Ax-Lindemann for $\mathcal{A}_g$

Abstract

We prove the Ax-Lindemann theorem for the coarse moduli space $\mathcal{A}_{g}$ of principally polarized abelian varieties of dimension $g\ge 1$. We affirm the André-Oort conjecture unconditionally for $\mathcal{A}_g$ for $g\le 6$, and under GRH for all $g$.

Note: To view the article, click on the URL link for the DOI number.

  • [ANDRE] Y. André, $G$-Functions and Geometry, Braunschweig: Friedr. Vieweg & Sohn, 1989, vol. E13.
    @book {ANDRE, MRKEY = {0990016},
      AUTHOR = {Andr{é},
      Yves},
      TITLE = {{$G$}-Functions and Geometry},
      SERIES = {Aspects Math.},
      VOLUME={E13},
      PUBLISHER = {Friedr. Vieweg \& Sohn},
      ADDRESS = {Braunschweig},
      YEAR = {1989},
      PAGES = {xii+229},
      ISBN = {3-528-06317-3},
      MRCLASS = {11J82 (11-02 11G10 11J87 12H25)},
      MRNUMBER = {0990016},
      MRREVIEWER = {C. L. Stewart},
      ZBLNUMBER = {0688.10032},
      }
  • [A] Go to document Y. André, "Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part," Compositio Math., vol. 82, iss. 1, pp. 1-24, 1992.
    @article {A, MRKEY = {1154159},
      AUTHOR = {Andr{é},
      Yves},
      TITLE = {Mumford-{T}ate groups of mixed {H}odge structures and the theorem of the fixed part},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {82},
      YEAR = {1992},
      NUMBER = {1},
      PAGES = {1--24},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14D07 (14C30)},
      MRNUMBER = {1154159},
      MRREVIEWER = {Min Ho Lee},
      ZBLNUMBER = {0770.14003},
      URL = {http://www.numdam.org/item?id=CM_1992__82_1_1_0},
      }
  • [AA] Go to document Y. André, "Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire," J. Reine Angew. Math., vol. 505, pp. 203-208, 1998.
    @article {AA, MRKEY = {1662256},
      AUTHOR = {Andr{é},
      Yves},
      TITLE = {Finitude des couples d'invariants modulaires singuliers sur une courbe algébrique plane non modulaire},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {505},
      YEAR = {1998},
      PAGES = {203--208},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {11G15 (11G18)},
      MRNUMBER = {1662256},
      MRREVIEWER = {Akio Tamagawa},
      ZBLNUMBER = {0918.14010},
      DOI = {10.1515/crll.1998.118},
      }
  • [AX] Go to document J. Ax, "On Schanuel’s conjectures," Ann. of Math., vol. 93, pp. 252-268, 1971.
    @article {AX, MRKEY = {0277482},
      AUTHOR = {Ax, James},
      TITLE = {On {S}chanuel's conjectures},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {93},
      YEAR = {1971},
      PAGES = {252--268},
      ISSN = {0003-486X},
      MRCLASS = {10.32 (12.00)},
      MRNUMBER = {0277482},
      MRREVIEWER = {J. L. Johnson},
      ZBLNUMBER = {0232.10026},
      DOI = {10.2307/1970774},
      }
  • [BILUMZ] Go to document Y. Bilu, D. Masser, and U. Zannier, "An effective “theorem of André” for $CM$-points on a plane curve," Math. Proc. Cambridge Philos. Soc., vol. 154, iss. 1, pp. 145-152, 2013.
    @article {BILUMZ, MRKEY = {3002589},
      AUTHOR = {Bilu, Yuri and Masser, David and Zannier, Umberto},
      TITLE = {An effective ``theorem of {A}ndré'' for {$CM$}-points on a plane curve},
      JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
      FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
      VOLUME = {154},
      YEAR = {2013},
      NUMBER = {1},
      PAGES = {145--152},
      ISSN = {0305-0041},
      CODEN = {MPCPCO},
      MRCLASS = {11G30 (11G15 11G18)},
      MRNUMBER = {3002589},
      MRREVIEWER = {Jordi Gu{à}rdia},
      ZBLNUMBER = {1263.14028},
      DOI = {10.1017/S0305004112000461},
      }
  • [CM] A. Borel, S. Chowla, C. S. Herz, K. Iwasawa, and J. -P. Serre, Seminar on Complex Multiplication, New York: Springer-Verlag, 1966, vol. 21.
    @book {CM, MRKEY = {0201394},
      AUTHOR = {Borel, A. and Chowla, S. and Herz, C. S. and Iwasawa, K. and Serre, J.-P.},
      TITLE = {Seminar on Complex Multiplication},
      SERIES = {Lecture Notes in Math.},
      VOLUME={21},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1966},
      PAGES = {ii+102 pp. (not consecutively paged)},
      MRCLASS = {10.68},
      MRNUMBER = {0201394},
      ZBLNUMBER = {0147.03902},
      }
  • [DAWYAFAEV] Go to document C. Daw and A. Yafaev, "An unconditional proof of the André-Oort conjecture for Hilbert modular surfaces," Manuscripta Math., vol. 135, iss. 1-2, pp. 263-271, 2011.
    @article {DAWYAFAEV, MRKEY = {2783398},
      AUTHOR = {Daw, Christopher and Yafaev, Andrei},
      TITLE = {An unconditional proof of the {A}ndré-{O}ort conjecture for {H}ilbert modular surfaces},
      JOURNAL = {Manuscripta Math.},
      FJOURNAL = {Manuscripta Mathematica},
      VOLUME = {135},
      YEAR = {2011},
      NUMBER = {1-2},
      PAGES = {263--271},
      ISSN = {0025-2611},
      CODEN = {MSMHB2},
      MRCLASS = {14G35 (11F41 11G15)},
      MRNUMBER = {2783398},
      MRREVIEWER = {Jeffrey D. Achter},
      ZBLNUMBER = {1215.14017},
      DOI = {10.1007/s00229-011-0445-x},
      }
  • [D1] Go to document P. Deligne, "Travaux de Shimura," in Séminaire Bourbaki, 23ème Année (1970/71), Exp. No. 389, New York: Springer-Verlag, 1971, vol. 244, pp. 123-165.
    @incollection {D1, MRKEY = {0498581},
      AUTHOR = {Deligne, Pierre},
      TITLE = {Travaux de {S}himura},
      BOOKTITLE = {Séminaire {B}ourbaki, 23ème Année (1970/71), {E}xp. {N}o. 389},
      PAGES = {123--165},
      SERIES={Lecture Notes in Math.},
      VOLUME={244},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1971},
      MRCLASS = {14G25 (10D25 14K15)},
      MRNUMBER = {0498581},
      MRREVIEWER = {Author's review},
      ZBLNUMBER = {1215.14017},
      DOI = {10.1007/BFb0058700},
      }
  • [D2] P. Deligne, "Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques," in Automorphic Forms, Representations and $L$-Functions, Providence, R.I.: Amer. Math. Soc., 1979, vol. XXXIII, pp. 247-289.
    @incollection {D2, MRKEY = {0546620},
      AUTHOR = {Deligne, Pierre},
      TITLE = {Variétés de {S}himura: interprétation modulaire, et techniques de construction de modèles canoniques},
      BOOKTITLE = {Automorphic Forms, Representations and {$L$}-Functions},
      VENUE={{P}roc. {S}ympos. {P}ure {M}ath., {O}regon {S}tate {U}niv., {C}orvallis, {O}re., 1977, {P}art 2},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME={XXXIII},
      PAGES = {247--289},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      YEAR = {1979},
      MRCLASS = {10D20 (14D20 14G25 14K15)},
      MRNUMBER = {0546620},
      MRREVIEWER = {J. S. Milne},
      ZBLNUMBER = {0437.14012},
      }
  • [DMM] Go to document L. van den Dries, A. Macintyre, and D. Marker, "The elementary theory of restricted analytic fields with exponentiation," Ann. of Math., vol. 140, iss. 1, pp. 183-205, 1994.
    @article {DMM, MRKEY = {1289495},
      AUTHOR = {van den Dries, Lou and Macintyre, Angus and Marker, David},
      TITLE = {The elementary theory of restricted analytic fields with exponentiation},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {140},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {183--205},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {12L12 (03C10 03C62)},
      MRNUMBER = {1289495},
      MRREVIEWER = {Thanases Pheidas},
      ZBLNUMBER = {0437.14012},
      DOI = {10.2307/2118545},
      }
  • [DM] Go to document L. van den Dries and C. Miller, "On the real exponential field with restricted analytic functions," Israel J. Math., vol. 85, iss. 1-3, pp. 19-56, 1994.
    @article {DM, MRKEY = {1264338},
      AUTHOR = {van den Dries, Lou and Miller, Chris},
      TITLE = {On the real exponential field with restricted analytic functions},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {85},
      YEAR = {1994},
      NUMBER = {1-3},
      PAGES = {19--56},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {03C10 (03C62 12J15)},
      MRNUMBER = {1264338},
      MRREVIEWER = {A. J. Wilkie},
      ZBLNUMBER = {0823.03017},
      DOI = {10.1007/BF02758635},
      }
  • [EMO] Go to document S. J. Edixhoven, B. J. J. Moonen, and F. Oort, "Open problems in algebraic geometry," Bull. Sci. Math., vol. 125, iss. 1, pp. 1-22, 2001.
    @article {EMO, MRKEY = {1812812},
      AUTHOR = {Edixhoven, S. J. and Moonen, B. J. J. and Oort, F.},
      TITLE = {Open problems in algebraic geometry},
      JOURNAL = {Bull. Sci. Math.},
      FJOURNAL = {Bulletin des Sciences Mathématiques},
      VOLUME = {125},
      YEAR = {2001},
      NUMBER = {1},
      PAGES = {1--22},
      ISSN = {0007-4497},
      CODEN = {BSMQA9},
      MRCLASS = {14-02 (11-02 11F41 11G10 11G18)},
      MRNUMBER = {1812812},
      MRREVIEWER = {Carlo Gasbarri},
      ZBLNUMBER = {1009.11002},
      DOI = {10.1016/S0007-4497(00)01075-7},
      }
  • [EMV] Go to document M. Einsiedler, G. Margulis, and A. Venkatesh, "Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces," Invent. Math., vol. 177, iss. 1, pp. 137-212, 2009.
    @article {EMV, MRKEY = {2507639},
      AUTHOR = {Einsiedler, M. and Margulis, G. and Venkatesh, A.},
      TITLE = {Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {177},
      YEAR = {2009},
      NUMBER = {1},
      PAGES = {137--212},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {22F30 (11F55 37A17 37A45)},
      MRNUMBER = {2507639},
      MRREVIEWER = {Elon Lindenstrauss},
      ZBLNUMBER = {1176.37003},
      DOI = {10.1007/s00222-009-0177-7},
      }
  • [EGH] P. Erdös, P. M. Gruber, and J. Hammer, Lattice Points, Harlow: Longman Scientific & Technical, 1989, vol. 39.
    @book {EGH, MRKEY = {1003606},
      AUTHOR = {Erd{ö}s, P. and Gruber, P. M. and Hammer, J.},
      TITLE = {Lattice Points},
      SERIES = {Pitman Monogr. Surveys Pure Appl. Math.},
      VOLUME = {39},
      PUBLISHER = {Longman Scientific \& Technical},
      ADDRESS = {Harlow},
      YEAR = {1989},
      PAGES = {viii+184},
      ISBN = {0-582-01478-6},
      MRCLASS = {11Hxx (05Bxx 11P21 52A43)},
      MRNUMBER = {1003606},
      MRREVIEWER = {T. H. Jackson},
      }
  • [GV] Go to document A. Galligo and N. Vorobjov, "Complexity of Finding Irreducible Components of a Semialgebraic Set," J. Complexity, vol. 11, iss. 1, pp. 174-193, 1995.
    @article {GV, MRKEY = {1319055},
      AUTHOR = {Galligo, Andr{é} and Vorobjov, Nicolai},
      TITLE = {Complexity of Finding Irreducible Components of a Semialgebraic Set},
      JOURNAL = {J. Complexity},
      FJOURNAL = {Journal of Complexity},
      VOLUME = {11},
      YEAR = {1995},
      NUMBER = {1},
      PAGES = {174--193},
      ISSN = {0885-064X},
      MRCLASS = {14P10 (13P10 68Q25)},
      MRNUMBER = {1319055},
      MRREVIEWER = {Henri Lombardi},
      ZBLNUBMER = {0821.14037},
      DOI = {10.1006/jcom.1995.1007},
      }
  • [GEER] Go to document G. van der Geer, "Siegel modular forms and their applications," in The 1-2-3 of Modular Forms, New York: Springer-Verlag, 2008, pp. 181-245.
    @incollection {GEER, MRKEY = {2409679},
      AUTHOR = {van der Geer, Gerard},
      TITLE = {Siegel modular forms and their applications},
      BOOKTITLE = {The 1-2-3 of Modular Forms},
      SERIES = {Universitext},
      PAGES = {181--245},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2008},
      MRCLASS = {11F46},
      MRNUMBER = {2409679},
      MRREVIEWER = {B. Ramakrishnan},
      ZBLNUMBER = {1259.11051},
      DOI = {10.1007/978-3-540-74119-0_3},
      }
  • [HP] Go to document P. Habegger and J. Pila, "Some unlikely intersections beyond André-Oort," Compos. Math., vol. 148, iss. 1, pp. 1-27, 2012.
    @article {HP, MRKEY = {2881307},
      AUTHOR = {Habegger, P. and Pila, J.},
      TITLE = {Some unlikely intersections beyond {A}ndré-{O}ort},
      JOURNAL = {Compos. Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {148},
      YEAR = {2012},
      NUMBER = {1},
      PAGES = {1--27},
      ISSN = {0010-437X},
      MRCLASS = {11G18 (03C64 11G15 11G50)},
      MRNUMBER = {2881307},
      MRREVIEWER = {Alexandra Shlapentokh},
      ZBLNUMBER = {06007108},
      DOI = {10.1112/S0010437X11005604},
      }
  • [IGUSA] . J-i. Igusa, Theta Functions, New York: Springer-Verlag, 1972, vol. 194.
    @book {IGUSA, MRKEY = {0325625},
      AUTHOR = {Igusa, {\relax J-i}},
      TITLE = {Theta Functions},
      SERIES = {Grundlehren Math. Wiss.},
      VOLUME={194},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1972},
      PAGES = {x+232},
      MRCLASS = {14K25 (32N05)},
      MRNUMBER = {0325625},
      MRREVIEWER = {H. Klingen},
      ZBLNUMBER = {0251.14016},
      }
  • [KYAO] B. Klingler and A. Yafaev, The André-Oort conjecture.
    @misc{KYAO,
      author={Klingler, B. and Yafaev, A.},
      TITLE={The {A}ndré-{O}ort conjecture},
      NOTE={\emph{Ann. of Math.} {\bf 180} (2014), to appear},
     }
  • [K] A. Krieg, "Primitive minima of positive definite quadratic forms," Acta Arith., vol. 63, iss. 1, pp. 91-96, 1993.
    @article {K, MRKEY = {1201621},
      AUTHOR = {Krieg, Aloys},
      TITLE = {Primitive minima of positive definite quadratic forms},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Acta Arithmetica},
      VOLUME = {63},
      YEAR = {1993},
      NUMBER = {1},
      PAGES = {91--96},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {11H55 (11H50)},
      MRNUMBER = {1201621},
      MRREVIEWER = {Thomas R. Shemanske},
      ZBLNUMBER = {0780.11034},
      }
  • [KUHNE] Go to document L. Kühne, "An effective result of André-Oort type," Ann. of Math., vol. 176, iss. 1, pp. 651-671, 2012.
    @article {KUHNE, MRKEY = {2925393},
      AUTHOR = {K{ü}hne, Lars},
      TITLE = {An effective result of {A}ndré-{O}ort type},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {176},
      YEAR = {2012},
      NUMBER = {1},
      PAGES = {651--671},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11G18 (11J86)},
      MRNUMBER = {2925393},
      MRREVIEWER = {Christopher Daw},
      ZBLNUMBER = {06074024},
      DOI = {10.4007/annals.2012.176.1.13},
      }
  • [LANDAU] E. Landau, "Bemerkung zum Heilbronnschen Satz," Acta Arithmetica, vol. 1, pp. 2-18, 1935.
    @article{LANDAU,
      author={Landau, E.},
      TITLE={Bemerkung zum {H}eilbronnschen {S}atz},
      JOURNAL={Acta Arithmetica},
      VOLUME={1},
      YEAR={1935},
      PAGES={2--18},
      ZBLNUMBER = {06074024},
      }
  • [LANG] S. Lang, Introduction to Transcendental Numbers, Reading, MA: Addison-Wesley Publishing Co., 1966.
    @book {LANG, MRKEY = {0214547},
      AUTHOR = {Lang, Serge},
      TITLE = {Introduction to Transcendental Numbers},
      PUBLISHER = {Addison-Wesley Publishing Co.},
      ADDRESS={Reading, MA},
      YEAR = {1966},
      PAGES = {vi+105},
      MRCLASS = {10.32},
      MRNUMBER = {0214547},
      MRREVIEWER = {B. Volkmann},
      iZBLNUMBER = {0144.04101},
      }
  • [MZ] D. Masser and U. Zannier, "Torsion anomalous points and families of elliptic curves," Amer. J. Math., vol. 132, iss. 6, pp. 1677-1691, 2010.
    @article {MZ, MRKEY = {2766181},
      AUTHOR = {Masser, D. and Zannier, U.},
      TITLE = {Torsion anomalous points and families of elliptic curves},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {132},
      YEAR = {2010},
      NUMBER = {6},
      PAGES = {1677--1691},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {11G05},
      MRNUMBER = {2766181},
      MRREVIEWER = {Hizuru Yamagishi},
      ZBLNUMBER = {1225.11078},
      }
  • [MOONEN] B. Moonen, "Linearity properties of Shimura varieties. I," J. Algebraic Geom., vol. 7, iss. 3, pp. 539-567, 1998.
    @article {MOONEN, MRKEY = {1618140},
      AUTHOR = {Moonen, Ben},
      TITLE = {Linearity properties of {S}himura varieties. {I}},
      JOURNAL = {J. Algebraic Geom.},
      FJOURNAL = {Journal of Algebraic Geometry},
      VOLUME = {7},
      YEAR = {1998},
      NUMBER = {3},
      PAGES = {539--567},
      ISSN = {1056-3911},
      MRCLASS = {14G35},
      MRNUMBER = {1618140},
      MRREVIEWER = {Conjeeveram S. Rajan},
      ZBLNUMBER = {0956.14016},
      }
  • [OORT] F. Oort, "Canonical lifts and dense sets of CM points," in Arithmetic Geometry, Catanese, F., Ed., Cambridge: Cambridge Univ. Press, 1997, vol. XXXVII, pp. 228-234.
    @incollection{OORT,
      author={Oort, F.},
      TITLE={Canonical lifts and dense sets of {CM} points},
      BOOKTITLE={Arithmetic Geometry},
      VENUE={Scuola Normale, Cortona, October 16--21, 1994},
      PAGES={228--234},
      PUBLISHER={Cambridge Univ. Press},
      ADDRESS={Cambridge},
      EDITOR={Catanese, F.},
      SERIES={Symposia. Math.},
      VOLUME={XXXVII},
      YEAR={1997},
      MRNUMBER = {1472488},
      ZBLNUMBER = {0864.00054},
      }
  • [PEST] Go to document Y. Peterzil and S. Starchenko, "Definability of restricted theta functions and families of abelian varieties," Duke Math. J., vol. 162, iss. 4, pp. 731-765, 2013.
    @article {PEST, MRKEY = {3039679},
      AUTHOR = {Peterzil, Ya'acov and Starchenko, Sergei},
      TITLE = {Definability of restricted theta functions and families of abelian varieties},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {162},
      YEAR = {2013},
      NUMBER = {4},
      PAGES = {731--765},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {03C64 (11F46 11G18 14G35 14K25)},
      MRNUMBER = {3039679},
      MRREVIEWER = {Jean-Philippe Rolin},
      ZBLNUMBER = {06157710},
      DOI = {10.1215/00127094-2080018},
      }
  • [PILAOAO] Go to document J. Pila, "O-minimality and the André-Oort conjecture for $\Bbb C^n$," Ann. of Math., vol. 173, iss. 3, pp. 1779-1840, 2011.
    @article {PILAOAO, MRKEY = {2800724},
      AUTHOR = {Pila, Jonathan},
      TITLE = {O-minimality and the {A}ndré-{O}ort conjecture for {$\Bbb C\sp n$}},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {173},
      YEAR = {2011},
      NUMBER = {3},
      PAGES = {1779--1840},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11G18 (03C64 11U09 14G35)},
      MRNUMBER = {2800724},
      MRREVIEWER = {Alexandra Shlapentokh},
      ZBLNUMBER = {1243.14022},
      DOI = {10.4007/annals.2011.173.3.11},
      }
  • [PT] Go to document J. Pila and J. Tsimerman, "The André-Oort conjecture for the moduli space of abelian surfaces," Compos. Math., vol. 149, iss. 2, pp. 204-216, 2013.
    @article {PT, MRKEY = {3020307},
      AUTHOR = {Pila, Jonathan and Tsimerman, Jacob},
      TITLE = {The {A}ndré-{O}ort conjecture for the moduli space of abelian surfaces},
      JOURNAL = {Compos. Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {149},
      YEAR = {2013},
      NUMBER = {2},
      PAGES = {204--216},
      ISSN = {0010-437X},
      MRCLASS = {11G18 (03C64 14K10)},
      MRNUMBER = {3020307},
      MRREVIEWER = {Jae-Hyun Yang},
      ZBLNUMBER = {06144656},
      DOI = {10.1112/S0010437X12000589},
      }
  • [PW] Go to document J. Pila and A. J. Wilkie, "The rational points of a definable set," Duke Math. J., vol. 133, iss. 3, pp. 591-616, 2006.
    @article {PW, MRKEY = {2228464},
      AUTHOR = {Pila, Jonathan and Wilkie, A. J.},
      TITLE = {The rational points of a definable set},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {133},
      YEAR = {2006},
      NUMBER = {3},
      PAGES = {591--616},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {03C64 (11G99 11U09)},
      MRNUMBER = {2228464},
      MRREVIEWER = {Alexandra Shlapentokh},
      ZBLNUMBER = {1129.11029},
      DOI = {10.1215/S0012-7094-06-13336-7},
      }
  • [PZ] Go to document J. Pila and U. Zannier, "Rational points in periodic analytic sets and the Manin-Mumford conjecture," Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., vol. 19, iss. 2, pp. 149-162, 2008.
    @article {PZ, MRKEY = {2411018},
      AUTHOR = {Pila, Jonathan and Zannier, Umberto},
      TITLE = {Rational points in periodic analytic sets and the {M}anin-{M}umford conjecture},
      JOURNAL = {Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl.},
      FJOURNAL = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni},
      VOLUME = {19},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {149--162},
      ISSN = {1120-6330},
      MRCLASS = {11J95 (11G10 14G05)},
      MRNUMBER = {2411018},
      MRREVIEWER = {Pavlos Tzermias},
      ZBLNUMBER = {1164.11029},
      DOI = {10.4171/RLM/514},
      }
  • [P] Go to document R. Pink, "A combination of the conjectures of Mordell-Lang and André-Oort," in Geometric Methods in Algebra and Number Theory, Boston, MA: Birkhäuser, 2005, vol. 235, pp. 251-282.
    @incollection {P, MRKEY = {2166087},
      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},
      ADDRESS = {Boston, MA},
      YEAR = {2005},
      MRCLASS = {11G10 (11G18 14G35)},
      MRNUMBER = {2166087},
      MRREVIEWER = {Mihran Papikian},
      ZBLNUMBER = {1200.11041},
      DOI = {10.1007/0-8176-4417-2_11},
      }
  • [PR] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Boston, MA: Academic Press, 1994, vol. 139.
    @book {PR, MRKEY = {1278263},
      AUTHOR = {Platonov, Vladimir and Rapinchuk, Andrei},
      TITLE = {Algebraic Groups and Number Theory},
      SERIES = {Pure Appl. Math.},
      VOLUME = {139},
      PUBLISHER = {Academic Press},
      ADDRESS = {Boston, MA},
      YEAR = {1994},
      PAGES = {xii+614},
      ISBN = {0-12-558180-7},
      MRCLASS = {11E57 (11-02 20Gxx)},
      MRNUMBER = {1278263},
      ZBLNUMBER = {0841.20046},
      }
  • [SANSUC] Go to document J. -J. Sansuc, "Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres," J. Reine Angew. Math., vol. 327, pp. 12-80, 1981.
    @article {SANSUC, MRKEY = {0631309},
      AUTHOR = {Sansuc, J.-J.},
      TITLE = {Groupe de {B}rauer et arithmétique des groupes algébriques linéaires sur un corps de nombres},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {327},
      YEAR = {1981},
      PAGES = {12--80},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {12G05 (10C30 14G25 20G30 20G35)},
      MRNUMBER = {0631309},
      MRREVIEWER = {David Goss},
      ZBLNUBMER = {0468.14007},
      DOI = {10.1515/crll.1981.327.12},
      }
  • [SAT] I. Satake, Algebraic Structures of Symmetric Domains, Tokyo: Iwanami Shoten, 1980, vol. 4.
    @book {SAT, MRKEY = {0591460},
      AUTHOR = {Satake, Ichir{ô}},
      TITLE = {Algebraic Structures of Symmetric Domains},
      SERIES = {Kanô Mem. Lect.},
      VOLUME = {4},
      PUBLISHER = {Iwanami Shoten},
      ADDRESS = {Tokyo},
      YEAR = {1980},
      PAGES = {xvi+321},
      MRCLASS = {32-02 (17C35 32Mxx 53C35)},
      MRNUMBER = {0591460},
      MRREVIEWER = {S. Murakami},
      ZBLNUMBER = {0483.32017},
      }
  • [SIEGELAA] C. L. Siegel, "Uber die Classenzahl quadratischer Zahlkorper," Acta Arithmetica, vol. 1, pp. 83-86, 1935.
    @article{SIEGELAA,
      author = {Siegel, Carl Ludwig},
      TITLE = {Uber die {C}lassenzahl quadratischer {Z}ahlkorper},
      JOURNAL={Acta Arithmetica},
      VOLUME={1},
      YEAR={1935},
      PAGES={83--86},
      ZBLNUMBER = {0011.00903},
      }
  • [SIEGEL] C. L. Siegel, Symplectic Geometry, New York: Academic Press, 1964.
    @book {SIEGEL, MRKEY = {0164063},
      AUTHOR = {Siegel, Carl Ludwig},
      TITLE = {Symplectic Geometry},
      PUBLISHER = {Academic Press},
      ADDRESS = {New York},
      YEAR = {1964},
      PAGES = {viii+86},
      MRCLASS = {10.23 (32.65)},
      MRNUMBER = {0164063},
      ZBLNUMBER = {0138.31403},
      }
  • [JT] Go to document J. Tsimerman, "Brauer-Siegel for arithmetic tori and lower bounds for Galois orbits of special points," J. Amer. Math. Soc., vol. 25, iss. 4, pp. 1091-1117, 2012.
    @article {JT, MRKEY = {2947946},
      AUTHOR = {Tsimerman, Jacob},
      TITLE = {Brauer-{S}iegel for arithmetic tori and lower bounds for {G}alois orbits of special points},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {25},
      YEAR = {2012},
      NUMBER = {4},
      PAGES = {1091--1117},
      ISSN = {0894-0347},
      MRCLASS = {11G15 (14K15)},
      MRNUMBER = {2947946},
      MRREVIEWER = {Christopher Daw},
      ZBLNUMBER = {06195219},
      DOI = {10.1090/S0894-0347-2012-00739-5},
      }
  • [ULLMO] E. Ullmo, Quelques applications du théorème de Ax-Lindemann hyperbolique.
    @misc{ULLMO,
      author={Ullmo, Emmanuel},
      TITLE={Quelques applications du théorème de {A}x-{L}indemann hyperbolique},
      NOTE={\emph{Compositio},
      to appear},
     }
  • [UYAO] E. Ullmo and A. Yafaev, Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture.
    @misc{UYAO,
      author={Ullmo, Emmanuel and Yafaev, Andrei},
      TITLE = {Galois orbits and equidistribution of special subvarieties: towards the {A}ndré-{O}ort conjecture},
      NOTE={\emph{Ann. of Math.} \textbf{180} (2014), to appear},
      SORTYEAR={2014},
     }
  • [UYS] Go to document E. Ullmo and A. Yafaev, "A characterization of special subvarieties," Mathematika, vol. 57, iss. 2, pp. 263-273, 2011.
    @article {UYS, MRKEY = {2825237},
      AUTHOR = {Ullmo, Emmanuel and Yafaev, Andrei},
      TITLE = {A characterization of special subvarieties},
      JOURNAL = {Mathematika},
      FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics},
      VOLUME = {57},
      YEAR = {2011},
      NUMBER = {2},
      PAGES = {263--273},
      ISSN = {0025-5793},
      MRCLASS = {14G35},
      MRNUMBER = {2825237},
      MRREVIEWER = {Lei Yang},
      ZBLNUMBER = {1236.14029},
      DOI = {10.1112/S0025579311001628},
      }
  • [UYGB] E. Ullmo and A. Yafaev, Nombre de classes des tores de multiplication complexe et bornes inférieures pour orbites Galoisiennes de points spéciaux.
    @misc{UYGB,
      author = {Ullmo, Emmanuel and Yafaev, Andrei},
      TITLE = {Nombre de classes des tores de multiplication complexe et bornes inférieures pour orbites {G}aloisiennes de points spéciaux},
      NOTE={\emph{Bulletin de la SMF},
      to appear},
      ARXIV={1209.0942},
      SORTYEAR={2012},
      }
  • [UYHYPERBOLIC] E. Ullmo and A. Yafaev, Hyperbolic Ax-Lindemann theorem in the cocompact case.
    @misc{UYHYPERBOLIC,
      author = {Ullmo, Emmanuel and Yafaev, Andrei},
      TITLE = {Hyperbolic {A}x-{L}indemann theorem in the cocompact case},
      SORTYEAR={2011},
      NOTE={to appear in \emph{Duke Math. J.}},
      }
  • [WILKIE] Go to document A. J. Wilkie, "Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function," J. Amer. Math. Soc., vol. 9, iss. 4, pp. 1051-1094, 1996.
    @article {WILKIE, MRKEY = {1398816},
      AUTHOR = {Wilkie, A. J.},
      TITLE = {Model completeness results for expansions of the ordered field of real numbers by restricted {P}faffian functions and the exponential function},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {9},
      YEAR = {1996},
      NUMBER = {4},
      PAGES = {1051--1094},
      ISSN = {0894-0347},
      MRCLASS = {03C62 (03C60 03C65 14P15)},
      MRNUMBER = {1398816},
      MRREVIEWER = {Luc B{é}lair},
      ZBLNUMBER = {0892.03013},
      DOI = {10.1090/S0894-0347-96-00216-0},
      }

Authors

Jonathan Pila

Mathematical Institute, University of Oxford, UK

Jacob Tsimerman

Harvard University, Cambridge, MA