The André-Oort conjecture

Abstract

In this paper we prove, assuming the Generalized Riemann Hypothesis, the André-Oort conjecture on the Zariski closure of sets of special points in a Shimura variety. In the case of sets of special points satisfying an additional assumption, we prove the conjecture without assuming the GRH.

  • [CDK] Go to document E. Cattani, P. Deligne, and A. Kaplan, "On the locus of Hodge classes," J. Amer. Math. Soc., vol. 8, iss. 2, pp. 483-506, 1995.
    @article{CDK,
      author = {Cattani, Eduardo and Deligne, Pierre and Kaplan, Aroldo},
      journal = {J. Amer. Math. Soc.},
      number = {2},
      pages = {483--506},
      title = {On the locus of {H}odge classes},
      volume = {8},
      year = {1995},
      doi = {10.2307/2152824},
      issn = {0894-0347},
      }
  • [Vo] Go to document C. Voisin, "Hodge loci and absolute Hodge classes," Compos. Math., vol. 143, iss. 4, pp. 945-958, 2007.
    @article{Vo,
      author = {Voisin, Claire},
      journal = {Compos. Math.},
      number = {4},
      pages = {945--958},
      title = {Hodge loci and absolute {H}odge classes},
      volume = {143},
      year = {2007},
      doi = {10.1112/S0010437X07002837},
      issn = {0010-437X},
      }
  • [De1] 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{De1, address = {New York},
      author = {Deligne, Pierre},
      booktitle = {Séminaire {B}ourbaki, 23ème Année (1970/71), {E}xp. {N}o. 389},
      pages = {123--165},
      publisher = {Springer-Verlag},
      series = {Lecture Notes in Math.},
      title = {Travaux de {S}himura},
      volume = {244},
      year = {1971},
      doi = {10.1007/BFb0058700},
      }
  • [De2] P. Deligne, "Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques," in Automorphic Forms, Representations and $L$-Functions, Part 2, Providence, RI: Amer. Math. Soc., 1979, vol. XXXIII, pp. 247-289.
    @incollection{De2, address = {Providence, RI},
      author = {Deligne, Pierre},
      booktitle = {Automorphic Forms, Representations and {$L$}-Functions, {P}art 2},
      pages = {247--289},
      publisher = {Amer. Math. Soc.},
      series = {Proc. Sympos. Pure Math.},
      title = {Variétés de {S}himura: interprétation modulaire, et techniques de construction de modèles canoniques},
      volume = {XXXIII},
      year = {1979},
      }
  • [Milne] Milne, J. S., Introduction to Shimura varietiesProvidence, RI: Amer. Math. Soc., 2005.
    @proceedings{Milne, address = {Providence, RI},
      author = {Milne, J. S.},
      booktitle = {Harmonic analysis, the trace formula, and {S}himura varieties},
      editor = {Arthur, J. and Ellwood, D. and R. Kottwitz},
      note = {An expanded version is available for free download on author’s webpage},
      number = {4},
      pages = {265--378},
      publisher = {Amer. Math. Soc.},
      series = {Clay Math. Proc.},
      title = {Introduction to {S}himura varieties},
      year = {2005},
      }
  • [UllmoYafaev] Go to document E. Ullmo and A. Yafaev, "Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture," Ann. of Math., vol. 180, pp. 823-865, 2014.
    @article{UllmoYafaev,
      author = {Ullmo, Emmanuel and Yafaev, Andrei},
      journal = {Ann. of Math.},
      pages = {823--865},
      title = {Galois orbits and equidistribution of special subvarieties: towards the {A}ndré-{O}ort conjecture},
      volume = {180},
      year = {2014},
      doi = {10.4007/annals.2014.180.3.1},
      }
  • [UllmoRata] N. Ratazzi and E. Ullmo, "Galois + équidistribution = Manin-Mumford," in Arithmetic Geometry, Providence, RI: Amer. Math. Soc., 2009, vol. 8, pp. 419-430.
    @incollection{UllmoRata, address = {Providence, RI},
      author = {Ratazzi, Nicolas and Ullmo, Emmanuel},
      booktitle = {Arithmetic Geometry},
      pages = {419--430},
      publisher = {Amer. Math. Soc.},
      series = {Clay Math. Proc.},
      title = {Galois + équidistribution = {M}anin-{M}umford},
      volume = {8},
      year = {2009},
      }
  • [EdHilbert] B. Edixhoven, "On the André-Oort conjecture for Hilbert modular surfaces," in Moduli of Abelian Varieties, Basel: Birkhäuser, 2001, vol. 195, pp. 133-155.
    @incollection{EdHilbert, address = {Basel},
      author = {Edixhoven, Bas},
      booktitle = {Moduli of Abelian Varieties},
      pages = {133--155},
      publisher = {Birkhäuser},
      series = {Progr. Math.},
      title = {On the {A}ndré-{O}ort conjecture for {H}ilbert modular surfaces},
      volume = {195},
      year = {2001},
      }
  • [CU1] Go to document L. Clozel and E. Ullmo, "Équidistribution de sous-variétés spéciales," Ann. of Math., vol. 161, iss. 3, pp. 1571-1588, 2005.
    @article{CU1,
      author = {Clozel, Laurent and Ullmo, Emmanuel},
      journal = {Ann. of Math.},
      number = {3},
      pages = {1571--1588},
      title = {\'{E}quidistribution de sous-variétés spéciales},
      volume = {161},
      year = {2005},
      doi = {10.4007/annals.2005.161.1571},
      issn = {0003-486X},
      }
  • [Ullmo] Go to document E. Ullmo, "Equidistribution de sous-variétés spéciales. II," J. Reine Angew. Math., vol. 606, pp. 193-216, 2007.
    @article{Ullmo,
      author = {Ullmo, Emmanuel},
      journal = {J. Reine Angew. Math.},
      pages = {193--216},
      title = {Equidistribution de sous-variétés spéciales. {II}},
      volume = {606},
      year = {2007},
      doi = {10.1515/CRELLE.2007.041},
      issn = {0075-4102},
      }
  • [EdYa] Go to document B. Edixhoven and A. Yafaev, "Subvarieties of Shimura varieties," Ann. of Math., vol. 157, iss. 2, pp. 621-645, 2003.
    @article{EdYa,
      author = {Edixhoven, Bas and Yafaev, Andrei},
      journal = {Ann. of Math.},
      number = {2},
      pages = {621--645},
      title = {Subvarieties of {S}himura varieties},
      volume = {157},
      year = {2003},
      doi = {10.4007/annals.2003.157.621},
      issn = {0003-486X},
      }
  • [Ya] Go to document A. Yafaev, "A conjecture of Yves André’s," Duke Math. J., vol. 132, iss. 3, pp. 393-407, 2006.
    @article{Ya,
      author = {Yafaev, Andrei},
      journal = {Duke Math. J.},
      number = {3},
      pages = {393--407},
      title = {A conjecture of {Y}ves {A}ndré's},
      volume = {132},
      year = {2006},
      doi = {10.1215/S0012-7094-06-13231-3},
      issn = {0012-7094},
      }
  • [Ed] Go to document B. Edixhoven, "Special points on products of modular curves," Duke Math. J., vol. 126, iss. 2, pp. 325-348, 2005.
    @article{Ed,
      author = {Edixhoven, Bas},
      journal = {Duke Math. J.},
      number = {2},
      pages = {325--348},
      title = {Special points on products of modular curves},
      volume = {126},
      year = {2005},
      doi = {10.1215/S0012-7094-04-12624-7},
      issn = {0012-7094},
      }
  • [YaMo] Go to document A. Yafaev, "On a result of Moonen on the moduli space of principally polarised abelian varieties," Compos. Math., vol. 141, iss. 5, pp. 1103-1108, 2005.
    @article{YaMo,
      author = {Yafaev, Andrei},
      journal = {Compos. Math.},
      number = {5},
      pages = {1103--1108},
      title = {On a result of {M}oonen on the moduli space of principally polarised abelian varieties},
      volume = {141},
      year = {2005},
      doi = {10.1112/S0010437X05000291},
      issn = {0010-437X},
      }
  • [Ratner] Go to document M. Ratner, "On Raghunathan’s measure conjecture," Ann. of Math., vol. 134, iss. 3, pp. 545-607, 1991.
    @article{Ratner,
      author = {Ratner, Marina},
      journal = {Ann. of Math.},
      number = {3},
      pages = {545--607},
      title = {On {R}aghunathan's measure conjecture},
      volume = {134},
      year = {1991},
      doi = {10.2307/2944357},
      issn = {0003-486X},
      }
  • [Dani-Margulis] Go to document S. G. Dani and G. A. Margulis, "Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces," Proc. Indian Acad. Sci. Math. Sci., vol. 101, iss. 1, pp. 1-17, 1991.
    @article{Dani-Margulis,
      author = {Dani, S. G. and Margulis, G. A.},
      journal = {Proc. Indian Acad. Sci. Math. Sci.},
      number = {1},
      pages = {1--17},
      title = {Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces},
      volume = {101},
      year = {1991},
      doi = {10.1007/BF02872005},
      issn = {0253-4142},
      }
  • [Zhang] Go to document S. Zhang, "Equidistribution of CM-points on quaternion Shimura varieties," Int. Math. Res. Not., vol. 59, pp. 3657-3689, 2005.
    @article{Zhang,
      author = {Zhang, Shou-Wu},
      journal = {Int. Math. Res. Not.},
      pages = {3657--3689},
      title = {Equidistribution of {CM}-points on quaternion {S}himura varieties},
      volume = {59},
      year = {2005},
      doi = {10.1155/IMRN.2005.3657},
      issn = {1073-7928},
      }
  • [UllmoMontreal] Go to document E. Ullmo, "Manin-Mumford, André-Oort, the equidistribution point of view," in Equidistribution in Number Theory, an Introduction, New York: Springer-Verlag, 2007, vol. 237, pp. 103-138.
    @incollection{UllmoMontreal, address = {New York},
      author = {Ullmo, Emmanuel},
      booktitle = {Equidistribution in Number Theory, an Introduction},
      pages = {103--138},
      publisher = {Springer-Verlag},
      series = {NATO Sci. Ser. II Math. Phys. Chem.},
      title = {Manin-{M}umford, {A}ndré-{O}ort, the equidistribution point of view},
      volume = {237},
      year = {2007},
      doi = {10.1007/978-1-4020-5404-4_7},
      }
  • [Ed2Curves] Go to document B. Edixhoven, "Special points on the product of two modular curves," Compositio Math., vol. 114, iss. 3, pp. 315-328, 1998.
    @article{Ed2Curves,
      author = {Edixhoven, Bas},
      journal = {Compositio Math.},
      number = {3},
      pages = {315--328},
      title = {Special points on the product of two modular curves},
      volume = {114},
      year = {1998},
      doi = {10.1023/A:1000539721162},
      issn = {0010-437X},
      }
  • [bor] A. Borel, Introduction aux Groupes Arithmétiques, Paris: Hermann, 1969, vol. XV.
    @book{bor, address = {Paris},
      author = {Borel, Armand},
      pages = {125},
      publisher = {Hermann},
      series = {Publ. l'Inst. Math. l'Univ. Strasbourg},
      title = {Introduction aux Groupes Arithmétiques},
      volume = {XV},
      year = {1969},
      }
  • [P] R. Pink, Arithmetical compactification of mixed Shimura varietiesBonn: Universität Bonn, Mathematisches Institut, 1990.
    @misc{P, address = {Bonn},
      author = {Pink, Richard},
      note = {dissertation, Rheinische Friedrich-Wilhelms-Universit{ä}t Bonn, Bonn, 1989},
      pages = {xviii+340},
      publisher = {Universität Bonn, Mathematisches Institut},
      series = {Bonner Math. Schriften},
      title = {Arithmetical compactification of mixed Shimura varieties},
      volume = {209},
      year = {1990},
      }
  • [SGA3] Go to document Schémas en groupes. III: Structure des schémas en groupes réductifs, Demazure, M. and Grothendieck, A., Eds., New York: Springer-Verlag, 1970, vol. 153.
    @book{SGA3, address = {New York},
      editor = {M. Demazure and A. Grothendieck},
      journal = {Lecture Notes in Mathematics},
      pages = {viii+529},
      publisher = {Springer-Verlag},
      series = {Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3)},
      title = {Schémas en groupes. {III}: {S}tructure des schémas en groupes réductifs},
      volume = {153},
      year = {1970},
      doi = {10.1007/BFb0059027},
      }
  • [weis] Go to document B. Weisfeiler, "Strong approximation for Zariski-dense subgroups of semisimple algebraic groups," Ann. of Math., vol. 120, iss. 2, pp. 271-315, 1984.
    @article{weis,
      author = {Weisfeiler, Boris},
      journal = {Ann. of Math.},
      number = {2},
      pages = {271--315},
      title = {Strong approximation for {Z}ariski-dense subgroups of semisimple algebraic groups},
      volume = {120},
      year = {1984},
      doi = {10.2307/2006943},
      issn = {0003-486X},
      }
  • [Nori] Go to document M. V. Nori, "On subgroups of ${ GL}_n({\bf {F}_p)}$," Invent. Math., vol. 88, iss. 2, pp. 257-275, 1987.
    @article{Nori,
      author = {Nori, Madhav V.},
      journal = {Invent. Math.},
      number = {2},
      pages = {257--275},
      title = {On subgroups of {${\rm GL}_n({\bf {F}_p)}$}},
      volume = {88},
      year = {1987},
      doi = {10.1007/BF01388909},
      issn = {0020-9910},
      }
  • [laz] R. Lazarsfeld, Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series, New York: Springer-Verlag, 2004, vol. 48.
    @book{laz, address = {New York},
      author = {Lazarsfeld, Robert},
      pages = {xviii+387},
      publisher = {Springer-Verlag},
      series = {Ergeb. Math. Grenzgeb.},
      title = {Positivity in Algebraic Geometry. {I}. Classical Setting: Line Bundles and Linear Series},
      volume = {48},
      year = {2004},
      isbn = {3-540-22533-1},
      }
  • [BB] Go to document W. L. Baily Jr. and A. Borel, "Compactification of arithmetic quotients of bounded symmetric domains," Ann. of Math., vol. 84, pp. 442-528, 1966.
    @article{BB,
      author = {Baily, Jr., W. L. and Borel, A.},
      journal = {Ann. of Math.},
      pages = {442--528},
      title = {Compactification of arithmetic quotients of bounded symmetric domains},
      volume = {84},
      year = {1966},
      doi = {10.2307/1970457},
      issn = {0003-486X},
      }
  • [Sat1] Go to document I. Satake, "A note on holomorphic imbeddings and compactification of symmetric domains," Amer. J. Math., vol. 90, pp. 231-247, 1968.
    @article{Sat1,
      author = {Satake, Ichir{ô}},
      journal = {Amer. J. Math.},
      pages = {231--247},
      title = {A note on holomorphic imbeddings and compactification of symmetric domains},
      volume = {90},
      year = {1968},
      doi = {10.2307/2373434},
      issn = {0002-9327},
      }
  • [Dy] Go to document E. B. Dynkin, "Semisimple subalgebras of semisimple Lie algebras," Mat. Sbornik N.S., vol. 30(72), pp. 349-462 (3 plates), 1952.
    @article{Dy,
      author = {Dynkin, E. B.},
      journal = {Mat. Sbornik N.S.},
      note = {translated in \emph{Amer. Math. Soc. Trans.} {\bf 6} (1957), 111--244},
      pages = {349--462 (3 plates)},
      title = {Semisimple subalgebras of semisimple {L}ie algebras},
      volume = {30(72)},
      year = {1952},
      url = {http://www.mathnet.ru/php/ archive.phtml?wshow=paper&jrnid=sm&paperid=5435&option_lang=eng},
      }
  • [ih] Go to document S. Ihara, "Holomorphic imbeddings of symmetric domains," J. Math. Soc. Japan, vol. 19, pp. 261-302, 1967.
    @article{ih,
      author = {Ihara, Shin-ichiro},
      journal = {J. Math. Soc. Japan},
      pages = {261--302},
      title = {Holomorphic imbeddings of symmetric domains},
      volume = {19},
      year = {1967},
      doi = {10.2969/jmsj/01930261},
      issn = {0025-5645},
      }
  • [Sat2] Go to document I. Satake, "On some properties of holomorphic imbeddings of symmetric domains," Amer. J. Math., vol. 91, pp. 289-305, 1969.
    @article{Sat2,
      author = {Satake, Ichir{ô}},
      journal = {Amer. J. Math.},
      pages = {289--305},
      title = {On some properties of holomorphic imbeddings of symmetric domains},
      volume = {91},
      year = {1969},
      doi = {10.2307/2373511},
      issn = {0002-9327},
      }
  • [MilneKazh] J. Milne, Kazhdan’s theorem on arithmetic varieties.
    @misc{MilneKazh,
      author = {Milne, J.},
      note = {1984 notes available for download from author's webpage},
      title = {Kazhdan's theorem on arithmetic varieties},
      }
  • [Mo] B. Moonen, "Linearity properties of Shimura varieties. I," J. Algebraic Geom., vol. 7, iss. 3, pp. 539-567, 1998.
    @article{Mo,
      author = {Moonen, Ben},
      journal = {J. Algebraic Geom.},
      number = {3},
      pages = {539--567},
      title = {Linearity properties of {S}himura varieties. {I}},
      volume = {7},
      year = {1998},
      issn = {1056-3911},
      }
  • [Pink] Go to document R. Pink, "Strong approximation for Zariski dense subgroups over arbitrary global fields," Comment. Math. Helv., vol. 75, iss. 4, pp. 608-643, 2000.
    @article{Pink,
      author = {Pink, Richard},
      journal = {Comment. Math. Helv.},
      number = {4},
      pages = {608--643},
      title = {Strong approximation for {Z}ariski dense subgroups over arbitrary global fields},
      volume = {75},
      year = {2000},
      doi = {10.1007/s000140050142},
      issn = {0010-2571},
      }
  • [PlaRap] V. Platonov and A. Rapinchuk, Algebraic groups and number theory, Boston: Academic Press, 1994.
    @book{PlaRap, address = {Boston},
      author = {Platonov, Vladimir and Rapinchuk, Andrei},
      note = {translated from the 1991 Russian original by Rachel Rowen},
      number = {139},
      pages = {xii+614},
      publisher = {Academic Press},
      series = {Pure Appl. Math.},
      title = {Algebraic groups and number theory},
      year = {1994},
      isbn = {0-12-558180-7},
      }
  • [bt] Go to document F. Bruhat and J. Tits, "Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée," Inst. Hautes Études Sci. Publ. Math., vol. 60, pp. 197-376, 1984.
    @article{bt,
      author = {Bruhat, F. and Tits, J.},
      journal = {Inst. Hautes Études Sci. Publ. Math.},
      pages = {197--376},
      title = {Groupes réductifs sur un corps local. {II}. {S}chémas en groupes. {E}xistence d'une donnée radicielle valuée},
      volume = {60},
      year = {1984},
      issn = {0073-8301},
      url = {http://www.numdam.org/item?id=PMIHES_1984__60__5_0},
      }
  • [bt1] F. Bruhat and J. Tits, "Groupes réductifs sur un corps local," Inst. Hautes Études Sci. Publ. Math., vol. 41, pp. 5-251, 1972.
    @article{bt1,
      author = {Bruhat, F. and Tits, J.},
      journal = {Inst. Hautes Études Sci. Publ. Math.},
      pages = {5--251},
      title = {Groupes réductifs sur un corps local},
      volume = {41},
      year = {1972},
      issn = {0073-8301},
      }
  • [car] P. Cartier, "Representations of $p$-adic groups: a survey," in Automorphic Forms, Representations and $L$-Functions, Part 1, Providence, RI: Amer. Math. Soc., 1979, vol. XXXIII, pp. 111-155.
    @incollection{car, address = {Providence, RI},
      author = {Cartier, P.},
      booktitle = {Automorphic Forms, Representations and {$L$}-Functions, {P}art 1},
      pages = {111--155},
      publisher = {Amer. Math. Soc.},
      series = {Proc. Sympos. Pure Math.},
      title = {Representations of {$p$}-adic groups: a survey},
      volume = {XXXIII},
      year = {1979},
      }
  • [Heckop] G. J. Heckman and E. M. Opdam, "Harmonic analysis for affine Hecke algebras," in Current Developments in Mathematics, 1996, Boston: Internat. Press, 1997, pp. 37-60.
    @incollection{Heckop, address = {Boston},
      author = {Heckman, G. J. and Opdam, E. M.},
      booktitle = {Current Developments in Mathematics, 1996},
      pages = {37--60},
      publisher = {Internat. Press},
      title = {Harmonic analysis for affine {H}ecke algebras},
      year = {1997},
      }
  • [tits] J. Tits, "Reductive groups over local fields," in Automorphic Forms, Representations and $L$-Functions, Part 1, Providence, RI: Amer. Math. Soc., 1979, vol. XXXIII, pp. 29-69.
    @incollection{tits, address = {Providence, RI},
      author = {Tits, J.},
      booktitle = {Automorphic Forms, Representations and {$L$}-Functions, {P}art 1},
      pages = {29--69},
      publisher = {Amer. Math. Soc.},
      series = {Proc. Sympos. Pure Math.},
      title = {Reductive groups over local fields},
      volume = {XXXIII},
      year = {1979},
      }
  • [bor1] Go to document A. Borel, "Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup," Invent. Math., vol. 35, pp. 233-259, 1976.
    @article{bor1,
      author = {Borel, Armand},
      journal = {Invent. Math.},
      pages = {233--259},
      title = {Admissible representations of a semi-simple group over a local field with vectors fixed under an {I}wahori subgroup},
      volume = {35},
      year = {1976},
      doi = {10.1007/BF01390139},
      issn = {0020-9910},
      }
  • [Fulton] Go to document W. Fulton, Intersection Theory, Second ed., New York: Springer-Verlag, 1998, vol. 2.
    @book{Fulton, address = {New York},
      author = {Fulton, William},
      edition = {Second},
      pages = {xiv+470},
      publisher = {Springer-Verlag},
      series = {Ergeb. Math. Grenzgeb.},
      title = {Intersection Theory},
      volume = {2},
      year = {1998},
      doi = {10.1007/978-1-4612-1700-8},
      isbn = {3-540-62046-X; 0-387-98549-2},
      }

Authors

Bruno Klingler

Institut de Mathématiques de Jussieu, Paris, France

Andrei Yafaev

University College London, London, UK