Mathematische Annalen

, Volume 365, Issue 3–4, pp 1305–1357 | Cite as

Heights of pre-special points of Shimura varieties



Let s be a special point on a Shimura variety, and x a pre-image of s in a fixed fundamental set of the associated Hermitian symmetric domain. We prove that the height of x is polynomially bounded with respect to the discriminant of the centre of the endomorphism ring of the corresponding \(\mathbb {Z}\)-Hodge structure. Our bound is the final step needed to complete a proof of the André–Oort conjecture under the conjectural lower bounds for the sizes of Galois orbits of special points, using a strategy of Pila and Zannier.

Mathematics Subject Classification



  1. 1.
    Atiyah, M.F., Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. (1969)Google Scholar
  2. 2.
    Bombieri, E., Masser, D., Zannier, U.: Anomalous subvarieties—structure theorems and applications. Int. Math. Res. Not. 2007 (2007)Google Scholar
  3. 3.
    Borel, A.: Introduction aux groupes arithmétiques. Publications de l’Institut de Mathématique de l’Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341. Hermann, Paris (1969)Google Scholar
  4. 4.
    Daw, C.: The André-Oort conjecture via o-minimality. In: G.O. Jones, A.J. Wilkie (eds.) O-minimality and Diophantine Geometry, London Mathematical Society Lecture Note Series 421. Cambridge University Press (2015)Google Scholar
  5. 5.
    Daw, C., Harris, A.: Categoricity of modular and Shimura curves. J. Inst. Math. Jussieu. (2015, To appear)Google Scholar
  6. 6.
    Daw, C., Yafaev, A.: An unconditional proof of the André-Oort conjecture for Hilbert modular surfaces. Manuscr. Math. 135, 263–271 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Deligne, P.: Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. In: Automorphic forms, representations and \(L\)-functions (Part 2), Proc. Sympos. Pure Math., XXXIII, pp. 247–289. Amer. Math. Soc., Providence (1979)Google Scholar
  8. 8.
    Demazure, M., Grothendieck, A.: Schémas en groupes, SGA 3. IHES: Exp . VIII-XIV, Fasc. 4 (1963)Google Scholar
  9. 9.
    Edixhoven, B., Yafaev, A.: Subvarieties of Shimura varieties. Ann. Math. 157(2), 621–645 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gao, Z.: About the mixed André–Oort conjecture: reduction to a lower bound for the pure case. Unpublished note.
  11. 11.
    Gao, Z.: Towards the André–Oort conjecture for mixed Shimura varieties: the Ax–Lindemann theorem and lower bounds for Galois orbits of special points. J. Reine Angew. Math. (2015, To appear)Google Scholar
  12. 12.
    Habegger, P.: Intersecting subvarieties of \({\mathbb{G}}_m^n\) with algebraic subgroups. Math. Ann. 342(2), 449–466 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Klingler, B., Ullmo, E., Yafaev, A.: The hyperbolic Ax–Lindemann–Weierstrass conjecture. arXiv:1307.3965
  14. 14.
    Klingler, B., Yafaev, A.: The André-Oort conjecture. Ann. Math. (2) 180(3), 867–925 (2014)Google Scholar
  15. 15.
    Nesterenko, J.V.: Estimates for the orders of zeros of functions of a certain class and applications in the theory of transcendental numbers. Math. USSR Izv. 11(2), 239–270 (1977)CrossRefMATHGoogle Scholar
  16. 16.
    Peterzil, Y., Starchenko, S.: Tame complex analysis and o-minimality. In: Proceedings of the ICM, Hyderabad (2010)Google Scholar
  17. 17.
    Philippon, P.: Sur des hauteurs alternatives. III. J. Math. Pures Appl. (9) 74(4), 345–365 (1995)Google Scholar
  18. 18.
    Pila, J.: Algebraic points of definable sets and results of André-Oort-Manin-Mumford type. IMRN 13, 2476–2507 (2009)MathSciNetMATHGoogle Scholar
  19. 19.
    Pila, J.: O-minimality and the André-Oort conjecture for \(\mathbb{C}^n\). Ann. Math. 173(3), 1779–1840 (2011)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Pila, J., Tsimerman, J.: The André-Oort conjecture for the moduli space of abelian surfaces. Composit. Math. 149(2), 204–216 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Pila, J., Tsimerman, J.: Ax–Lindemann for \({\cal A}_{g}\). Ann. Math. (2) 179(2), 659–681 (2014)Google Scholar
  22. 22.
    Pila, J., Wilkie, A.: The rational points of a definable set. Duke Math. J. 133, 591–616 (2006)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Pila, J., Zannier, U.: Rational points in periodic analytic sets and the Manin-Mumford conjecture. Rend. Mat. Acc. Lincei 19, 149–162 (2008)MathSciNetMATHGoogle Scholar
  24. 24.
    Ratazzi, N., Ullmo, E.: Galois+equidistribution=Manin-Mumford. In: Ecole d’été arithmetic geometry, Clay Math. Proc. Amer. Math. Soc. (2009)Google Scholar
  25. 25.
    Tsimerman, J.: A proof of the André-Oort conjecture for \({\cal A}_{g}\). arXiv:1506.01466
  26. 26.
    Tsimerman, J.: Brauer-Siegel for arithmetic tori and lower bounds for Galois orbits of special points. J. Am. Math. Soc. 25, 1091–1117 (2012)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Ullmo, E.: Applications du théorème d’Ax-Lindemann hyperbolique. Compos. Math. 150(2), 175–190 (2014)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Ullmo, E., Yafaev, A.: A characterisation of special subvarieties. Mathematika 57(2), 263–273 (2011)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Ullmo, E., Yafaev, A.: Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture. Ann. Math. (2) 180(3), 823–865 (2014)Google Scholar
  30. 30.
    Ullmo, E., Yafaev, A.: The hyperbolic Ax-Lindemann theorem in the compact case. Duke Math. J. 163(2), 267–463 (2014)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Ullmo, E., Yafaev, A.: Nombre de classes des tores de multiplication complexe et bornes inférieures pour les orbites galoisiennes de points spéciaux. Bull. Soc. Math. France 143(1), 197–228 (2015)MathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance
  2. 2.Department of MathematicsImperial College LondonLondonUK

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