Uniformity in Mordell–Lang for curves

Abstract

Consider a smooth, geometrically irreducible, projective curve of genus $g\ge 2$ defined over a number field of degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of $g$, $d$, and the Mordell–Weil rank of the curve’s Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounds, in $g$ and $d$, for the number of geometric torsion points of the Jacobian which lie in the image of an Abel–Jacobi map. Both estimates generalize our previous work for one-parameter families. Our proof uses Vojta’s approach to the Mordell Conjecture, and the key new ingredient is the generalization of a height inequality due to the second- and third-named authors.

  • [ACG:Curve] Go to document E. Arbarello, M. Cornalba, and P. A. Griffiths, Geometry of Agebraic Curves. Volume II, Springer, Heidelberg, 2011, vol. 268.
    @BOOK{ACG:Curve,
      author = {Arbarello, Enrico and Cornalba, Maurizio and Griffiths, Phillip A.},
      title = {Geometry of Agebraic Curves. {V}olume {II}},
      series = {Grundlehren Math. Wissen.},
      volume = {268},
      note = {with a contribution by Joseph Daniel Harris},
      publisher = {Springer, Heidelberg},
      year = {2011},
      pages = {xxx+963},
      isbn = {978-3-540-42688-2},
      mrclass = {14H10 (32G15)},
      mrnumber = {2807457},
      mrreviewer = {E. Looijenga},
      doi = {10.1007/978-3-540-69392-5},
      url = {https://doi.org/10.1007/978-3-540-69392-5},
      zblnumber = {1235.14002},
      }
  • [ACZBetti] Go to document Y. André, P. Corvaja, and U. Zannier, "The Betti map associated to a section of an abelian scheme," Invent. Math., vol. 222, iss. 1, pp. 161-202, 2020.
    @ARTICLE{ACZBetti,
      author = {André,
      Y. and Corvaja, P. and Zannier, U.},
      title = {The {B}etti map associated to a section of an abelian scheme},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {222},
      year = {2020},
      number = {1},
      pages = {161--202},
      issn = {0020-9910},
      mrclass = {14K10 (11G10 11J89)},
      mrnumber = {4145789},
      doi = {10.1007/s00222-020-00963-w},
      url = {https://doi.org/10.1007/s00222-020-00963-w},
      zblnumber = {07259017},
      }
  • [AlpogeRatPt] L. H. A. Alpoge, The average number of rational points on genus two curves is bounded, 2018.
    @MISC{AlpogeRatPt,
      author = {Alpoge, Levent Hasan Ali},
      title = {The average number of rational points on genus two curves is bounded},
      arxiv = {1804.05859},
      year = {2018},
      zblnumber = {},
      }
  • [AlpogeThesis] Go to document L. H. A. Alpoge, Points on Curves, ProQuest LLC, Ann Arbor, MI, 2020.
    @BOOK{AlpogeThesis,
      author = {Alpoge, Levent Hasan Ali},
      title = {Points on {C}urves},
      note = {Thesis (Ph.D.)--Princeton Univ.},
      publisher = {ProQuest LLC, Ann Arbor, MI},
      year = {2020},
      pages = {277},
      isbn = {979-8557-04473-8},
      mrclass = {Thesis},
      mrnumber = {4209988},
      url = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28024911},
      zblnumber = {},
      }
  • [BG] Go to document E. Bombieri and W. Gubler, Heights in Diophantine Geometry, Cambridge Univ. Press, Cambridge, 2006, vol. 4.
    @BOOK{BG,
      author = {Bombieri, Enrico and Gubler, Walter},
      title = {Heights in {D}iophantine Geometry},
      series = {New Math. Monogr.},
      volume = {4},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {2006},
      pages = {xvi+652},
      isbn = {978-0-521-84615-8; 0-521-84615-3},
      mrclass = {11G50 (11-02 11G10 11G30 11J68 14G40)},
      mrnumber = {2216774},
      mrreviewer = {Yuri Bilu},
      doi = {10.1017/CBO9780511542879},
      url = {https://doi.org/10.1017/CBO9780511542879},
      zblnumber = {1115.11034},
      }
  • [BGS] Go to document J. -B. Bost, H. Gillet, and C. Soulé, "Heights of projective varieties and positive Green forms," J. Amer. Math. Soc., vol. 7, iss. 4, pp. 903-1027, 1994.
    @ARTICLE{BGS,
      author = {Bost, J.-B. and Gillet, H. and Soulé,
      C.},
      title = {Heights of projective varieties and positive {G}reen forms},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {7},
      year = {1994},
      number = {4},
      pages = {903--1027},
      issn = {0894-0347},
      mrclass = {14G40 (11G35 14C17)},
      mrnumber = {1260106},
      mrreviewer = {Yoichi Miyaoka},
      doi = {10.2307/2152736},
      url = {https://doi.org/10.2307/2152736},
      zblnumber = {0973.14013},
      }
  • [NeronModels] Go to document S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, Springer-Verlag, Berlin, 1990, vol. 21.
    @BOOK{NeronModels,
      author = {Bosch, Siegfried and Lütkebohmert, Werner and Raynaud, Michel},
      title = {Néron Models},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {21},
      publisher = {Springer-Verlag, Berlin},
      year = {1990},
      pages = {x+325},
      isbn = {3-540-50587-3},
      mrclass = {14K15 (11G10 14L15)},
      mrnumber = {1045822},
      mrreviewer = {James Milne},
      doi = {10.1007/978-3-642-51438-8},
      url = {https://doi.org/10.1007/978-3-642-51438-8},
      zblnumber = {0705.14001},
      }
  • [Bombieri:Mordell] Go to document E. Bombieri, "The Mordell conjecture revisited," Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), vol. 17, iss. 4, pp. 615-640, 1990.
    @ARTICLE{Bombieri:Mordell,
      author = {Bombieri, Enrico},
      title = {The {M}ordell conjecture revisited},
      journal = {Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)},
      fjournal = {Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV},
      volume = {17},
      year = {1990},
      number = {4},
      pages = {615--640},
      issn = {0391-173X},
      mrclass = {11G30 (11D41 14H25)},
      mrnumber = {1093712},
      mrreviewer = {Joseph H. Silverman},
      url = {http://www.numdam.org/item?id=ASNSP_1990_4_17_4_615_0},
      zblnumber = {0722.14010},
      }
  • [CGHX:18] Go to document S. Cantat, Z. Gao, P. Habegger, and J. Xie, "The geometric Bogomolov conjecture," Duke Math. J., vol. 170, iss. 2, pp. 247-277, 2021.
    @ARTICLE{CGHX:18,
      author = {Cantat, Serge and Gao, Ziyang and Habegger, Philipp and Xie, Junyi},
      title = {The geometric {B}ogomolov conjecture},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {170},
      year = {2021},
      number = {2},
      pages = {247--277},
      issn = {0012-7094},
      mrclass = {14G25 (14G40)},
      mrnumber = {4202494},
      doi = {10.1215/00127094-2020-0044},
      url = {https://doi.org/10.1215/00127094-2020-0044},
      zblnumber = {07322638},
      }
  • [Chabauty] 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{Chabauty,
      author = {Chabauty, Claude},
      title = {Sur les points rationnels des courbes algébriques de genre supérieur à l'unité},
      journal = {C. R. Acad. Sci. Paris},
      fjournal = {Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences},
      volume = {212},
      year = {1941},
      pages = {882--885},
      issn = {0001-4036},
      mrclass = {14.0X},
      mrnumber = {0004484},
      mrreviewer = {O. F. G. Schilling},
      zblnumber = {0025.24902},
      }
  • [Coleman:effCha] Go to document R. F. Coleman, "Effective Chabauty," Duke Math. J., vol. 52, iss. 3, pp. 765-770, 1985.
    @ARTICLE{Coleman:effCha,
      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},
      }
  • [CVV:17] Go to document S. Checcoli, F. Veneziano, and E. Viada, "On the explicit torsion anomalous conjecture," Trans. Amer. Math. Soc., vol. 369, iss. 9, pp. 6465-6491, 2017.
    @ARTICLE{CVV:17,
      author = {Checcoli, S. and Veneziano, F. and Viada, E.},
      title = {On the explicit torsion anomalous conjecture},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the American Mathematical Society},
      volume = {369},
      year = {2017},
      number = {9},
      pages = {6465--6491},
      issn = {0002-9947},
      mrclass = {11G50 (11G05 14G40)},
      mrnumber = {3660229},
      mrreviewer = {Yu Yasufuku},
      doi = {10.1090/tran/6893},
      url = {https://doi.org/10.1090/tran/6893},
      zblnumber = {1428.11124},
      }
  • [deDiego:97] Go to document T. de Diego, "Points rationnels sur les familles de courbes de genre au moins 2," J. Number Theory, vol. 67, iss. 1, pp. 85-114, 1997.
    @ARTICLE{deDiego:97,
      author = {de Diego, Teresa},
      title = {Points rationnels sur les familles de courbes de genre au moins 2},
      journal = {J. Number Theory},
      fjournal = {Journal of Number Theory},
      volume = {67},
      year = {1997},
      number = {1},
      pages = {85--114},
      issn = {0022-314X},
      mrclass = {11G30 (14G05 14G25)},
      mrnumber = {1485428},
      mrreviewer = {David Harari},
      doi = {10.1006/jnth.1997.2146},
      url = {https://doi.org/10.1006/jnth.1997.2146},
      zblnumber = {0896.11025},
      }
  • [Demailly] J. P. Demailly, Complex analytic and differential geometry, 2012.
    @MISC{Demailly,
      author = {Demailly, J.P.},
      title = {Complex analytic and differential geometry},
      year = {2012},
      note = {available on author's website},
      zblnumber = {},
      }
  • [DGH1p] Go to document V. Dimitrov, Z. Gao, and P. Habegger, "Uniform bound for the number of rational points on a pencil of curves," Int. Math. Res. Not. IMRN, iss. 2, pp. 1138-1159, 2021.
    @ARTICLE{DGH1p,
      author = {Dimitrov, Vesselin and Gao, Ziyang and Habegger, Philipp},
      title = {Uniform bound for the number of rational points on a pencil of curves},
      journal = {Int. Math. Res. Not. IMRN},
      fjournal = {International Mathematics Research Notices. IMRN},
      year = {2021},
      number = {2},
      pages = {1138--1159},
      issn = {1073-7928},
      mrclass = {14G05 (14H10)},
      mrnumber = {4201963},
      doi = {10.1093/imrn/rnz248},
      url = {https://doi.org/10.1093/imrn/rnz248},
      zblnumber = {},
      }
  • [DGHBog] Go to document V. Dimitrov, Z. Gao, and P. Habegger, "A consequence of the relative Bogomolov conjecture," Journal of Number Theory, 2021.
    @article{DGHBog,
      author = {Dimitrov, Vesselin and Gao, Ziyang and Habegger, Philipp},
      title = {A consequence of the relative {B}ogomolov conjecture},
      journal = {Journal of Number Theory},
      year = {2021},
      issn = {0022-314X},
      note = {available online 1 June 2021},
      doi = {10.1016/j.jnt.2021.03.028},
      url = {https://doi.org/10.1016/j.jnt.2021.03.028},
      }
  • [DeMarcoKriegerYeUniManinMumford] Go to document L. DeMarco, H. Krieger, and H. Ye, "Uniform Manin-Mumford for a family of genus 2 curves," Ann. of Math. (2), vol. 191, iss. 3, pp. 949-1001, 2020.
    @ARTICLE{DeMarcoKriegerYeUniManinMumford,
      author = {DeMarco, Laura and Krieger, Holly and Ye, Hexi},
      title = {Uniform {M}anin-{M}umford for a family of genus 2 curves},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {191},
      year = {2020},
      number = {3},
      pages = {949--1001},
      issn = {0003-486X},
      mrclass = {14G40 (11G50 14H40 37F44 37P50)},
      mrnumber = {4088354},
      mrreviewer = {Yu Yasufuku},
      doi = {10.4007/annals.2020.191.3.5},
      url = {https://doi.org/10.4007/annals.2020.191.3.5},
      zblnumber = {1452.14027},
      }
  • [DM:irreducibility] Go to document P. Deligne and D. Mumford, "The irreducibility of the space of curves of given genus," Inst. Hautes Études Sci. Publ. Math., iss. 36, pp. 75-109, 1969.
    @ARTICLE{DM:irreducibility,
      author = {Deligne, P. and Mumford, D.},
      title = {The irreducibility of the space of curves of given genus},
      journal = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
      fjournal = {Institut des Hautes \'{E}tudes Scientifiques. Publications Mathématiques},
      number = {36},
      year = {1969},
      pages = {75--109},
      issn = {0073-8301},
      mrclass = {14.20},
      mrnumber = {0262240},
      mrreviewer = {Manfred Herrmann},
      url = {http://www.numdam.org/item?id=PMIHES_1969__36__75_0},
      zblnumber = {0181.48803},
      }
  • [DaNaPh:07] S. David, M. Nakamaye, and P. Philippon, "Bornes uniformes pour le nombre de points rationnels de certaines courbes," in Diophantine Geometry, Ed. Norm., Pisa, 2007, vol. 4, pp. 143-164.
    @INCOLLECTION{DaNaPh:07,
      author = {David, Sinnou and Nakamaye, Michael and Philippon, Patrice},
      title = {Bornes uniformes pour le nombre de points rationnels de certaines courbes},
      booktitle = {Diophantine Geometry},
      series = {CRM Series},
      volume = {4},
      pages = {143--164},
      publisher = {Ed. Norm., Pisa},
      year = {2007},
      mrclass = {11G10 (11G50 14G05 14G40)},
      mrnumber = {2349652},
      mrreviewer = {Sylvain Duquesne},
      zblnumber = {1230.14031},
      }
  • [DPvarabII] Go to document S. David and P. Philippon, "Minorations des hauteurs normalisées des sous-variétés de variétés abeliennes. II," Comment. Math. Helv., vol. 77, iss. 4, pp. 639-700, 2002.
    @ARTICLE{DPvarabII,
      author = {David, Sinnou and Philippon, Patrice},
      title = {Minorations des hauteurs normalisées des sous-variétés de variétés abeliennes. {II}},
      journal = {Comment. Math. Helv.},
      fjournal = {Commentarii Mathematici Helvetici},
      volume = {77},
      year = {2002},
      number = {4},
      pages = {639--700},
      issn = {0010-2571},
      mrclass = {11G50 (11G10 11J95 14G40)},
      mrnumber = {1949109},
      mrreviewer = {Timothy D. Browning},
      doi = {10.1007/PL00012437},
      url = {https://doi.org/10.1007/PL00012437},
      zblnumber = {1030.11026},
      }
  • [DaPh:07] Go to document S. David and P. Philippon, "Minorations des hauteurs normalisées des sous-variétés des puissances des courbes elliptiques," Int. Math. Res. Pap. IMRP, iss. 3, p. I, 2007.
    @ARTICLE{DaPh:07,
      author = {David, Sinnou and Philippon, Patrice},
      title = {Minorations des hauteurs normalisées des sous-variétés des puissances des courbes elliptiques},
      journal = {Int. Math. Res. Pap. IMRP},
      fjournal = {International Mathematics Research Papers. IMRP},
      year = {2007},
      number = {3},
      pages = {Art. ID rpm006, 113},
      issn = {1687-3017},
      mrclass = {11G50 (11G10)},
      mrnumber = {2355454},
      mrreviewer = {Timothy D. Browning},
      doi = {10.1093/imrp/rpm006},
      url = {https://doi.org/10.1093/imrp/rpm006},
      zblnumber = {1163.11049},
      }
  • [Faltings:ES] Go to document G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern," Invent. Math., vol. 73, iss. 3, pp. 349-366, 1983.
    @ARTICLE{Faltings:ES,
      author = {Faltings, Gerd},
      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},
      mrclass = {11D41 (11G30 14G25)},
      mrnumber = {0718935},
      mrreviewer = {James Milne},
      doi = {10.1007/BF01388432},
      url = {https://doi.org/10.1007/BF01388432},
      zblnumber = {0588.14026},
      }
  • [Faltings:DAAV] Go to document G. Faltings, "Diophantine approximation on abelian varieties," Ann. of Math. (2), vol. 133, iss. 3, pp. 549-576, 1991.
    @ARTICLE{Faltings:DAAV,
      author = {Faltings, Gerd},
      title = {Diophantine approximation on abelian varieties},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {133},
      year = {1991},
      number = {3},
      pages = {549--576},
      issn = {0003-486X},
      mrclass = {11G10 (11J99)},
      mrnumber = {1109353},
      mrreviewer = {Joseph H. Silverman},
      doi = {10.2307/2944319},
      url = {https://doi.org/10.2307/2944319},
      zblnumber = {0734.14007},
      }
  • [FaltingsChai] Go to document G. Faltings and C. Chai, Degeneration of Abelian Varieties, Springer-Verlag, Berlin, 1990, vol. 22.
    @BOOK{FaltingsChai,
      author = {Faltings, Gerd and Chai, Ching-Li},
      title = {Degeneration of Abelian Varieties},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {22},
      note = {with an appendix by David Mumford},
      publisher = {Springer-Verlag, Berlin},
      year = {1990},
      pages = {xii+316},
      isbn = {3-540-52015-5},
      mrclass = {14K10 (11G10 14D20 14K25)},
      mrnumber = {1083353},
      mrreviewer = {Min Ho Lee},
      doi = {10.1007/978-3-662-02632-8},
      url = {https://doi.org/10.1007/978-3-662-02632-8},
      zblnumber = {0744.14031},
      }
  • [Fulton] Go to document W. Fulton, Intersection Theory, Second ed., Springer-Verlag, Berlin, 1998, vol. 2.
    @BOOK{Fulton,
      author = {Fulton, William},
      title = {Intersection {T}heory},
      series = {Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]},
      volume = {2},
      edition = {Second},
      publisher = {Springer-Verlag, Berlin},
      year = {1998},
      pages = {xiv+470},
      isbn = {3-540-62046-X; 0-387-98549-2},
      mrclass = {14C17 (14-02)},
      mrnumber = {1644323},
      doi = {10.1007/978-1-4612-1700-8},
      url = {https://doi.org/10.1007/978-1-4612-1700-8},
      zblnumber = {0885.14002},
      }
  • [GaoBettiRank] Go to document Z. Gao, "Generic rank of Betti map and unlikely intersections," Compos. Math., vol. 156, iss. 12, pp. 2469-2509, 2020.
    @ARTICLE{GaoBettiRank,
      author = {Gao, Ziyang},
      title = {Generic rank of {B}etti map and unlikely intersections},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {156},
      year = {2020},
      number = {12},
      pages = {2469--2509},
      issn = {0010-437X},
      mrclass = {14K15 (11G10 11G50 14G25 14G35)},
      mrnumber = {4202453},
      doi = {10.1112/s0010437x20007435},
      url = {https://doi.org/10.1112/s0010437x20007435},
      zblnumber = {07308572},
      }
  • [GaoMixedAS] Go to document Z. Gao, "Mixed Ax-Schanuel for the universal abelian varieties and some applications," Compos. Math., vol. 156, iss. 11, pp. 2263-2297, 2020.
    @ARTICLE{GaoMixedAS,
      author = {Gao, Ziyang},
      title = {Mixed {A}x-{S}chanuel for the universal abelian varieties and some applications},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {156},
      year = {2020},
      number = {11},
      pages = {2263--2297},
      issn = {0010-437X},
      mrclass = {14G35 (11G10 11G18)},
      mrnumber = {4190045},
      doi = {10.1112/S0010437X20007447},
      url = {https://doi.org/10.1112/S0010437X20007447},
      zblnumber = {07305885},
      }
  • [GaoSurveyUML] Z. Gao, Recent developments of the uniform Mordell–Lang conjecture, 2021.
    @MISC{GaoSurveyUML,
      author = {Gao, Ziyang},
      title = {Recent developments of the uniform {M}ordell--{L}ang conjecture},
      year = {2021},
      arxiv = {2104.03431},
      }
  • [GaoHab] Go to document Z. Gao and P. Habegger, "Heights in families of abelian varieties and the geometric Bogomolov conjecture," Ann. of Math. (2), vol. 189, iss. 2, pp. 527-604, 2019.
    @ARTICLE{GaoHab,
      author = {Gao, Ziyang and Habegger, Philipp},
      title = {Heights in families of abelian varieties and the geometric {B}ogomolov conjecture},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {189},
      year = {2019},
      number = {2},
      pages = {527--604},
      issn = {0003-486X},
      mrclass = {11G10 (11G50 14G25 14G40 14K15)},
      mrnumber = {3922127},
      mrreviewer = {John L. Boxall},
      doi = {10.4007/annals.2019.189.2.3},
      url = {https://doi.org/10.4007/annals.2019.189.2.3},
      zblnumber = {1432.11060},
      }
  • [GenestierNgo] Go to document A. Genestier and B. C. Ngô, "Lectures on Shimura varieties," in Autour des Motifs—École d’été Franco-Asiatique de Géométrie Algébrique et de Théorie des Nombres/Asian-French Summer School on Algebraic Geometry and Number Theory. Volume I, Soc. Math. France, Paris, 2009, vol. 29, pp. 187-236.
    @INCOLLECTION{GenestierNgo,
      author = {Genestier, Alain and Ngô,
      Bao Ch\^{a}u},
      title = {Lectures on {S}himura varieties},
      booktitle = {Autour des Motifs---\'{E}cole d'été {F}ranco-{A}siatique de {G}éométrie {A}lgébrique et de {T}héorie des {N}ombres/{A}sian-{F}rench {S}ummer {S}chool on {A}lgebraic {G}eometry and {N}umber {T}heory. {V}olume {I}},
      series = {Panor. Synthèses},
      volume = {29},
      pages = {187--236},
      publisher = {Soc. Math. France, Paris},
      year = {2009},
      mrclass = {14G35 (11G18 14C30 14K05)},
      mrnumber = {2730658},
      mrreviewer = {Eva Viehmann},
      url = {https://smf.emath.fr/publications/lectures-shimura-varieties},
      zblnumber = {1215.14018},
      }
  • [EGAII] Go to document A. Grothendieck, "Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes," Inst. Hautes Études Sci. Publ. Math., iss. 8, p. 222, 1961.
    @ARTICLE{EGAII,
      author = {Grothendieck, A.},
      title = {\'{E}léments de géométrie algébrique. {II}. \'{E}tude globale élémentaire de quelques classes de morphismes},
      journal = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
      fjournal = {Institut des Hautes \'{E}tudes Scientifiques. Publications Mathématiques},
      number = {8},
      year = {1961},
      pages = {222},
      issn = {0073-8301},
      mrclass = {14.55},
      mrnumber = {0217084},
      doi = {10.1007/BF02699291},
      url = {https://doi.org/10.1007/BF02699291},
      zblnumber = {},
      }
  • [EGAIV] Go to document A. Grothendieck, "Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV," Inst. Hautes Études Sci. Publ. Math., iss. 32, p. 361, 1967.
    @ARTICLE{EGAIV,
      author = {Grothendieck, A.},
      title = {\'{E}léments de géométrie algébrique. {IV}. \'{E}tude locale des schémas et des morphismes de schémas {IV}},
      journal = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
      fjournal = {Institut des Hautes \'{E}tudes Scientifiques. Publications Mathématiques},
      number = {32},
      year = {1967},
      pages = {361},
      issn = {0073-8301},
      mrclass = {14.55},
      mrnumber = {0238860},
      mrreviewer = {J. P. Murre},
      doi = {10.1007/BF02732123},
      url = {https://doi.org/10.1007/BF02732123},
      zblnumber = {},
      }
  • [GoertzWedhorn] Go to document U. Görtz and T. Wedhorn, Algebraic Geometry I. Schemes with Examples and Exercises, Vieweg + Teubner, Wiesbaden, 2010.
    @BOOK{GoertzWedhorn,
      author = {Görtz, Ulrich and Wedhorn, Torsten},
      title = {Algebraic Geometry {I}. Schemes with Examples and Exercises},
      series = {Adv. Lect. Math.},
      publisher = {Vieweg + Teubner, Wiesbaden},
      year = {2010},
      pages = {viii+615},
      isbn = {978-3-8348-0676-5},
      mrclass = {14-01},
      mrnumber = {2675155},
      mrreviewer = {C\'ıcero Carvalho},
      doi = {10.1007/978-3-8348-9722-0},
      url = {https://doi.org/10.1007/978-3-8348-9722-0},
      zblnumber = {1213.14001},
      }
  • [habegger:imrn] Go to document P. Habegger, "On the bounded height conjecture," Int. Math. Res. Not. IMRN, iss. 5, pp. 860-886, 2009.
    @ARTICLE{habegger:imrn,
      author = {Habegger, Philipp},
      title = {On the bounded height conjecture},
      journal = {Int. Math. Res. Not. IMRN},
      fjournal = {International Mathematics Research Notices. IMRN},
      year = {2009},
      number = {5},
      pages = {860--886},
      issn = {1073-7928},
      mrclass = {14G25 (11G50 14G40)},
      mrnumber = {2482128},
      mrreviewer = {Dimitrios Poulakis},
      doi = {10.1093/imrn/rnn149},
      url = {https://doi.org/10.1093/imrn/rnn149},
      zblnumber = {1239.11070},
      }
  • [Hab:Special] Go to document P. Habegger, "Special points on fibered powers of elliptic surfaces," J. Reine Angew. Math., vol. 685, pp. 143-179, 2013.
    @ARTICLE{Hab:Special,
      author = {Habegger, Philipp},
      title = {Special points on fibered powers of elliptic surfaces},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {685},
      year = {2013},
      pages = {143--179},
      issn = {0075-4102},
      mrclass = {11G10 (11G18 14J27)},
      mrnumber = {3181568},
      mrreviewer = {Mihran Papikian},
      doi = {10.1515/crelle-2012-0007},
      url = {https://doi.org/10.1515/crelle-2012-0007},
      zblnumber = {1318.14023},
      }
  • [HabeggerPilaENS] Go to document P. Habegger and J. Pila, "O-minimality and certain atypical intersections," Ann. Sci. Éc. Norm. Supér. (4), vol. 49, iss. 4, pp. 813-858, 2016.
    @ARTICLE{HabeggerPilaENS,
      author = {Habegger, Philipp and Pila, Jonathan},
      title = {O-minimality and certain atypical intersections},
      journal = {Ann. Sci. \'{E}c. Norm. Supér. (4)},
      fjournal = {Annales Scientifiques de l'\'{E}cole Normale Supérieure. Quatrième Série},
      volume = {49},
      year = {2016},
      number = {4},
      pages = {813--858},
      issn = {0012-9593},
      mrclass = {14G35 (03C64 14K12)},
      mrnumber = {3552014},
      mrreviewer = {Fabrizio Barroero},
      doi = {10.24033/asens.2296},
      url = {https://doi.org/10.24033/asens.2296},
      zblnumber = {1364.11110},
      }
  • [Hurwitz1892] Go to document A. Hurwitz, "Ueber algebraische Gebilde mit eindeutigen Transformationen in sich," Math. Ann., vol. 41, iss. 3, pp. 403-442, 1892.
    @ARTICLE{Hurwitz1892,
      author = {Hurwitz, A.},
      title = {Ueber algebraische {G}ebilde mit eindeutigen {T}ransformationen in sich},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {41},
      year = {1892},
      number = {3},
      pages = {403--442},
      issn = {0025-5831},
      mrclass = {DML},
      mrnumber = {1510753},
      doi = {10.1007/BF01443420},
      url = {https://doi.org/10.1007/BF01443420},
      jfmnumber = {24.0380.02},
      }
  • [KatzRabinoffZB] 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{KatzRabinoffZB,
      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 = {1428.11121},
      }
  • [kuehne:semiabelianbhc] Go to document L. Kühne, "The bounded height conjecture for semiabelian varieties," Compos. Math., vol. 156, iss. 7, pp. 1405-1456, 2020.
    @ARTICLE{kuehne:semiabelianbhc,
      author = {K{ü}hne, Lars},
      title = {The bounded height conjecture for semiabelian varieties},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {156},
      year = {2020},
      number = {7},
      pages = {1405--1456},
      issn = {0010-437X},
      mrclass = {11G50 (14G40 14K15)},
      mrnumber = {4120167},
      mrreviewer = {Ariyan Javanpeykar},
      doi = {10.1112/s0010437x20007198},
      url = {https://doi.org/10.1112/s0010437x20007198},
      zblnumber = {07321639},
      }
  • [K21] L. Kühne, Equidistribution in families of Abelian varieties and uniformity, 2021.
    @MISC{K21,
      author = {K{ü}hne, Lars},
      title = {{E}quidistribution in families of {A}belian varieties and uniformity},
      arxiv = {2101.10272},
      year = {2021},
      zblnumber = {},
      }
  • [Lang:Division] Go to document S. Lang, "Division points on curves," Ann. Mat. Pura Appl. (4), vol. 70, pp. 229-234, 1965.
    @ARTICLE{Lang:Division,
      author = {Lang, Serge},
      title = {Division points on curves},
      journal = {Ann. Mat. Pura Appl. (4)},
      fjournal = {Annali di Matematica Pura ed Applicata. Serie Quarta},
      volume = {70},
      year = {1965},
      pages = {229--234},
      issn = {0003-4622},
      mrclass = {14.40 (10.10)},
      mrnumber = {0190146},
      mrreviewer = {B. Segre},
      doi = {10.1007/BF02410091},
      url = {https://doi.org/10.1007/BF02410091},
      zblnumber = {0151.27401},
      }
  • [Lang:EllipticCurves78] Go to document S. Lang, Elliptic Curves: Diophantine Analysis, Springer-Verlag, New York, 1978, vol. 231.
    @BOOK{Lang:EllipticCurves78,
      author = {Lang, Serge},
      title = {Elliptic {C}urves: {D}iophantine Analysis},
      series = {Grundlehren Math. Wissen.},
      volume = {231},
      publisher = {Springer-Verlag, New York},
      year = {1978},
      pages = {xi+261},
      isbn = {3-540-08489-4},
      mrclass = {10B15 (10Fxx 14G25)},
      mrnumber = {0518817},
      mrreviewer = {A. J. van der Poorten},
      doi = {10.1007/978-3-662-07010-9},
      url = {https://doi.org/10.1007/978-3-662-07010-9},
      zblnumber = {0388.10001},
      }
  • [PosAlgGeom] Go to document R. Lazarsfeld, Positivity in Algebraic Geometry. I, Springer-Verlag, Berlin, 2004, vol. 48.
    @BOOK{PosAlgGeom,
      author = {Lazarsfeld, Robert},
      title = {Positivity in Algebraic Geometry. {I}},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {48},
      publisher = {Springer-Verlag, Berlin},
      year = {2004},
      isbn = {3-540-22533-1},
      mrnumber = {2095471},
      mrclass = {14-02 (14C20)},
      mrreviewer = {Mihnea Popa},
      doi = {10.1007/978-3-642-18808-4},
      url = {https://doi.org/10.1007/978-3-642-18808-4},
      zblnumber = {1066.14021},
      }
  • [Lelong] Go to document P. Lelong, "Intégration sur un ensemble analytique complexe," Bull. Soc. Math. France, vol. 85, pp. 239-262, 1957.
    @ARTICLE{Lelong,
      author = {Lelong, Pierre},
      title = {Intégration sur un ensemble analytique complexe},
      journal = {Bull. Soc. Math. France},
      fjournal = {Bulletin de la Société Mathématique de France},
      volume = {85},
      year = {1957},
      pages = {239--262},
      issn = {0037-9484},
      mrclass = {32.00},
      mrnumber = {0095967},
      mrreviewer = {H. Grauert},
      url = {http://www.numdam.org/item?id=BSMF_1957__85__239_0},
      zblnumber = {0079.30901},
      }
  • [mazur1986arithmetic] Go to document B. Mazur, "Arithmetic on curves," Bull. Amer. Math. Soc. (N.S.), vol. 14, iss. 2, pp. 207-259, 1986.
    @ARTICLE{mazur1986arithmetic,
      author = {Mazur, Barry},
      title = {Arithmetic on curves},
      journal = {Bull. Amer. Math. Soc. (N.S.)},
      fjournal = {American Mathematical Society. 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},
      }
  • [Mazur:00] B. Mazur, "Abelian varieties and the Mordell-Lang conjecture," in Model Theory, Algebra, and Geometry, Cambridge Univ. Press, Cambridge, 2000, vol. 39, pp. 199-227.
    @INCOLLECTION{Mazur:00,
      author = {Mazur, Barry},
      title = {Abelian varieties and the {M}ordell-{L}ang conjecture},
      booktitle = {Model {T}heory, {A}lgebra, and {G}eometry},
      series = {Math. Sci. Res. Inst. Publ.},
      volume = {39},
      pages = {199--227},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {2000},
      mrclass = {11G10 (11G35 14K05)},
      mrnumber = {1773708},
      mrreviewer = {Dan Abramovich},
      zblnumber = {0990.14006},
      }
  • [MFK:GIT94] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, Third ed., Springer-Verlag, Berlin, 1994, vol. 34.
    @BOOK{MFK:GIT94,
      author = {Mumford, David and Fogarty, J. and Kirwan, F.},
      title = {Geometric invariant theory},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {34},
      edition = {Third},
      publisher = {Springer-Verlag, Berlin},
      year = {1994},
      pages = {xiv+292},
      isbn = {3-540-56963-4},
      mrclass = {14D25 (58E05 58F05)},
      mrnumber = {1304906},
      mrreviewer = {Yi Hu},
      zblnumber = {0797.14004},
      }
  • [Mok11Form] Go to document N. Mok, "Aspects of Kähler geometry on arithmetic varieties," in Several Complex Variables and Complex Geometry, Part 2, Amer. Math. Soc., Providence, RI, 1991, vol. 52, pp. 335-396.
    @INCOLLECTION{Mok11Form,
      author = {Mok, Ngaiming},
      title = {Aspects of {K}ähler geometry on arithmetic varieties},
      booktitle = {Several Complex Variables and Complex Geometry, {P}art 2},
      venue = {{S}anta {C}ruz, {CA},
      1989},
      series = {Proc. Sympos. Pure Math.},
      volume = {52},
      pages = {335--396},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1991},
      mrclass = {32C17 (14K99 22E40 32L07 32L20 53C55)},
      mrnumber = {1128556},
      mrreviewer = {Zong Yuan Yao},
      doi = {10.1090/pspum/052.2/1128556},
      url = {https://doi.org/10.1090/pspum/052.2/1128556},
      zblnumber = {0736.53048},
      }
  • [Mumford:quadeq] D. Mumford, "Varieties defined by quadratic equations," in Questions on Algebraic Varieties, Edizioni Cremonese, Rome, 1970, pp. 29-100.
    @INCOLLECTION{Mumford:quadeq,
      author = {Mumford, David},
      title = {Varieties defined by quadratic equations},
      booktitle = {Questions on {A}lgebraic {V}arieties},
      venue = {{C}.{I}.{M}.{E}., {III} {C}iclo, {V}arenna, 1969},
      pages = {29--100},
      publisher = {Edizioni Cremonese, Rome},
      year = {1970},
      mrclass = {14.01},
      mrnumber = {0282975},
      mrreviewer = {Y. Nakai},
      zblnumber = {0198.25801},
      }
  • [MumfordAbVar] D. Mumford, Abelian Varieties, 2nd ed., Oxford Univ. Press, London, 1974.
    @BOOK{MumfordAbVar,
      author = {Mumford, David},
      title = {Abelian Varieties},
      edition = {2nd},
      publisher = {Oxford Univ. Press, London},
      year = {1974},
      zblnumber = {0326.14012},
      }
  • [OortSteenbrink] F. Oort and J. Steenbrink, "The local Torelli problem for algebraic curves," in Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 157-204.
    @INCOLLECTION{OortSteenbrink,
      author = {Oort, Frans and Steenbrink, Joseph},
      title = {The local {T}orelli problem for algebraic curves},
      booktitle = {Journées de {G}éometrie {A}lgébrique d'{A}ngers, {J}uillet 1979/{A}lgebraic {G}eometry, {A}ngers, 1979},
      pages = {157--204},
      publisher = {Sijthoff \& Noordhoff, Alphen aan den Rijn---Germantown, Md.},
      year = {1980},
      mrclass = {14H10 (14K10)},
      mrnumber = {0605341},
      mrreviewer = {A. Duma},
      zblnumber = {0444.14007},
      }
  • [Pazuki:12] Go to document F. Pazuki, "Theta height and Faltings height," Bull. Soc. Math. France, vol. 140, iss. 1, pp. 19-49, 2012.
    @ARTICLE{Pazuki:12,
      author = {Pazuki, Fabien},
      title = {Theta height and {F}altings height},
      journal = {Bull. Soc. Math. France},
      fjournal = {Bulletin de la Société Mathématique de France},
      volume = {140},
      year = {2012},
      number = {1},
      pages = {19--49},
      issn = {0037-9484},
      mrclass = {11G50 (14G40)},
      mrnumber = {2903770},
      mrreviewer = {Robin de Jong},
      doi = {10.24033/bsmf.2623},
      url = {https://doi.org/10.24033/bsmf.2623},
      zblnumber = {1245.14029},
      }
  • [Philippon] Go to document P. Philippon, "Lemmes de zéros dans les groupes algébriques commutatifs," Bull. Soc. Math. France, vol. 114, iss. 3, pp. 355-383, 1986.
    @ARTICLE{Philippon,
      author = {Philippon, Patrice},
      title = {Lemmes de zéros dans les groupes algébriques commutatifs},
      journal = {Bull. Soc. Math. France},
      fjournal = {Bulletin de la Société Mathématique de France},
      volume = {114},
      year = {1986},
      number = {3},
      pages = {355--383},
      issn = {0037-9484},
      mrclass = {11J89},
      mrnumber = {0878242},
      mrreviewer = {D. W. Masser},
      url = {http://www.numdam.org/item?id=BSMF_1986__114__355_0},
      zblnumber = {0617.14001},
      }
  • [HauteursAlt3] P. Philippon, "Sur des hauteurs alternatives. III," J. Math. Pures Appl. (9), vol. 74, iss. 4, pp. 345-365, 1995.
    @ARTICLE{HauteursAlt3,
      author = {Philippon, Patrice},
      title = {Sur des hauteurs alternatives. {III}},
      journal = {J. Math. Pures Appl. (9)},
      fjournal = {Journal de Mathématiques Pures et Appliquées. Neuvième Série},
      volume = {74},
      year = {1995},
      number = {4},
      pages = {345--365},
      issn = {0021-7824},
      mrclass = {11G35 (11G10 14G40 14K15)},
      mrnumber = {1341770},
      mrreviewer = {Gregory S. Call},
      zblnumber = {0878.11025},
      }
  • [PinkThesis] R. Pink, Arithmetical Compactification of Mixed Shimura Varieties, Universität Bonn, Mathematisches Institut, Bonn, 1990, vol. 209.
    @BOOK{PinkThesis,
      author = {Pink, Richard},
      title = {Arithmetical Compactification of Mixed {S}himura Varieties},
      series = {Bonner Math. Schriften},
      volume = {209},
      note = {Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1989},
      publisher = {Universität Bonn, Mathematisches Institut, Bonn},
      year = {1990},
      pages = {xviii+340},
      mrclass = {11G18 (14G35)},
      mrnumber = {1128753},
      mrreviewer = {Min Ho Lee},
      zblnumber = {0748.14007},
      }
  • [Pink05] Go to document R. Pink, "A combination of the conjectures of Mordell-Lang and André-Oort," in Geometric Methods in Algebra and Number Theory, Birkhäuser Boston, Boston, MA, 2005, vol. 235, pp. 251-282.
    @INCOLLECTION{Pink05,
      author = {Pink, Richard},
      title = {A combination of the conjectures of {M}ordell-{L}ang and {A}ndré-{O}ort},
      booktitle = {Geometric Methods in Algebra and Number Theory},
      series = {Progr. Math.},
      volume = {235},
      pages = {251--282},
      publisher = {Birkhäuser Boston, Boston, MA},
      year = {2005},
      mrclass = {11G10 (11G18 14G35)},
      mrnumber = {2166087},
      mrreviewer = {Mihran Papikian},
      doi = {10.1007/0-8176-4417-2\_11},
      url = {https://doi.org/10.1007/0-8176-4417-2_11},
      zblnumber = {1200.11041},
      }
  • [Pink] R. Pink, A common generalization of the conjectures of André-Oort, Manin-Mumford, and Mordell-Lang, 2005.
    @MISC{Pink,
      author = {Pink, Richard},
      title = {A common generalization of the conjectures of {A}ndré-{O}ort, {M}anin-{M}umford, and {M}ordell-{L}ang},
      note = {preprint, 13pp., Available on author's website},
      year = {2005},
      zblnumber = {},
      }
  • [LNM119] Go to document M. Raynaud, Faisceaux Amples sur les Schémas en Groupes et les Espaces Homogènes, Springer-Verlag, New York, 1970, vol. 119.
    @BOOK{LNM119,
      author = {Raynaud, Michel},
      title = {Faisceaux Amples sur les Schémas en Groupes et les Espaces Homogènes},
      series = {Lecture Notes in Math.},
      volume = {119},
      publisher = {Springer-Verlag, New York},
      year = {1970},
      pages = {ii+218},
      mrclass = {14.55},
      mrnumber = {0260758},
      mrreviewer = {D. Gieseker},
      doi = {10.1007/BFb0059504},
      url = {https://doi.org/10.1007/BFb0059504},
      zblnumber = {0195.22701},
      }
  • [Raynaud:MMcurves] 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:MMcurves,
      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},
      }
  • [Remond:Decompte] Go to document G. Rémond, "Décompte dans une conjecture de Lang," Invent. Math., vol. 142, iss. 3, pp. 513-545, 2000.
    @ARTICLE{Remond:Decompte,
      author = {R{é}mond, Gaël},
      title = {Décompte dans une conjecture de {L}ang},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {142},
      year = {2000},
      number = {3},
      pages = {513--545},
      issn = {0020-9910},
      mrclass = {14K15 (11G10 14G05)},
      mrnumber = {1804159},
      mrreviewer = {Klaus Künnemann},
      doi = {10.1007/s002220000095},
      url = {https://doi.org/10.1007/s002220000095},
      zblnumber = {0972.11054},
      }
  • [remond:vojtasup] Go to document G. Rémond, "Inégalité de Vojta en dimension supérieure," Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), vol. 29, iss. 1, pp. 101-151, 2000.
    @ARTICLE{remond:vojtasup,
      author = {R{é}mond, Gaël},
      title = {Inégalité de {V}ojta en dimension supérieure},
      journal = {Ann. Scuola~Norm.~Sup. Pisa Cl. Sci. (4)},
      fjournal = {Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV},
      volume = {29},
      year = {2000},
      number = {1},
      pages = {101--151},
      issn = {0391-173X},
      mrclass = {11J95 (11G10 11G50 14G05 14K15)},
      mrnumber = {1765539},
      mrreviewer = {Carlo Gasbarri},
      url = {http://www.numdam.org/item?id=ASNSP_2000_4_29_1_101_0},
      zblnumber = {},
      }
  • [Silverman] Go to document J. H. Silverman, "Heights and the specialization map for families of abelian varieties," J. Reine Angew. Math., vol. 342, pp. 197-211, 1983.
    @ARTICLE{Silverman,
      author = {Silverman, Joseph H.},
      title = {Heights and the specialization map for families of abelian varieties},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {342},
      year = {1983},
      pages = {197--211},
      issn = {0075-4102},
      mrclass = {14K15 (14D10 14G25)},
      mrnumber = {0703488},
      mrreviewer = {Gerd Faltings},
      doi = {10.1515/crll.1983.342.197},
      url = {https://doi.org/10.1515/crll.1983.342.197},
      zblnumber = {0505.14035},
      }
  • [Silverman:twists] Go to document J. H. Silverman, "A uniform bound for rational points on twists of a given curve," J. London Math. Soc. (2), vol. 47, iss. 3, pp. 385-394, 1993.
    @ARTICLE{Silverman:twists,
      author = {Silverman, Joseph H.},
      title = {A uniform bound for rational points on twists of a given curve},
      journal = {J. London Math. Soc. (2)},
      fjournal = {Journal of the London Mathematical Society. Second Series},
      volume = {47},
      year = {1993},
      number = {3},
      pages = {385--394},
      issn = {0024-6107},
      mrclass = {11G30},
      mrnumber = {1214903},
      mrreviewer = {Philippe Satgé},
      doi = {10.1112/jlms/s2-47.3.385},
      url = {https://doi.org/10.1112/jlms/s2-47.3.385},
      zblnumber = {0811.11046},
      }
  • [Silverman:heightest11] J. H. Silverman, "Height estimates for equidimensional dominant rational maps," J. Ramanujan Math. Soc., vol. 26, iss. 2, pp. 145-163, 2011.
    @ARTICLE{Silverman:heightest11,
      author = {Silverman, Joseph H.},
      title = {Height estimates for equidimensional dominant rational maps},
      journal = {J. Ramanujan Math. Soc.},
      fjournal = {Journal of the Ramanujan Mathematical Society},
      volume = {26},
      year = {2011},
      number = {2},
      pages = {145--163},
      issn = {0970-1249},
      mrclass = {11G50 (14G40 37P30)},
      mrnumber = {2816785},
      mrreviewer = {Thomas Ward},
      zblnumber = {1311.11057},
      }
  • [Stoll:Uniform] Go to document M. Stoll, "Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell-Weil rank," J. Eur. Math. Soc. (JEMS), vol. 21, iss. 3, pp. 923-956, 2019.
    @ARTICLE{Stoll:Uniform,
      author = {Stoll, Michael},
      title = {Uniform bounds for the number of rational points on hyperelliptic curves of small {M}ordell-{W}eil rank},
      journal = {J. Eur. Math. Soc. (JEMS)},
      fjournal = {Journal of the European Mathematical Society (JEMS)},
      volume = {21},
      year = {2019},
      number = {3},
      pages = {923--956},
      issn = {1435-9855},
      mrclass = {11G30 (14G05 14H25 14H40)},
      mrnumber = {3908770},
      mrreviewer = {Sajad Salami},
      doi = {10.4171/JEMS/857},
      url = {https://doi.org/10.4171/JEMS/857},
      zblnumber = {1428.11122},
      }
  • [voisin:hodge1] Go to document C. Voisin, Hodge Theory and Complex Algebraic Geometry. I, Cambridge Univ. Press, Cambridge, 2002, vol. 76.
    @BOOK{voisin:hodge1,
      author = {Voisin, Claire},
      title = {Hodge Theory and Complex Algebraic Geometry. {I}},
      series = {Cambridge Stud. Adv. Math.},
      volume = {76},
      note = {translated from the French original by Leila Schneps},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {2002},
      pages = {x+322},
      isbn = {0-521-80260-1},
      mrclass = {32J25 (14C30 14D07 32G20)},
      mrnumber = {1967689},
      mrreviewer = {Javier A. Fern\'{a}ndez},
      doi = {10.1017/CBO9780511615344},
      url = {https://doi.org/10.1017/CBO9780511615344},
      zblnumber = {1005.14002},
      }
  • [Vojta:siegelcompact] Go to document P. Vojta, "Siegel’s theorem in the compact case," Ann. of Math. (2), vol. 133, iss. 3, pp. 509-548, 1991.
    @ARTICLE{Vojta:siegelcompact,
      author = {Vojta, Paul},
      title = {Siegel's theorem in the compact case},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {133},
      year = {1991},
      number = {3},
      pages = {509--548},
      issn = {0003-486X},
      mrclass = {11G30 (11J68 14G25)},
      mrnumber = {1109352},
      mrreviewer = {Joseph H. Silverman},
      doi = {10.2307/2944318},
      url = {https://doi.org/10.2307/2944318},
      zblnumber = {0774.14019},
      }
  • [Ast6970] M. Waldschmidt, "Nombres transcendants et groupes algébriques," Astérisque, vol. 69-70, p. 218, 1987.
    @ARTICLE{Ast6970,
      author = {Waldschmidt, Michel},
      title = {Nombres transcendants et groupes algébriques},
      edition = {Second},
      note = {with appendices by Daniel Bertrand and Jean-Pierre Serre},
      journal = {Astérisque},
      fjournal = {Astérisque},
      volume = {69-70},
      year = {1987},
      pages = {218},
      issn = {0303-1179},
      mrclass = {11J81 (11J89)},
      mrnumber = {0890168},
      mrreviewer = {D. W. Masser},
      zblnumber = {0621.10022},
      }
  • [ZannierBook] U. Zannier, Some Problems of Unlikely Intersections in Arithmetic and Geometry, Princeton Univ. Press, Princeton, NJ, 2012, vol. 181.
    @BOOK{ZannierBook,
      author = {Zannier, Umberto},
      title = {Some Problems of Unlikely Intersections in Arithmetic and Geometry},
      series = {Ann. of Math. Stud.},
      volume = {181},
      note = {with appendixes by David Masser},
      publisher = {Princeton Univ. Press, Princeton, NJ},
      year = {2012},
      pages = {xiv+160},
      isbn = {978-0-691-15371-1},
      mrclass = {11G35 (11G15 11G18 11G50 14G05)},
      mrnumber = {2918151},
      mrreviewer = {Yuri Bilu},
      zblnumber = {1246.14003},
      }
  • [ZhangEquidist] 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{ZhangEquidist,
      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},
      }

Authors

Vesselin Dimitrov

University of Toronto, Toronto, Ontario, Canada

Ziyang Gao

CNRS and Institut de Mathématiques de Jussieu-Paris Rive Gauche, Paris, France

Philipp Habegger

University of Basel, Basel, Switzerland