Mathematische Annalen

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

Heights of pre-special points of Shimura varieties

Article

Abstract

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

11G18 

Notes

Acknowledgments

Both authors would like to thank Emmanuel Ullmo and Andrei Yafaev for suggesting that they work together on this problem and for numerous conversations regarding the subject of this paper. The authors also owe a special thank you to Philipp Habegger who suggested the use of Chow polynomials to prove Proposition 3.11. They are grateful to Ziyang Gao for pointing out the issue with quantifiers in Theorem 1.4 which is needed for it to imply Theorem 1.2. Both authors would like to thank the referee for their reading of the manuscript and their helpful comments. The first author is indebted to the Engineering and Physical Sciences Research Council and the Institut des Hautes Études Scientifiques for their financial support. The second author was funded by European Research Council Grant 307364 “Some problems in Geometry of Shimura varieties”.

References

  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. http://www.math.u-psud.fr/~gao
  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

Copyright information

© 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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