadratic {C}habauty and rational points, {II}: Generalised height functions on {S}elmer varieties},
      arxiv={1705.00401},
      year={2017},
      }
  • [BDCKW] Go to document J. S. Balakrishnan, I. Dan-Cohen, M. Kim, and S. Wewers, "A non-abelian conjecture of Tate-Shafarevich type for hyperbolic curves," Math. Ann., vol. 372, iss. 1-2, pp. 369-428, 2018.
    @ARTICLE{BDCKW,
      author = {Balakrishnan, Jennifer S. and Dan-Cohen, Ishai and Kim, Minhyong and Wewers, Stefan},
      title = {A non-abelian conjecture of {T}ate-{S}hafarevich type for hyperbolic curves},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {372},
      year = {2018},
      number = {1-2},
      pages = {369--428},
      issn = {0025-5831},
      mrclass = {11D45 (11G50 14F35 14H52)},
      mrnumber = {3856816},
      doi = {10.1007/s00208-018-1684-x},
      url = {https://doi.org/10.1007/s00208-018-1684-x},
      zblnumber = {06943973},
      }
  • [CartanCode] Go to document J. S. Balakrishnan, N. Dogra, J. S. Müller, J. Tuitman, and J. Vonk, Magma code.
    @MISC{CartanCode,
      author = {Balakrishnan, Jennifer S. and Dogra, N. and Müller, J. S. and Tuitman, J. and Vonk, J.},
      title = {Magma code},
      url = {https://github.com/jtuitman/Cartan13},
      zblnumber = {},
      }
  • [Ber96] P. Berthelot, Cohomologie rigide et cohomologie rigide à supports propres, 1996.
    @misc{Ber96,
      author = {P. Berthelot},
      note = {preprint},
      Title={Cohomologie rigide et cohomologie rigide à supports propres},
      Year={1996},
     }
  • [Bes02] Go to document A. Besser, "Coleman integration using the Tannakian formalism," Math. Ann., vol. 322, iss. 1, pp. 19-48, 2002.
    @ARTICLE{Bes02,
      author = {Besser, Amnon},
      title = {Coleman integration using the {T}annakian formalism},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {322},
      year = {2002},
      number = {1},
      pages = {19--48},
      issn = {0025-5831},
      mrclass = {11S80 (11G25 14F30)},
      mrnumber = {1883387},
      mrreviewer = {Bruno Chiarellotto},
      doi = {10.1007/s002080100263},
      url = {https://doi.org/10.1007/s002080100263},
      zblnumber = {1013.11028},
      }
  • [Bes04] Go to document A. Besser, "The $p$-adic height pairings of Coleman-Gross and of Neková\vr," in Number Theory, Amer. Math. Soc., Providence, RI, 2004, vol. 36, pp. 13-25.
    @INCOLLECTION{Bes04,
      author = {Besser, Amnon},
      title = {The {$p$}-adic height pairings of {C}oleman-{G}ross and of {N}ekov\'{a}\v{r}},
      booktitle = {Number Theory},
      series = {CRM Proc. Lecture Notes},
      volume = {36},
      pages = {13--25},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2004},
      mrclass = {11G50 (11G25 11G30)},
      mrnumber = {2076563},
      mrreviewer = {Mark Kisin},
      zblnumber = {1153.11316},
      doi = {10.1090/crmp/036/02},
     }
  • [Bet17] A. L. Betts, The motivic anabelian geometry of local heights on abelian varieties, 2017.
    @MISC{Bet17,
      author = {Betts, L. Alexander},
      title = {The motivic anabelian geometry of local heights on abelian varieties},
      arxiv = {1706.04850},
      year = {2017},
      zblnumber = {},
      }
  • [BK90] Go to document S. Bloch and K. Kato, "$L$-functions and Tamagawa numbers of motives," in The Grothendieck Festschrift, Vol. I, Birkhäuser Boston, Boston, MA, 1990, vol. 86, pp. 333-400.
    @INCOLLECTION{BK90,
      author = {Bloch, Spencer and Kato, Kazuya},
      title = {{$L$}-functions and {T}amagawa numbers of motives},
      booktitle = {The {G}rothendieck {F}estschrift, {V}ol. {I}},
      series = {Progr. Math.},
      volume = {86},
      pages = {333--400},
      publisher = {Birkhäuser Boston, Boston, MA},
      year = {1990},
      mrclass = {11G40 (11G09 14C35 14F30 14G10)},
      mrnumber = {1086888},
      mrreviewer = {Ehud de Shalit},
      zblnumber = {0768.14001},
      doi = {10.1007/978-0-8176-4574-8_9},
     }
  • [BL04] Go to document C. Birkenhake and H. Lange, Complex Abelian Barieties, Second ed., Springer-Verlag, Berlin, 2004, vol. 302.
    @BOOK{BL04,
      author = {Birkenhake, Christina and Lange, Herbert},
      title = {Complex Abelian Barieties},
      series = {Grundlehren Math. Wissen.},
      volume = {302},
      edition = {Second},
      publisher = {Springer-Verlag, Berlin},
      year = {2004},
      pages = {xii+635},
      isbn = {3-540-20488-1},
      mrclass = {14-02 (14H37 14Kxx 32G20)},
      mrnumber = {2062673},
      mrreviewer = {Fumio Hazama},
      doi = {10.1007/978-3-662-06307-1},
      url = {https://doi.org/10.1007/978-3-662-06307-1},
      zblnumber = {1056.14063},
      }
  • [BO83] Go to document P. Berthelot and A. Ogus, "$F$-isocrystals and de Rham cohomology. I," Invent. Math., vol. 72, iss. 2, pp. 159-199, 1983.
    @ARTICLE{BO83,
      author = {Berthelot, P. and Ogus, A.},
      title = {{$F$}-isocrystals and de {R}ham cohomology. {I}},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {72},
      year = {1983},
      number = {2},
      pages = {159--199},
      issn = {0020-9910},
      mrclass = {14F30},
      mrnumber = {0700767},
      mrreviewer = {G. Horrocks},
      doi = {10.1007/BF01389319},
      url = {https://doi.org/10.1007/BF01389319},
      zblnumber = {0516.14017},
      }
  • [BP11] Go to document Y. Bilu and P. Parent, "Serre’s uniformity problem in the split Cartan case," Ann. of Math. (2), vol. 173, iss. 1, pp. 569-584, 2011.
    @ARTICLE{BP11,
      author = {Bilu, Yuri and Parent, Pierre},
      title = {Serre's uniformity problem in the split {C}artan case},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {173},
      year = {2011},
      number = {1},
      pages = {569--584},
      issn = {0003-486X},
      mrclass = {11G15 (11G05)},
      mrnumber = {2753610},
      mrreviewer = {Damian Rössler},
      doi = {10.4007/annals.2011.173.1.13},
      url = {https://doi.org/10.4007/annals.2011.173.1.13},
      zblnumber = {1278.11065},
      }
  • [BPR13] Go to document Y. Bilu, P. Parent, and M. Rebolledo, "Rational points on $X^+_0(p^r)$," Ann. Inst. Fourier (Grenoble), vol. 63, iss. 3, pp. 957-984, 2013.
    @ARTICLE{BPR13,
      author = {Bilu, Yuri and Parent, Pierre and Rebolledo, Marusia},
      title = {Rational points on {$X^+_0(p^r)$}},
      journal = {Ann. Inst. Fourier (Grenoble)},
      fjournal = {Université de Grenoble. Annales de l'Institut Fourier},
      volume = {63},
      year = {2013},
      number = {3},
      pages = {957--984},
      issn = {0373-0956},
      mrclass = {11G18 (11G05 11G16)},
      mrnumber = {3137477},
      doi = {10.5802/aif.2781},
      url = {https://doi.org/10.5802/aif.2781},
      zblnumber = {1307.11075},
      }
  • [BPS16] Go to document N. Bruin, B. Poonen, and M. Stoll, "Generalized explicit descent and its application to curves of genus 3," Forum Math. Sigma, vol. 4, p. 6, 2016.
    @ARTICLE{BPS16,
      author = {Bruin, Nils and Poonen, Bjorn and Stoll, Michael},
      title = {Generalized explicit descent and its application to curves of genus 3},
      journal = {Forum Math. Sigma},
      fjournal = {Forum of Mathematics. Sigma},
      volume = {4},
      year = {2016},
      pages = {e6, 80},
      issn = {2050-5094},
      mrclass = {11G30 (11G10 14G25 14H45)},
      mrnumber = {3482281},
      mrreviewer = {Joseph H. Silverman},
      doi = {10.1017/fms.2016.1},
      url = {https://doi.org/10.1017/fms.2016.1},
      zblnumber = {06554135},
      }
  • [Bru03] Go to document N. Bruin, "Chabauty methods using elliptic curves," J. Reine Angew. Math., vol. 562, pp. 27-49, 2003.
    @ARTICLE{Bru03,
      author = {Bruin, Nils},
      title = {Chabauty methods using elliptic curves},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {562},
      year = {2003},
      pages = {27--49},
      issn = {0075-4102},
      mrclass = {11G05},
      mrnumber = {2011330},
      mrreviewer = {Matthew H. Baker},
      doi = {10.1515/crll.2003.076},
      url = {https://doi.org/10.1515/crll.2003.076},
      zblnumber = {1135.11320},
      }
  • [BS09] Go to document N. Bruin and M. Stoll, "Two-cover descent on hyperelliptic curves," Math. Comp., vol. 78, iss. 268, pp. 2347-2370, 2009.
    @ARTICLE{BS09,
      author = {Bruin, Nils and Stoll, Michael},
      title = {Two-cover descent on hyperelliptic curves},
      journal = {Math. Comp.},
      fjournal = {Mathematics of Computation},
      volume = {78},
      year = {2009},
      number = {268},
      pages = {2347--2370},
      issn = {0025-5718},
      mrclass = {11G30},
      mrnumber = {2521292},
      mrreviewer = {Pavlos Tzermias},
      doi = {10.1090/S0025-5718-09-02255-8},
      url = {https://doi.org/10.1090/S0025-5718-09-02255-8},
      zblnumber = {1208.11078},
      }
  • [BT] J. S. Balakrishnan and J. Tuitman, Explicit Coleman integration for curves, 2017.
    @MISC{BT,
      author = {Balakrishnan, Jennifer S. and Tuitman, Jan},
      title = {Explicit {C}oleman integration for curves},
      note = {preprint},
      year = {2017},
      zblnumber = {},
      arxiv = {1710.01673},
      }
  • [CG89] Go to document R. F. Coleman and B. H. Gross, "$p$-adic heights on curves," in Algebraic Number Theory, Academic Press, Boston, MA, 1989, vol. 17, pp. 73-81.
    @INCOLLECTION{CG89,
      author = {Coleman, Robert F. and Gross, Benedict H.},
      title = {{$p$}-adic heights on curves},
      booktitle = {Algebraic Number Theory},
      series = {Adv. Stud. Pure Math.},
      volume = {17},
      pages = {73--81},
      publisher = {Academic Press, Boston, MA},
      year = {1989},
      mrclass = {11F85 (11G10 11G30)},
      mrnumber = {1097610},
      mrreviewer = {Dipendra Prasad},
      zblnumber = {0758.14009},
      doi = {10.2969/aspm/01710073},
      }
  • [Cha41] C. Chabauty, "Sur les points rationnels des courbes algébriques de genre supérieur à l’unité," C. R. Acad. Sci. Paris, vol. 212, pp. 882-885, 1941.
    @ARTICLE{Cha41,
      author = {Chabauty, Claude},
      title = {Sur les points rationnels des courbes algébriques de genre supérieur à l'unité},
      journal = {C. R. Acad. Sci. Paris},
      volume = {212},
      year = {1941},
      pages = {882--885},
      mrclass = {14.0X},
      mrnumber = {0004484},
      mrreviewer = {O. F. G. Schilling},
      zblnumber = {0025.24902},
      }
  • [C98] Go to document I. Chen, "The Jacobians of non-split Cartan modular curves," Proc. London Math. Soc. (3), vol. 77, iss. 1, pp. 1-38, 1998.
    @ARTICLE{C98,
      author = {Chen, Imin},
      title = {The {J}acobians of non-split {C}artan modular curves},
      journal = {Proc. London Math. Soc. (3)},
      fjournal = {Proceedings of the London Mathematical Society. Third Series},
      volume = {77},
      year = {1998},
      number = {1},
      pages = {1--38},
      issn = {0024-6115},
      mrclass = {11G18 (11F72)},
      mrnumber = {1625491},
      mrreviewer = {Antoine Chambert-Loir},
      doi = {10.1112/S0024611598000392},
      url = {https://doi.org/10.1112/S0024611598000392},
      zblnumber = {0903.11019},
      }
  • [CK10] Go to document J. Coates and M. Kim, "Selmer varieties for curves with CM Jacobians," Kyoto J. Math., vol. 50, iss. 4, pp. 827-852, 2010.
    @ARTICLE{CK10,
      author = {Coates, John and Kim, Minhyong},
      title = {Selmer varieties for curves with {CM} {J}acobians},
      journal = {Kyoto J. Math.},
      fjournal = {Kyoto Journal of Mathematics},
      volume = {50},
      year = {2010},
      number = {4},
      pages = {827--852},
      issn = {2156-2261},
      mrclass = {11G30 (11G15 11R23)},
      mrnumber = {2740695},
      mrreviewer = {Dipendra Prasad},
      doi = {10.1215/0023608X-2010-015},
      url = {https://doi.org/10.1215/0023608X-2010-015},
      zblnumber = {1283.11092},
      }
  • [Col85] Go to document R. F. Coleman, "Effective Chabauty," Duke Math. J., vol. 52, iss. 3, pp. 765-770, 1985.
    @ARTICLE{Col85,
      author = {Coleman, Robert F.},
      title = {Effective {C}habauty},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {52},
      year = {1985},
      number = {3},
      pages = {765--770},
      issn = {0012-7094},
      mrclass = {11G30 (14G25 14K15)},
      mrnumber = {0808103},
      mrreviewer = {Loren D. Olson},
      doi = {10.1215/S0012-7094-85-05240-8},
      url = {https://doi.org/10.1215/S0012-7094-85-05240-8},
      zblnumber = {0588.14015},
      }
  • [CLS99] Go to document B. Chiarellotto and B. Le Stum, "$F$-isocristaux unipotents," Compositio Math., vol. 116, iss. 1, pp. 81-110, 1999.
    @ARTICLE{CLS99,
      author = {Chiarellotto, Bruno and Le Stum, Bernard},
      title = {{$F$}-isocristaux unipotents},
      journal = {Compositio Math.},
      fjournal = {Compositio Mathematica},
      volume = {116},
      year = {1999},
      number = {1},
      pages = {81--110},
      issn = {0010-437X},
      mrclass = {14F30},
      mrnumber = {1669440},
      mrreviewer = {Abdellah Mokrane},
      doi = {10.1023/A:1000602824628},
      url = {https://doi.org/10.1023/A:1000602824628},
      zblnumber = {0936.14017},
      }
  • [CT03] Go to document B. Chiarellotto and N. Tsuzuki, "Cohomological descent of rigid cohomology for étale coverings," Rend. Sem. Mat. Univ. Padova, vol. 109, pp. 63-215, 2003.
    @ARTICLE{CT03,
      author = {Chiarellotto, Bruno and Tsuzuki, Nobuo},
      title = {Cohomological descent of rigid cohomology for étale coverings},
      journal = {Rend. Sem. Mat. Univ. Padova},
      fjournal = {Rendiconti del Seminario Matematico della Università di Padova. The Mathematical Journal of the University of Padova},
      volume = {109},
      year = {2003},
      pages = {63--215},
      issn = {0041-8994},
      mrclass = {14F30 (14F43 14G22)},
      mrnumber = {1997987},
      mrreviewer = {Elmar Grosse-Klönne},
      zblnumber = {1167.14306},
      url = {http://www.numdam.org/item/?id=RSMUP_2003__109__63_0},
      }
  • [DC] I. Dan-Cohen, Mixed Tate motives and the unit equation II, 2018.
    @MISC{DC,
      author = {Dan-Cohen, Ishai},
      title = {Mixed {T}ate motives and the unit equation {II}},
      note = {preprint},
      year = {2018},
      arxiv = {1510.01362},
      }
  • [DCW15] Go to document I. Dan-Cohen and S. Wewers, "Explicit Chabauty-Kim theory for the thrice punctured line in depth 2," Proc. Lond. Math. Soc. (3), vol. 110, iss. 1, pp. 133-171, 2015.
    @ARTICLE{DCW15,
      author = {Dan-Cohen, Ishai and Wewers, Stefan},
      title = {Explicit {C}habauty-{K}im theory for the thrice punctured line in depth 2},
      journal = {Proc. Lond. Math. Soc. (3)},
      fjournal = {Proceedings of the London Mathematical Society. Third Series},
      volume = {110},
      year = {2015},
      number = {1},
      pages = {133--171},
      issn = {0024-6115},
      mrclass = {11G55 (11D45 14F42)},
      mrnumber = {3299602},
      mrreviewer = {Piotr Krasoń},
      doi = {10.1112/plms/pdu034},
      url = {https://doi.org/10.1112/plms/pdu034},
      zblnumber = {1379.11068},
      }
  • [DCW16] Go to document I. Dan-Cohen and S. Wewers, "Mixed Tate motives and the unit equation," Int. Math. Res. Not. IMRN, iss. 17, pp. 5291-5354, 2016.
    @ARTICLE{DCW16,
      author = {Dan-Cohen, Ishai and Wewers, Stefan},
      title = {Mixed {T}ate motives and the unit equation},
      journal = {Int. Math. Res. Not. IMRN},
      fjournal = {International Mathematics Research Notices. IMRN},
      year = {2016},
      number = {17},
      pages = {5291--5354},
      issn = {1073-7928},
      mrclass = {14F42 (11D45 11G55 14H30)},
      mrnumber = {3556439},
      mrreviewer = {Piotr Krasoń},
      doi = {10.1093/imrn/rnv239},
      url = {https://doi.org/10.1093/imrn/rnv239},
      zblnumber = {1404.11093},
      }
  • [darmon2015algorithms] Go to document H. Darmon, M. Daub, S. Lichtenstein, and V. Rotger, "Algorithms for Chow-Heegner points via iterated integrals," Math. Comp., vol. 84, iss. 295, pp. 2505-2547, 2015.
    @ARTICLE{darmon2015algorithms,
      author = {Darmon, Henri and Daub, Michael and Lichtenstein, Sam and Rotger, Victor},
      title = {Algorithms for {C}how-{H}eegner points via iterated integrals},
      journal = {Math. Comp.},
      fjournal = {Mathematics of Computation},
      volume = {84},
      year = {2015},
      number = {295},
      pages = {2505--2547},
      issn = {0025-5718},
      mrclass = {11F67 (11G05 11Y16 14C15)},
      mrnumber = {3356037},
      mrreviewer = {Andrea Mori},
      doi = {10.1090/S0025-5718-2015-02927-5},
      url = {https://doi.org/10.1090/S0025-5718-2015-02927-5},
      zblnumber = {1378.11059},
      }
  • [Del89] Go to document P. Deligne, "Le groupe fondamental de la droite projective moins trois points," in Galois Groups over ${\bf Q}$, Springer, New York, 1989, vol. 16, pp. 79-297.
    @INCOLLECTION{Del89,
      author = {Deligne, P.},
      title = {Le groupe fondamental de la droite projective moins trois points},
      booktitle = {Galois Groups over {${\bf Q}$}},
      venue = {{B}erkeley, {CA},
      1987},
      series = {Math. Sci. Res. Inst. Publ.},
      volume = {16},
      pages = {79--297},
      publisher = {Springer, New York},
      year = {1989},
      mrclass = {14G25 (11G35 11M06 11R70 14F35 19E99 19F27)},
      mrnumber = {1012168},
      mrreviewer = {James Milne},
      doi = {10.1007/978-1-4613-9649-9_3},
      url = {https://doi.org/10.1007/978-1-4613-9649-9_3},
      zblnumber = {0742.14022},
      }
  • [Del90] Go to document P. Deligne, "Catégories tannakiennes," in The Grothendieck Festschrift, Vol. II, Birkhäuser Boston, Boston, MA, 1990, vol. 87, pp. 111-195.
    @INCOLLECTION{Del90,
      author = {Deligne, P.},
      title = {Catégories tannakiennes},
      booktitle = {The {G}rothendieck {F}estschrift, {V}ol. {II}},
      series = {Progr. Math.},
      volume = {87},
      pages = {111--195},
      publisher = {Birkhäuser Boston, Boston, MA},
      year = {1990},
      mrclass = {14A99 (12H05 18A99)},
      mrnumber = {1106898},
      mrreviewer = {James Milne},
      zblnumber = {},
      doi = {10.1007/978-0-8176-4575-5_3},
      }
  • [DRS12] H. Darmon, V. Rotger, and I. Sols, "Iterated integrals, diagonal cycles and rational points on elliptic curves," in Publications Mathématiques de Besançon. Algèbre et Théorie des Nombres, 2012/2, Presses Univ. Franche-Comté, Besançon, 2012, vol. 2012/, pp. 19-46.
    @INCOLLECTION{DRS12,
      author = {Darmon, Henri and Rotger, Victor and Sols, Ignacio},
      title = {Iterated integrals, diagonal cycles and rational points on elliptic curves},
      booktitle = {Publications Mathématiques de {B}esançon. {A}lgèbre et Théorie des Nombres, 2012/2},
      series = {Publ. Math. Besançon Algèbre Théorie Nr.},
      volume = {2012/},
      pages = {19--46},
      publisher = {Presses Univ. Franche-Comté,
      Besançon},
      year = {2012},
      mrclass = {11F67 (11G05 11G40 14C15 14C25 14G05)},
      mrnumber = {3074917},
      mrreviewer = {Rolf Berndt},
      zblnumber = {1332.11054},
      }
  • [Edi89] B. Edixhoven, Stable models of modular curves and applications, 1989.
    @MISC{Edi89,
      author = {Edixhoven, Bas},
      title = {Stable models of modular curves and applications},
      note = {Ph.D. thesis, Utrecht},
      year={1989},
      }
  • [Edi90] Go to document B. Edixhoven, "Minimal resolution and stable reduction of $X_0(N)$," Ann. Inst. Fourier (Grenoble), vol. 40, iss. 1, pp. 31-67, 1990.
    @ARTICLE{Edi90,
      author = {Edixhoven, Bas},
      title = {Minimal resolution and stable reduction of {$X_0(N)$}},
      journal = {Ann. Inst. Fourier (Grenoble)},
      fjournal = {Université de Grenoble. Annales de l'Institut Fourier},
      volume = {40},
      year = {1990},
      number = {1},
      pages = {31--67},
      issn = {0373-0956},
      mrclass = {11G18 (14G35)},
      mrnumber = {1056773},
      mrreviewer = {Autorreferat},
      url = {http://www.numdam.org/item?id=AIF_1990__40_1_31_0},
      zblnumber = {0679.14009},
      }
  • [EH] J. S. Ellenberg and D. R. Hast, Rational points on solvable curves over $\mathbb{Q}$ via non-abelian Chabauty, 2017.
    @MISC{EH,
      author = {Ellenberg, Jordan S. and Hast, Daniel Rayor},
      title = {Rational points on solvable curves over {$\mathbb{Q}$} via non-abelian {C}habauty},
      note = {preprint},
      year = {2017},
      arxiv = {1706.00525},
      }
  • [faltings89] G. Faltings, "Crystalline cohomology and $p$-adic Galois-representations," in Algebraic Analysis, Geometry, and Number Theory, Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 25-80.
    @INCOLLECTION{faltings89,
      author = {Faltings, Gerd},
      title = {Crystalline cohomology and {$p$}-adic {G}alois-representations},
      booktitle = {Algebraic Analysis, Geometry, and Number Theory},
      venue = {Baltimore, {MD},
      1988},
      pages = {25--80},
      publisher = {Johns Hopkins Univ. Press, Baltimore, MD},
      year = {1989},
      mrclass = {14F30 (14F40)},
      mrnumber = {1463696},
      mrreviewer = {Abdellah Mokrane},
      zblnumber = {0805.14008},
      }
  • [FPR94] Go to document J. Fontaine and B. Perrin-Riou, "Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions $L$," in Motives, Amer. Math. Soc., Providence, RI, 1994, vol. 55, pp. 599-706.
    @INCOLLECTION{FPR94,
      author = {Fontaine, Jean-Marc and Perrin-Riou, Bernadette},
      title = {Autour des conjectures de {B}loch et {K}ato: cohomologie galoisienne et valeurs de fonctions {$L$}},
      booktitle = {Motives},
      venue = {{S}eattle, {WA},
      1991},
      series = {Proc. Sympos. Pure Math.},
      volume = {55},
      pages = {599--706},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1994},
      mrclass = {11F85 (11F33 11F67 11G09 11G40 14G10 19F27)},
      mrnumber = {1265546},
      mrreviewer = {Alexey A. Panchishkin},
      zblnumber = {0821.14013},
      doi = {10.1090/pspum/055.1/1265546},
     }
  • [FW99] Go to document V. E. Flynn and J. L. Wetherell, "Finding rational points on bielliptic genus 2 curves," Manuscripta Math., vol. 100, iss. 4, pp. 519-533, 1999.
    @ARTICLE{FW99,
      author = {Flynn, E. Victor and Wetherell, Joseph L.},
      title = {Finding rational points on bielliptic genus 2 curves},
      journal = {Manuscripta Math.},
      fjournal = {Manuscripta Mathematica},
      volume = {100},
      year = {1999},
      number = {4},
      pages = {519--533},
      issn = {0025-2611},
      mrclass = {11G30 (14G05)},
      mrnumber = {1734798},
      mrreviewer = {Samir Siksek},
      doi = {10.1007/s002290050215},
      url = {https://doi.org/10.1007/s002290050215},
      zblnumber = {1029.11024},
      }
  • [Gal02] Go to document S. D. Galbraith, "Rational points on $X^+_0(N)$ and quadratic $\Bbb Q$-curves," J. Théor. Nombres Bordeaux, vol. 14, iss. 1, pp. 205-219, 2002.
    @ARTICLE{Gal02,
      author = {Galbraith, Steven D.},
      title = {Rational points on {$X^+_0(N)$} and quadratic {$\Bbb Q$}-curves},
      journal = {J. Théor. Nombres Bordeaux},
      fjournal = {Journal de Théorie des Nombres de Bordeaux},
      volume = {14},
      year = {2002},
      number = {1},
      pages = {205--219},
      issn = {1246-7405},
      mrclass = {11G18 (14G05)},
      mrnumber = {1925998},
      mrreviewer = {Thomas A. Weston},
      url = {http://jtnb.cedram.org/item?id=JTNB_2002__14_1_205_0},
      zblnumber = {1035.14008},
      }
  • [SGAVII_1] A. Grothendieck, Groupes de Monodromie en Géométrie Algébrique, , 1972, vol. 288.
    @BOOK{SGAVII_1,
      author = {Grothendieck, A.},
      title = {Groupes de Monodromie en Géométrie Algébrique},
      volume = {288},
      year = {1972},
      zblnumber = {},
      }
  • [GZ86] Go to document B. H. Gross and D. B. Zagier, "Heegner points and derivatives of $L$-series," Invent. Math., vol. 84, iss. 2, pp. 225-320, 1986.
    @ARTICLE{GZ86,
      author = {Gross, Benedict H. and Zagier, Don B.},
      title = {Heegner points and derivatives of {$L$}-series},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {84},
      year = {1986},
      number = {2},
      pages = {225--320},
      issn = {0020-9910},
      mrclass = {11G40 (11F11 11G05 14G10)},
      mrnumber = {0833192},
      mrreviewer = {Loren D. Olson},
      doi = {10.1007/BF01388809},
      url = {https://doi.org/10.1007/BF01388809},
      zblnumber = {0608.14019},
      }
  • [H11] Go to document M. Hadian, "Motivic fundamental groups and integral points," Duke Math. J., vol. 160, iss. 3, pp. 503-565, 2011.
    @ARTICLE{H11,
      author = {Hadian, Majid},
      title = {Motivic fundamental groups and integral points},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {160},
      year = {2011},
      number = {3},
      pages = {503--565},
      issn = {0012-7094},
      mrclass = {19E20 (11G35 14F42 14G05)},
      mrnumber = {2852368},
      mrreviewer = {Matthias Wendt},
      doi = {10.1215/00127094-1444296},
      url = {https://doi.org/10.1215/00127094-1444296},
      zblnumber = {1234.14020},
      }
  • [Kim05] Go to document M. Kim, "The motivic fundamental group of $\Bbb P^1-\{0,1,\infty\}$ and the theorem of Siegel," Invent. Math., vol. 161, iss. 3, pp. 629-656, 2005.
    @ARTICLE{Kim05,
      author = {Kim, Minhyong},
      title = {The motivic fundamental group of {$\bold P^1-\{0,1,\infty\}$} and the theorem of {S}iegel},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {161},
      year = {2005},
      number = {3},
      pages = {629--656},
      issn = {0020-9910},
      mrclass = {11G30 (11G55 14F30 14F42)},
      mrnumber = {2181717},
      mrreviewer = {Tam\'{a}s Szamuely},
      doi = {10.1007/s00222-004-0433-9},
      url = {https://doi.org/10.1007/s00222-004-0433-9},
      zblnumber = {1090.14006},
      }
  • [Kim09] Go to document M. Kim, "The unipotent Albanese map and Selmer varieties for curves," Publ. Res. Inst. Math. Sci., vol. 45, iss. 1, pp. 89-133, 2009.
    @ARTICLE{Kim09,
      author = {Kim, Minhyong},
      title = {The unipotent {A}lbanese map and {S}elmer varieties for curves},
      journal = {Publ. Res. Inst. Math. Sci.},
      fjournal = {Kyoto University. Research Institute for Mathematical Sciences. Publications},
      volume = {45},
      year = {2009},
      number = {1},
      pages = {89--133},
      issn = {0034-5318},
      mrclass = {14G05 (11G30 11G40 14F35)},
      mrnumber = {2512779},
      mrreviewer = {Ramdorai Sujatha},
      doi = {10.2977/prims/1234361156},
      url = {https://doi.org/10.2977/prims/1234361156},
      zblnumber = {1165.14020},
      }
  • [KL89] Go to document V. A. Kolyvagin and Y. D. Logachëv, "Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties," Algebra i Analiz, vol. 1, iss. 5, pp. 171-196, 1989.
    @ARTICLE{KL89,
      author = {Kolyvagin, V. A. and Logachëv, D. Yu.},
      title = {Finiteness of the {S}hafarevich-{T}ate group and the group of rational points for some modular abelian varieties},
      journal = {Algebra i Analiz},
      fjournal = {Algebra i Analiz},
      volume = {1},
      year = {1989},
      number = {5},
      pages = {171--196},
      issn = {0234-0852},
      mrclass = {11G10 (11G40 14K15)},
      mrnumber = {1036843},
      mrreviewer = {Takeshi Ooe},
      zblnumber = {0728.14026},
      url = {http://mi.mathnet.ru/eng/aa47},
      }
  • [KM85] Go to document N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Princeton Univ. Press, Princeton, NJ, 1985, vol. 108.
    @BOOK{KM85,
      author = {Katz, Nicholas M. and Mazur, Barry},
      title = {Arithmetic Moduli of Elliptic Curves},
      series = {Ann. of Math. Stud.},
      volume = {108},
      publisher = {Princeton Univ. Press, Princeton, NJ},
      year = {1985},
      pages = {xiv+514},
      isbn = {0-691-08349-5; 0-691-08352-5},
      mrclass = {11G05 (11F11 14G25 14K15)},
      mrnumber = {0772569},
      mrreviewer = {Kenneth A. Ribet},
      doi = {10.1515/9781400881710},
      url = {https://doi.org/10.1515/9781400881710},
      zblnumber = {0576.14026},
      }
  • [KT08] Go to document M. Kim and A. Tamagawa, "The $l$-component of the unipotent Albanese map," Math. Ann., vol. 340, iss. 1, pp. 223-235, 2008.
    @ARTICLE{KT08,
      author = {Kim, Minhyong and Tamagawa, Akio},
      title = {The {$l$}-component of the unipotent {A}lbanese map},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {340},
      year = {2008},
      number = {1},
      pages = {223--235},
      issn = {0025-5831},
      mrclass = {11G30 (11G25 11G55 14F30 14G20)},
      mrnumber = {2349775},
      mrreviewer = {Dipendra Prasad},
      doi = {10.1007/s00208-007-0151-x},
      url = {https://doi.org/10.1007/s00208-007-0151-x},
      zblnumber = {1126.14035},
      }
  • [LS07] Go to document B. Le Stum, Rigid Cohomology, Cambridge Univ. Press, Cambridge, 2007, vol. 172.
    @BOOK{LS07,
      author = {Le Stum, Bernard},
      title = {Rigid Cohomology},
      series = {Cambridge Tracts in Math.},
      volume = {172},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {2007},
      pages = {xvi+319},
      isbn = {978-0-521-87524-0},
      mrclass = {14F30 (14G22)},
      mrnumber = {2358812},
      mrreviewer = {Lorenzo Ramero},
      doi = {10.1017/CBO9780511543128},
      url = {https://doi.org/10.1017/CBO9780511543128},
      zblnumber = {1131.14001},
      }
  • [LV18] B. Lawrence and A. Venkatesh, Diophantine problems and $p$-adic period mappings, 2018.
    @MISC{LV18,
      author = {Lawrence, B. and Venkatesh, A.},
      title = {Diophantine problems and $p$-adic period mappings},
      arxiv = {1807.02721},
      year = {2018},
      zblnumber = {},
      }
  • [Maz77] 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{Maz77,
      author = {Mazur, B.},
      title = {Modular curves and the {E}isenstein ideal},
      journal = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
      fjournal = {Institut des Hautes \'{E}tudes Scientifiques. Publications Mathématiques},
      number = {47},
      year = {1977},
      pages = {33--186 (1978)},
      issn = {0073-8301},
      mrclass = {14G25 (10D05)},
      mrnumber = {0488287},
      mrreviewer = {M. Ohta},
      url = {http://www.numdam.org/item?id=PMIHES_1977__47__33_0},
      zblnumber = {0394.14008},
      }
  • [Maz78] 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{Maz78,
      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},
      mrclass = {14K07 (10D35 14G25)},
      mrnumber = {0482230},
      mrreviewer = {V. V. Shokurov},
      doi = {10.1007/BF01390348},
      url = {https://doi.org/10.1007/BF01390348},
      zblnumber = {0386.14009},
      }
  • [Mer96] Go to document L. Merel, "Bornes pour la torsion des courbes elliptiques sur les corps de nombres," Invent. Math., vol. 124, iss. 1-3, pp. 437-449, 1996.
    @ARTICLE{Mer96,
      author = {Merel, Lo{\"{i}}c},
      title = {Bornes pour la torsion des courbes elliptiques sur les corps de nombres},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {124},
      year = {1996},
      number = {1-3},
      pages = {437--449},
      issn = {0020-9910},
      mrclass = {11G05 (11L05)},
      mrnumber = {1369424},
      mrreviewer = {Henri Darmon},
      doi = {10.1007/s002220050059},
      url = {https://doi.org/10.1007/s002220050059},
      zblnumber = {0936.11037},
      }
  • [Mil80] Go to document J. S. Milne, Étale Cohomology, Princeton Univ. Press, Princeton, N.J., 1980, vol. 33.
    @BOOK{Mil80,
      author = {Milne, James S.},
      title = {\'{E}tale Cohomology},
      series = {Princeton Math. Ser.},
      volume = {33},
      publisher = {Princeton Univ. Press, Princeton, N.J.},
      year = {1980},
      pages = {xiii+323},
      isbn = {0-691-08238-3},
      mrclass = {14-02 (14F20 18F99)},
      mrnumber = {0559531},
      mrreviewer = {G. Horrocks},
      zblnumber = {0433.14012},
      doi = {10.1515/9781400883981},
      }
  • [Mum70] D. Mumford, Abelian Varieties, published for the Tata Institute of Fundamental Research, Bombay; Oxford Univ. Press, London, 1970, vol. 5.
    @BOOK{Mum70,
      author = {Mumford, David},
      title = {Abelian Varieties},
      series = {Tata Inst. Fund. Res. Stud. Math.},
      volume = {5 },
      publisher = {published for the Tata Institute of Fundamental Research, Bombay; Oxford Univ. Press, London},
      year = {1970},
      pages = {viii+242},
      mrclass = {14.51},
      mrnumber = {0282985},
      mrreviewer = {J. S. Milne},
      zblnumber = {0223.14022},
      }
  • [Nek93] Go to document J. Nekovávr, "On $p$-adic height pairings," in Séminaire de Théorie des Nombres, Paris, 1990–91, Birkhäuser Boston, Boston, MA, 1993, vol. 108, pp. 127-202.
    @INCOLLECTION{Nek93,
      author = {Nekov\'{a}\v{r},
      Jan},
      title = {On {$p$}-adic height pairings},
      booktitle = {Séminaire de {T}héorie des {N}ombres, {P}aris, 1990--91},
      series = {Progr. Math.},
      volume = {108},
      pages = {127--202},
      publisher = {Birkhäuser Boston, Boston, MA},
      year = {1993},
      mrclass = {11G09 (11F85 14F30)},
      mrnumber = {1263527},
      mrreviewer = {Abdellah Mokrane},
      doi = {10.1007/s10107-005-0696-y},
      url = {https://doi.org/10.1007/s10107-005-0696-y},
      zblnumber = {0859.11038},
      }
  • [Ols11] Go to document M. C. Olsson, "Towards non-abelian $p$-adic Hodge theory in the good reduction case," Mem. Amer. Math. Soc., vol. 210, iss. 990, p. vi, 2011.
    @ARTICLE{Ols11,
      author = {Olsson, Martin C.},
      title = {Towards non-abelian {$p$}-adic {H}odge theory in the good reduction case},
      journal = {Mem. Amer. Math. Soc.},
      fjournal = {Memoirs of the American Mathematical Society},
      volume = {210},
      year = {2011},
      number = {990},
      pages = {vi+157},
      issn = {0065-9266},
      isbn = {978-0-8218-5240-8},
      mrclass = {14G17 (11G25 14C30 14D23)},
      mrnumber = {2791384},
      mrreviewer = {Fabrizio Andreatta},
      doi = {10.1090/S0065-9266-2010-00625-2},
      url = {https://doi.org/10.1090/S0065-9266-2010-00625-2},
      zblnumber = {1213.14002},
      }
  • [Ray90] Go to document M. Raynaud, "$p$-groupes et réduction semi-stable des courbes," in The Grothendieck Festschrift, Vol. III, Birkhäuser Boston, Boston, MA, 1990, vol. 88, pp. 179-197.
    @INCOLLECTION{Ray90,
      author = {Raynaud, Michel},
      title = {{$p$}-groupes et réduction semi-stable des courbes},
      booktitle = {The {G}rothendieck {F}estschrift, {V}ol. {III}},
      series = {Progr. Math.},
      volume = {88},
      pages = {179--197},
      publisher = {Birkhäuser Boston, Boston, MA},
      year = {1990},
      mrclass = {14E20 (14F30 14G20 14H30)},
      mrnumber = {1106915},
      mrreviewer = {Takeshi Ooe},
      doi = {10.1007/978-0-8176-4576-2_7},
      url = {https://doi.org/10.1007/978-0-8176-4576-2_7},
      zblnumber = {0722.14013},
     }
  • [Rib80] Go to document K. A. Ribet, "Twists of modular forms and endomorphisms of abelian varieties," Math. Ann., vol. 253, iss. 1, pp. 43-62, 1980.
    @ARTICLE{Rib80,
      author = {Ribet, Kenneth A.},
      title = {Twists of modular forms and endomorphisms of abelian varieties},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {253},
      year = {1980},
      number = {1},
      pages = {43--62},
      issn = {0025-5831},
      mrclass = {10D15 (14G13)},
      mrnumber = {0594532},
      mrreviewer = {Wen Ch'ing Winnie Li},
      doi = {10.1007/BF01457819},
      url = {https://doi.org/10.1007/BF01457819},
      zblnumber = {0421.14008},
      }
  • [Sch12] R. Schoof, The Mordell-Weil group of a modular curve of level 13, 2012.
    @MISC{Sch12,
      author = {Schoof, R.},
      title = {The {M}ordell-{W}eil group of a modular curve of level 13},
      note = {unpublished manuscript},
      year = {2012},
      zblnumber = {},
      }
  • [Ser72] Go to document . J-P. Serre, "Propriétés galoisiennes des points d’ordre fini des courbes elliptiques," Invent. Math., vol. 15, iss. 4, pp. 259-331, 1972.
    @ARTICLE{Ser72,
      author = {Serre, {\relax J-P}},
      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 = {0387283},
      mrreviewer = {J. W. S. Cassels},
      doi = {10.1007/BF01405086},
      url = {https://doi.org/10.1007/BF01405086},
      zblnumber = {0235.14012},
     }
  • [Ser97] Go to document . J-P. Serre, Lectures on the Mordell-Weil Theorem, Third ed., Friedr. Vieweg & Sohn, Braunschweig, 1997.
    @BOOK{Ser97,
      author = {Serre, {\relax J-P}},
      title = {Lectures on the {M}ordell-{W}eil Theorem},
      series = {Aspects Math.},
      edition = {Third},
      note = {translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, with a foreword by Brown and Serre},
      publisher = {Friedr. Vieweg \& Sohn, Braunschweig},
      year = {1997},
      pages = {x+218},
      isbn = {3-528-28968-6},
      mrclass = {11G10 (11D41 11G30 14G25)},
      mrnumber = {1757192},
      doi = {10.1007/978-3-663-10632-6},
      url = {https://doi.org/10.1007/978-3-663-10632-6},
      zblnumber = {0863.14013},
     }
  • [Ser02] . J-P. Serre, Galois Cohomology, English ed., Springer-Verlag, Berlin, 2002.
    @BOOK{Ser02,
      author = {Serre, {\relax J-P}},
      title = {Galois Cohomology},
      series = {Springer Monogr. Math.},
      edition = {English},
      note = {translated from the French by Patrick Ion and revised by the author},
      publisher = {Springer-Verlag, Berlin},
      year = {2002},
      pages = {x+210},
      isbn = {3-540-42192-0},
      mrclass = {12G05 (11R34)},
      mrnumber = {1867431},
      zblnumber = {1004.12003},
      }
  • [Shi70] Go to document G. Shimura, "On canonical models of arithmetic quotients of bounded symmetric domains," Ann. of Math. (2), vol. 91, pp. 144-222, 1970.
    @ARTICLE{Shi70,
      author = {Shimura, Goro},
      title = {On canonical models of arithmetic quotients of bounded symmetric domains},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {91},
      year = {1970},
      pages = {144--222},
      issn = {0003-486X},
      mrclass = {10.68},
      mrnumber = {0257031},
      mrreviewer = {K. Iyanaga},
      doi = {10.2307/1970604},
      url = {https://doi.org/10.2307/1970604},
      zblnumber = {0237.14009},
      }
  • [Sik] S. Siksek, Quadratic Chabauty for modular curves, 2017.
    @MISC{Sik,
      author = {Siksek, S.},
      title = {Quadratic {C}habauty for modular curves},
      note = {preprint},
      year = {2017},
      zblnumber = {},
      arxiv = {1704.00473},
      }
  • [Tui16] Go to document J. Tuitman, "Counting points on curves using a map to $\bold{P}^1$," Math. Comp., vol. 85, iss. 298, pp. 961-981, 2016.
    @ARTICLE{Tui16,
      author = {Tuitman, Jan},
      title = {Counting points on curves using a map to {$\bold{P}^1$}},
      journal = {Math. Comp.},
      fjournal = {Mathematics of Computation},
      volume = {85},
      year = {2016},
      number = {298},
      pages = {961--981},
      issn = {0025-5718},
      mrclass = {11M38 (11Y16)},
      mrnumber = {3434890},
      mrreviewer = {Giuseppe Molteni},
      doi = {10.1090/mcom/2996},
      url = {https://doi.org/10.1090/mcom/2996},
      zblnumber = {1402.11096},
      }
  • [Tui17] Go to document J. Tuitman, "Counting points on curves using a map to $\bold{P}^1$, II," Finite Fields Appl., vol. 45, pp. 301-322, 2017.
    @ARTICLE{Tui17,
      author = {Tuitman, Jan},
      title = {Counting points on curves using a map to {$\bold{P}^1$},
      {II}},
      journal = {Finite Fields Appl.},
      fjournal = {Finite Fields and their Applications},
      volume = {45},
      year = {2017},
      pages = {301--322},
      issn = {1071-5797},
      mrclass = {11G20 (11M38 11Y16 14G10)},
      mrnumber = {3631366},
      mrreviewer = {Steven D. Galbraith},
      doi = {10.1016/j.ffa.2016.12.008},
      url = {https://doi.org/10.1016/j.ffa.2016.12.008},
      zblnumber = {1402.11097},
      }

Authors

Jennifer S. Balakrishnan

Department of Mathematics and Statistics, Boston University, Boston, MA

Netan Dogra

Jesus College, University of Oxford, Oxford, UK

J. Steffen Müller

Bernoulli Institute, University of Groningen, Groningen, The Netherlands

Jan Tuitman

Departement Wiskunde, KU Leuven, Leuven, Belgium

Jan Vonk

Mathematical Institute, University of Oxford, Oxford, UK