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]
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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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},
}
Full Article