Uniform Manin-Mumford for a family of genus 2 curves

Abstract

We introduce a general strategy for proving quantitative and uniform bounds on the number of common points of height zero for a pair of inequivalent height functions on $\mathbb {P}^1(\overline {\mathbb {Q}}).$ We apply this strategy to prove a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds on the number of common torsion points of elliptic curves in the case of two Legendre curves over $\mathbb {C}$. As a consequence, we obtain two uniform bounds for a two-dimensional family of genus 2 curves: a uniform Manin-Mumford bound for the family over $\mathbb {C}$, and a uniform Bogomolov bound for the family over $\overline {\mathbb {Q}}.$

  • [BRbook] Go to document M. Baker and R. Rumely, Potential Theory and Dynamics on the Berkovich Projective Line, Amer. Math. Soc., Providence, RI, 2010, vol. 159.
    @BOOK{BRbook,
      author = {Baker, Matthew and Rumely, Robert},
      title = {Potential Theory and Dynamics on the {B}erkovich Projective Line},
      series = {Math. Surveys Monogr.},
      volume = {159},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2010},
      pages = {xxxiv+428},
      isbn = {978-0-8218-4924-8},
      mrclass = {37P50 (14G20 14G22 31C15 31C45 37P40)},
      mrnumber = {2599526},
      mrreviewer = {Charles Favre},
      doi = {10.1090/surv/159},
      url = {https://doi.org/10.1090/surv/159},
      zblnumber = {1196.14002},
      }
  • [Baker:Petsche] Go to document M. Baker and C. Petsche, "Global discrepancy and small points on elliptic curves," Int. Math. Res. Not., vol. 61, pp. 3791-3834, 2005.
    @ARTICLE{Baker:Petsche,
      author = {Baker, Matthew and Petsche, Clayton},
      title = {Global discrepancy and small points on elliptic curves},
      journal = {Int. Math. Res. Not.},
      fjournal = {International Mathematics Research Notices},
      year = {2005},
      volume = {61},
      pages = {3791--3834},
      issn = {1073-7928},
      mrclass = {11G50 (11G05 11K38 14G25)},
      mrnumber = {2205235},
      doi = {10.1155/IMRN.2005.3791},
      url = {https://doi.org/10.1155/IMRN.2005.3791},
      zblnumber = {1120.11025},
      }
  • [Baker:Poonen:torsion] Go to document M. Baker and B. Poonen, "Torsion packets on curves," Compositio Math., vol. 127, iss. 1, pp. 109-116, 2001.
    @ARTICLE{Baker:Poonen:torsion,
      author = {Baker, Matthew and Poonen, Bjorn},
      title = {Torsion packets on curves},
      journal = {Compositio Math.},
      fjournal = {Compositio Mathematica},
      volume = {127},
      year = {2001},
      number = {1},
      pages = {109--116},
      issn = {0010-437X},
      mrclass = {14H40},
      mrnumber = {1832989},
      mrreviewer = {Dino J. Lorenzini},
      doi = {10.1023/A:1017557230785},
      url = {https://doi.org/10.1023/A:1017557230785},
      zblnumber = {0987.14020},
      }
  • [Berkovichbook] Go to document V. G. Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Amer. Math. Soc., Providence, RI, 1990, vol. 33.
    @BOOK{Berkovichbook,
      author = {Berkovich, Vladimir G.},
      title = {Spectral Theory and Analytic Geometry over Non-{A}rchimedean Fields},
      series = {Math. Surveys Monogr.},
      volume = {33},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1990},
      pages = {x+169},
      isbn = {0-8218-1534-2},
      mrclass = {32P05 (32C15 32C37 46S10 47S10)},
      mrnumber = {1070709},
      mrreviewer = {W. Bartenwerfer},
      zblnumber = {0715.14013},
      doi = {10.1090/surv/033},
      url = {https://doi.org/10.1090/surv/033},
      }
  • [BFT:torsion] Go to document F. Bogomolov, H. Fu, and Y. Tschinkel, "Torsion of elliptic curves and unlikely intersections," in Geometry and Physics. Vol. I, Oxford Univ. Press, Oxford, 2018, pp. 19-37.
    @INCOLLECTION{BFT:torsion,
      author = {Bogomolov, Fedor and Fu, Hang and Tschinkel, Yuri},
      title = {Torsion of elliptic curves and unlikely intersections},
      booktitle = {Geometry and Physics. {V}ol. {I}},
      pages = {19--37},
      publisher = {Oxford Univ. Press, Oxford},
      year = {2018},
      mrclass = {14H10 (11G05 14G35 14H52)},
      mrnumber = {3932255},
      doi = {10.1093/oso/9780198802013.003.0002},
      url = {https://doi.org/10.1093/oso/9780198802013.003.0002},
      zblnumber = {1423.14214},
      }
  • [Bogomolov:Tschinkel:small] F. Bogomolov and Y. Tschinkel, "Algebraic varieties over small fields," in Diophantine Geometry, Ed. Norm., Pisa, 2007, vol. 4, pp. 73-91.
    @INCOLLECTION{Bogomolov:Tschinkel:small,
      author = {Bogomolov, Fedor and Tschinkel, Yuri},
      title = {Algebraic varieties over small fields},
      booktitle = {Diophantine Geometry},
      series = {CRM Series},
      volume = {4},
      pages = {73--91},
      publisher = {Ed. Norm., Pisa},
      year = {2007},
      mrclass = {14G25 (11G25)},
      mrnumber = {2349648},
      mrreviewer = {Marco A. Garuti},
      zblnumber = {1142.14016},
      }
  • [Boucksom:Jonsson:hybrid] Go to document S. Boucksom and M. Jonsson, "Tropical and non-Archimedean limits of degenerating families of volume forms," J. Éc. Polytech. Math., vol. 4, pp. 87-139, 2017.
    @ARTICLE{Boucksom:Jonsson:hybrid,
      author = {Boucksom, Sébastien and Jonsson, Mattias},
      title = {Tropical and non-{A}rchimedean limits of degenerating families of volume forms},
      journal = {J. \'{E}c. Polytech. Math.},
      fjournal = {Journal de l'\'{E}cole polytechnique. Mathématiques},
      volume = {4},
      year = {2017},
      pages = {87--139},
      issn = {2429-7100},
      mrclass = {32Q25 (14G22 14J32 14T05 32P05 53C23)},
      mrnumber = {3611100},
      mrreviewer = {Dan Abramovich},
      doi = {10.5802/jep.39},
      url = {https://doi.org/10.5802/jep.39},
      zblnumber = {1401.32019},
      }
  • [Buium] Go to document A. Buium, "Geometry of $p$-jets," Duke Math. J., vol. 82, iss. 2, pp. 349-367, 1996.
    @ARTICLE{Buium,
      author = {Buium, Alexandru},
      title = {Geometry of {$p$}-jets},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {82},
      year = {1996},
      number = {2},
      pages = {349--367},
      issn = {0012-7094},
      mrclass = {14H25 (14G25 14H40)},
      mrnumber = {1387233},
      mrreviewer = {Fumio Hazama},
      doi = {10.1215/S0012-7094-96-08216-2},
      url = {https://doi.org/10.1215/S0012-7094-96-08216-2},
      zblnumber = {0882.14007},
      }
  • [Call:Silverman] Go to document G. S. Call and J. H. Silverman, "Canonical heights on varieties with morphisms," Compositio Math., vol. 89, iss. 2, pp. 163-205, 1993.
    @ARTICLE{Call:Silverman,
      author = {Call, Gregory S. and Silverman, Joseph H.},
      title = {Canonical heights on varieties with morphisms},
      journal = {Compositio Math.},
      fjournal = {Compositio Mathematica},
      volume = {89},
      year = {1993},
      number = {2},
      pages = {163--205},
      issn = {0010-437X},
      mrclass = {11G35 (14G25)},
      mrnumber = {1255693},
      mrreviewer = {Takeshi Ooe},
      url = {http://www.numdam.org/item?id=CM_1993__89_2_163_0},
      zblnumber = {0826.14015},
      }
  • [ChambertLoir:equidistribution] Go to document A. Chambert-Loir, "Mesures et équidistribution sur les espaces de Berkovich," J. Reine Angew. Math., vol. 595, pp. 215-235, 2006.
    @ARTICLE{ChambertLoir:equidistribution,
      author = {Chambert-Loir, Antoine},
      title = {Mesures et équidistribution sur les espaces de {B}erkovich},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {595},
      year = {2006},
      pages = {215--235},
      issn = {0075-4102},
      mrclass = {14G40 (14G25)},
      mrnumber = {2244803},
      doi = {10.1515/CRELLE.2006.049},
      url = {https://doi.org/10.1515/CRELLE.2006.049},
      zblnumber = {1112.14022},
      }
  • [ChambertLoir:survey] Go to document A. Chambert-Loir, "Heights and measures on analytic spaces. A survey of recent results, and some remarks," in Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry. Vol. II, Cambridge Univ. Press, Cambridge, 2011, vol. 384, pp. 1-50.
    @INCOLLECTION{ChambertLoir:survey,
      author = {Chambert-Loir, Antoine},
      title = {Heights and measures on analytic spaces. {A} survey of recent results, and some remarks},
      booktitle = {Motivic Integration and its Interactions with Model Theory and Non-{A}rchimedean Geometry. {V}ol. {II}},
      series = {London Math. Soc. Lecture Note Ser.},
      volume = {384},
      pages = {1--50},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {2011},
      mrclass = {14G22},
      mrnumber = {2885340},
      doi = {10.1017/CBO9780511984433.002},
      url = {≈},
      zblnumber = {1279.14027},
      }
  • [Coleman] Go to document R. F. Coleman, "Torsion points on curves and $p$-adic abelian integrals," Ann. of Math. (2), vol. 121, iss. 1, pp. 111-168, 1985.
    @ARTICLE{Coleman,
      author = {Coleman, Robert F.},
      title = {Torsion points on curves and {$p$}-adic abelian integrals},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {121},
      year = {1985},
      number = {1},
      pages = {111--168},
      issn = {0003-486X},
      mrclass = {14G20 (11G10 11G20 14K15 14K22)},
      mrnumber = {0782557},
      mrreviewer = {Reinhard Bölling},
      doi = {10.2307/1971194},
      url = {https://doi.org/10.2307/1971194},
      zblnumber = {0578.14038},
      }
  • [D:measures] Go to document L. DeMarco, "Iteration at the boundary of the space of rational maps," Duke Math. J., vol. 130, iss. 1, pp. 169-197, 2005.
    @ARTICLE{D:measures,
      author = {DeMarco, Laura},
      title = {Iteration at the boundary of the space of rational maps},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {130},
      year = {2005},
      number = {1},
      pages = {169--197},
      issn = {0012-7094},
      mrclass = {37F45 (32H50 37F10 37F35 37F40)},
      mrnumber = {2176550},
      mrreviewer = {Kevin M. Pilgrim},
      doi = {10.1215/S0012-7094-05-13015-0},
      url = {https://doi.org/10.1215/S0012-7094-05-13015-0},
      zblnumber = {1183.37086},
      }
  • [DF:degenerations] Go to document L. De Marco and X. Faber, "Degenerations of complex dynamical systems," Forum Math. Sigma, vol. 2, p. 6, 2014.
    @ARTICLE{DF:degenerations,
      author = {De Marco, Laura and Faber, Xander},
      title = {Degenerations of complex dynamical systems},
      journal = {Forum Math. Sigma},
      fjournal = {Forum of Mathematics. Sigma},
      volume = {2},
      year = {2014},
      pages = {e6, 36},
      mrclass = {37F10 (37F45 37P50)},
      mrnumber = {3264250},
      mrreviewer = {Xavier Jarque},
      doi = {10.1017/fms.2014.8},
      url = {https://doi.org/10.1017/fms.2014.8},
      zblnumber = {1308.37023},
      }
  • [DF:degenerations2] Go to document L. DeMarco and X. Faber, "Degenerations of complex dynamical systems II: analytic and algebraic stability," Math. Ann., vol. 365, iss. 3-4, pp. 1669-1699, 2016.
    @ARTICLE{DF:degenerations2,
      author = {DeMarco, Laura and Faber, Xander},
      title = {Degenerations of complex dynamical systems {II}: analytic and algebraic stability},
      note = {With an appendix by Jan Kiwi},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {365},
      year = {2016},
      number = {3-4},
      pages = {1669--1699},
      issn = {0025-5831},
      mrclass = {37F10 (37F45 37P50)},
      mrnumber = {3521102},
      mrreviewer = {Xavier Jarque},
      doi = {10.1007/s00208-015-1331-8},
      url = {https://doi.org/10.1007/s00208-015-1331-8},
      zblnumber = {1343.37089},
      }
  • [DKY:quadpoly] L. DeMarco, H. Krieger, and H. Ye, Common preperiodic points of quadratic polynomial, 2019.
    @MISC{DKY:quadpoly,
      author = {DeMarco, Laura and Krieger, Holly and Ye, Hexi},
      title = {Common preperiodic points of quadratic polynomial},
      arxiv = {1911.02458v1},
      year = {2019},
      zblnumber = {},
      }
  • [DWY:Lattes] Go to document L. DeMarco, X. Wang, and H. Ye, "Torsion points and the Lattès family," Amer. J. Math., vol. 138, iss. 3, pp. 697-732, 2016.
    @ARTICLE{DWY:Lattes,
      author = {DeMarco, Laura and Wang, Xiaoguang and Ye, Hexi},
      title = {Torsion points and the {L}attès family},
      journal = {Amer. J. Math.},
      fjournal = {American Journal of Mathematics},
      volume = {138},
      year = {2016},
      number = {3},
      pages = {697--732},
      issn = {0002-9327},
      mrclass = {37P55 (11G05 31C15 37P30)},
      mrnumber = {3506383},
      mrreviewer = {Joseph H. Silverman},
      doi = {10.1353/ajm.2016.0026},
      url = {https://doi.org/10.1353/ajm.2016.0026},
      zblnumber = {1364.37101},
      }
  • [Favre:degenerations] C. Favre, Degeneration of endomorphisms of the complex projective space in the hybrid space, 2016.
    @MISC{Favre:degenerations,
      author = {Favre, Charles},
      title = {Degeneration of endomorphisms of the complex projective space in the hybrid space},
      arxiv = {1611.08490v1},
      year = {2016},
      zblnumber = {},
      }
  • [FRL:equidistribution] Go to document C. Favre and J. Rivera-Letelier, "Équidistribution quantitative des points de petite hauteur sur la droite projective," Math. Ann., vol. 335, iss. 2, pp. 311-361, 2006.
    @ARTICLE{FRL:equidistribution,
      author = {Favre, Charles and Rivera-Letelier, Juan},
      title = {\'{E}quidistribution quantitative des points de petite hauteur sur la droite projective},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {335},
      year = {2006},
      number = {2},
      pages = {311--361},
      issn = {0025-5831},
      mrclass = {11G50 (37F10)},
      mrnumber = {2221116},
      mrreviewer = {Matthew H. Baker},
      doi = {10.1007/s00208-006-0751-x},
      url = {https://doi.org/10.1007/s00208-006-0751-x},
      zblnumber = {1175.11029},
      }
  • [FRL:ergodic] Go to document C. Favre and J. Rivera-Letelier, "Théorie ergodique des fractions rationnelles sur un corps ultramétrique," Proc. Lond. Math. Soc. (3), vol. 100, iss. 1, pp. 116-154, 2010.
    @ARTICLE{FRL:ergodic,
      author = {Favre, Charles and Rivera-Letelier, Juan},
      title = {Théorie ergodique des fractions rationnelles sur un corps ultramétrique},
      journal = {Proc. Lond. Math. Soc. (3)},
      fjournal = {Proceedings of the London Mathematical Society. Third Series},
      volume = {100},
      year = {2010},
      number = {1},
      pages = {116--154},
      issn = {0024-6115},
      mrclass = {37P50 (11S82 12J25 37A25 37A35 37B40 37F10)},
      mrnumber = {2578470},
      mrreviewer = {Mattias Jonsson},
      doi = {10.1112/plms/pdp022},
      url = {https://doi.org/10.1112/plms/pdp022},
      zblnumber = {1254.37064},
      }
  • [Fili:energy] P. Fili, A metric of mutual energy and unlikely intersections for dynamical systems, 2017.
    @MISC{Fili:energy,
      author = {Fili, Paul},
      title = {A metric of mutual energy and unlikely intersections for dynamical systems},
      arxiv = {1708.08403v1},
      year = {2017},
      zblnumber = {},
      }
  • [Hindry:Silverman] Go to document M. Hindry and J. H. Silverman, Diophantine Geometry, Springer-Verlag, New York, 2000, vol. 201.
    @BOOK{Hindry:Silverman,
      author = {Hindry, Marc and Silverman, Joseph H.},
      title = {Diophantine Geometry},
      series = {Grad. Texts in Mathematics},
      volume = {201},
      note = {An introduction},
      publisher = {Springer-Verlag, New York},
      year = {2000},
      pages = {xiv+558},
      isbn = {0-387-98975-7; 0-387-98981-1},
      mrclass = {11Gxx (11-02 11G10 11G30 11G50 14G25)},
      mrnumber = {1745599},
      mrreviewer = {Dino J. Lorenzini},
      doi = {10.1007/978-1-4612-1210-2},
      url = {https://doi.org/10.1007/978-1-4612-1210-2},
      zblnumber = {0948.11023},
      }
  • [Hrushovski] Go to document E. Hrushovski, "The Manin–Mumford conjecture and the model theory of difference fields," Ann. Pure Appl. Logic, vol. 112, iss. 1, pp. 43-115, 2001.
    @ARTICLE{Hrushovski,
      author = {Hrushovski, Ehud},
      title = {The {M}anin--{M}umford conjecture and the model theory of difference fields},
      journal = {Ann. Pure Appl. Logic},
      fjournal = {Annals of Pure and Applied Logic},
      volume = {112},
      year = {2001},
      number = {1},
      pages = {43--115},
      issn = {0168-0072},
      mrclass = {03C60 (11G10 11U09 12L12)},
      mrnumber = {1854232},
      mrreviewer = {C. Toffalori},
      doi = {10.1016/S0168-0072(01)00096-3},
      url = {https://doi.org/10.1016/S0168-0072(01)00096-3},
      zblnumber = {0987.03036},
      }
  • [Jacobi:review] Go to document C. G. J. Jacobi, "Nachrichten von Büchern," J. Reine Angew. Math., vol. 8, pp. 413-417, 1832.
    @ARTICLE{Jacobi:review,
      author = {Jacobi, C. G. J.},
      title = {Nachrichten von {B}üchern},
      journal = {J. Reine Angew. Math.},
      volume = {8},
      year = {1832},
      pages = {413--417},
      doi = {10.1515/crll.1832.8.413},
      url = {https://doi.org/10.1515/crll.1832.8.413},
      }
  • [KRZB] Go to document E. Katz, J. Rabinoff, and D. Zureick-Brown, "Uniform bounds for the number of rational points on curves of small Mordell-Weil rank," Duke Math. J., vol. 165, iss. 16, pp. 3189-3240, 2016.
    @ARTICLE{KRZB,
      author = {Katz, Eric and Rabinoff, Joseph and Zureick-Brown, David},
      title = {Uniform bounds for the number of rational points on curves of small {M}ordell-{W}eil rank},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {165},
      year = {2016},
      number = {16},
      pages = {3189--3240},
      issn = {0012-7094},
      mrclass = {14G05 (14H25 14T05)},
      mrnumber = {3566201},
      mrreviewer = {Nurdagül Anbar},
      doi = {10.1215/00127094-3673558},
      url = {https://doi.org/10.1215/00127094-3673558},
      zblnumber = {06666955},
      }
  • [Kawaguchi:Silverman:pairing] Go to document S. Kawaguchi and J. H. Silverman, "Canonical heights and the arithmetic complexity of morphisms on projective space," Pure Appl. Math. Q., vol. 5, iss. 4, Special Issue: In honor of John Tate. Part 1, pp. 1201-1217, 2009.
    @ARTICLE{Kawaguchi:Silverman:pairing,
      author = {Kawaguchi, Shu and Silverman, Joseph H.},
      title = {Canonical heights and the arithmetic complexity of morphisms on projective space},
      journal = {Pure Appl. Math. Q.},
      fjournal = {Pure and Applied Mathematics Quarterly},
      volume = {5},
      year = {2009},
      number = {4, Special Issue: In honor of John Tate. Part 1},
      pages = {1201--1217},
      issn = {1558-8599},
      mrclass = {11G50 (14G40 37P30)},
      mrnumber = {2560315},
      mrreviewer = {Philipp Habegger},
      doi = {10.4310/PAMQ.2009.v5.n4.a2},
      url = {https://doi.org/10.4310/PAMQ.2009.v5.n4.a2},
      zblnumber = {1221.11152},
      }
  • [Mazur:curves] Go to document B. Mazur, "Arithmetic on curves," Bull. Amer. Math. Soc. (N.S.), vol. 14, iss. 2, pp. 207-259, 1986.
    @ARTICLE{Mazur:curves,
      author = {Mazur, Barry},
      title = {Arithmetic on curves},
      journal = {Bull. Amer. Math. Soc. (N.S.)},
      fjournal = {Amer. Math. Soc.. Bulletin. New Series},
      volume = {14},
      year = {1986},
      number = {2},
      pages = {207--259},
      issn = {0273-0979},
      mrclass = {11G30 (11D41 14Gxx 14K15)},
      mrnumber = {0828821},
      mrreviewer = {C.-G. Schmidt},
      doi = {10.1090/S0273-0979-1986-15430-3},
      url = {https://doi.org/10.1090/S0273-0979-1986-15430-3},
      zblnumber = {0593.14021},
      }
  • [McMullen:genus2] Go to document C. T. McMullen, "Dynamics of ${ SL}_2(\Bbb R)$ over moduli space in genus two," Ann. of Math. (2), vol. 165, iss. 2, pp. 397-456, 2007.
    @ARTICLE{McMullen:genus2,
      author = {McMullen, Curtis T.},
      title = {Dynamics of {${\rm SL}_2(\Bbb R)$} over moduli space in genus two},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {165},
      year = {2007},
      number = {2},
      pages = {397--456},
      issn = {0003-486X},
      mrclass = {32G15 (37C85 37F99)},
      mrnumber = {2299738},
      mrreviewer = {Serge L. Tabachnikov},
      doi = {10.4007/annals.2007.165.397},
      url = {https://doi.org/10.4007/annals.2007.165.397},
      zblnumber = {1131.14027},
      }
  • [Milnor:Lattes] Go to document J. Milnor, "On Lattès maps," in Dynamics on the Riemann Sphere, Eur. Math. Soc., Zürich, 2006, pp. 9-43.
    @INCOLLECTION{Milnor:Lattes,
      author = {Milnor, John},
      title = {On {L}attès maps},
      booktitle = {Dynamics on the {R}iemann Sphere},
      pages = {9--43},
      publisher = {Eur. Math. Soc., Zürich},
      year = {2006},
      mrclass = {37F10 (30D05 37-02)},
      mrnumber = {2348953},
      mrreviewer = {Feliks Przytycki},
      doi = {10.4171/011-1/1},
      url = {https://doi.org/10.4171/011-1/1},
      zblnumber = {1235.37015},
      }
  • [PST:pairing] Go to document C. Petsche, L. Szpiro, and T. J. Tucker, "A dynamical pairing between two rational maps," Trans. Amer. Math. Soc., vol. 364, iss. 4, pp. 1687-1710, 2012.
    @ARTICLE{PST:pairing,
      author = {Petsche, Clayton and Szpiro, Lucien and Tucker, Thomas J.},
      title = {A dynamical pairing between two rational maps},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the Amer. Math. Soc.},
      volume = {364},
      year = {2012},
      number = {4},
      pages = {1687--1710},
      issn = {0002-9947},
      mrclass = {37P30 (11G50 14G40 37P05)},
      mrnumber = {2869188},
      mrreviewer = {Kevin Doerksen},
      doi = {10.1090/S0002-9947-2011-05350-X},
      url = {https://doi.org/10.1090/S0002-9947-2011-05350-X},
      zblnumber = {1300.37052},
      }
  • [Poonen:1] Go to document B. Poonen, "Genus-two curves with 22 torsion points," C. R. Acad. Sci. Paris Sér. I Math., vol. 330, iss. 7, pp. 573-576, 2000.
    @ARTICLE{Poonen:1,
      author = {Poonen, Bjorn},
      title = {Genus-two curves with 22 torsion points},
      journal = {C. R. Acad. Sci. Paris Sér. I Math.},
      fjournal = {Comptes Rendus de l'Académie des Sciences. Série I. Mathématique},
      volume = {330},
      year = {2000},
      number = {7},
      pages = {573--576},
      issn = {0764-4442},
      mrclass = {11G30 (14H25)},
      mrnumber = {1760441},
      mrreviewer = {Matthew H. Baker},
      doi = {10.1016/S0764-4442(00)00222-6},
      url = {https://doi.org/10.1016/S0764-4442(00)00222-6},
      zblnumber = {0966.11026},
      }
  • [Raynaud:1] Go to document M. Raynaud, "Courbes sur une variété abélienne et points de torsion," Invent. Math., vol. 71, iss. 1, pp. 207-233, 1983.
    @ARTICLE{Raynaud:1,
      author = {Raynaud, M.},
      title = {Courbes sur une variété abélienne et points de torsion},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {71},
      year = {1983},
      number = {1},
      pages = {207--233},
      issn = {0020-9910},
      mrclass = {14H25 (14G25 14K15)},
      mrnumber = {0688265},
      mrreviewer = {F. Oort},
      doi = {10.1007/BF01393342},
      url = {https://doi.org/10.1007/BF01393342},
      zblnumber = {0564.14020},
      }
  • [Shaska:Volklein] Go to document T. Shaska and H. Völklein, "Elliptic subfields and automorphisms of genus 2 function fields," in Algebra, Arithmetic and Geometry with Applications, Springer, Berlin, 2004, pp. 703-723.
    @INCOLLECTION{Shaska:Volklein,
      author = {Shaska, Tanush and Völklein, Helmut},
      title = {Elliptic subfields and automorphisms of genus 2 function fields},
      booktitle = {Algebra, Arithmetic and Geometry with Applications},
      venue = {{W}est {L}afayette, {IN},
      2000},
      pages = {703--723},
      publisher = {Springer, Berlin},
      year = {2004},
      mrclass = {14H05 (12F99 14H10 14H37 14H52)},
      mrnumber = {2037120},
      mrreviewer = {Vasil\cprime Ī. Andr\=ıĭchuk},
      zblnumber = {1093.14036},
      doi = {10.1007/978-3-642-18487-1_42},
      url = {https://doi.org/10.1007/978-3-642-18487-1_42},
      }
  • [Silverman:elliptic] Go to document J. H. Silverman, The Arithmetic of Elliptic Curves, Second ed., Springer, Dordrecht, 2009, vol. 106.
    @BOOK{Silverman:elliptic,
      author = {Silverman, Joseph H.},
      title = {The Arithmetic of Elliptic Curves},
      series = {Grad. Texts in Math.},
      volume = {106},
      edition = {Second},
      publisher = {Springer, Dordrecht},
      year = {2009},
      pages = {xx+513},
      isbn = {978-0-387-09493-9},
      mrclass = {11-02 (11G05 11G20 14H52 14K15)},
      mrnumber = {2514094},
      mrreviewer = {Vasil\cprime Ī. Andr\=ıĭchuk},
      doi = {10.1007/978-0-387-09494-6},
      url = {https://doi.org/10.1007/978-0-387-09494-6},
      zblnumber = {1194.11005},
      }
  • [Stoll:example] M. Stoll, Another New Record.
    @MISC{Stoll:example,
      author = {Stoll, Michael},
      title = {Another {N}ew {R}ecord},
      note={available on the author's website},
      zblnumber = {},
      }
  • [Szpiro:Ullmo:Zhang] Go to document L. Szpiro, E. Ullmo, and S. Zhang, "Équirépartition des petits points," Invent. Math., vol. 127, iss. 2, pp. 337-347, 1997.
    @ARTICLE{Szpiro:Ullmo:Zhang,
      author = {Szpiro, L. and Ullmo, E. and Zhang, S.},
      title = {\'{E}quirépartition des petits points},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {127},
      year = {1997},
      number = {2},
      pages = {337--347},
      issn = {0020-9910},
      mrclass = {14G40 (11G35 14G05)},
      mrnumber = {1427622},
      mrreviewer = {Antoine Chambert-Loir},
      doi = {10.1007/s002220050123},
      url = {https://doi.org/10.1007/s002220050123},
      zblnumber = {0991.11035},
      }
  • [Ullmo:Bogomolov] Go to document E. Ullmo, "Positivité et discrétion des points algébriques des courbes," Ann. of Math. (2), vol. 147, iss. 1, pp. 167-179, 1998.
    @ARTICLE{Ullmo:Bogomolov,
      author = {Ullmo, Emmanuel},
      title = {Positivité et discrétion des points algébriques des courbes},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {147},
      year = {1998},
      number = {1},
      pages = {167--179},
      issn = {0003-486X},
      mrclass = {14G40 (11G30 14H25 14H40)},
      mrnumber = {1609514},
      mrreviewer = {Yuichiro Takeda},
      doi = {10.2307/120987},
      url = {https://doi.org/10.2307/120987},
      zblnumber = {0934.14013},
      }
  • [Zhang:Bogomolov] Go to document S. Zhang, "Equidistribution of small points on abelian varieties," Ann. of Math. (2), vol. 147, iss. 1, pp. 159-165, 1998.
    @ARTICLE{Zhang:Bogomolov,
      author = {Zhang, Shou-Wu},
      title = {Equidistribution of small points on abelian varieties},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {147},
      year = {1998},
      number = {1},
      pages = {159--165},
      issn = {0003-486X},
      mrclass = {14G40 (11G10 14G05 14K15)},
      mrnumber = {1609518},
      mrreviewer = {Yuichiro Takeda},
      doi = {10.2307/120986},
      url = {https://doi.org/10.2307/120986},
      zblnumber = {0991.11034},
      }
  • [Zhang:adelic] S. Zhang, "Small points and adelic metrics," J. Algebraic Geom., vol. 4, iss. 2, pp. 281-300, 1995.
    @ARTICLE{Zhang:adelic,
      author = {Zhang, Shouwu},
      title = {Small points and adelic metrics},
      journal = {J. Algebraic Geom.},
      fjournal = {Journal of Algebraic Geometry},
      volume = {4},
      year = {1995},
      number = {2},
      pages = {281--300},
      issn = {1056-3911},
      mrclass = {14G40},
      mrnumber = {1311351},
      mrreviewer = {Antoine Chambert-Loir},
      zblnumber = {0861.14019},
      }

Authors

Laura DeMarco

Northwestern University, Evanston, IL, USA

Holly Krieger

University of Cambridge, Cambridge, UK

Hexi Ye

Zhejiang University, Hangzhou, China