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A CHARACTERIZATION OF SPECIAL SUBVARIETIES

Part of: Arithmetic problems. Diophantine geometry Global differential geometry

Published online by Cambridge University Press:  07 March 2011

Emmanuel Ullmo  and
Andrei Yafaev
Emmanuel Ullmo
Affiliation:
Departement de Mathématique, Université de Paris-Sud, Orsay, France (email: [email protected])
Andrei Yafaev
Affiliation:
Department of Mathematics, University College London, U.K. (email: [email protected])
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Abstract

We prove that an algebraic subvariety of a Shimura variety is weakly special if and only if analytic components of its preimage in the symmetric space are algebraic. We also prove an analogous result in the case of abelian varieties.

MSC classification

Secondary: 14G35: Modular and Shimura varieties 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture) 53C35: Symmetric spaces
Type
Research Article
Information
Mathematika , Volume 57 , Issue 2 , July 2011 , pp. 263 - 273
Copyright
Copyright © University College London 2011

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References

[1]André, Y., Mumford–Tate groups of mixed Hodge structures and the theorem of the fixed part. Compositio Math. 82(1) (1992), 1–24.Google Scholar
[2]Cohen, P., Humbert surfaces and transcendence properties of automorphic functions. Rocky Mountain J. Math. 26(3) (1996), 987–1001, Symposium on Diophantine Problems (Boulder, CO, 1994).CrossRefGoogle Scholar
[3]Deligne, P., Travaux de Shimura. In Séminaire Bourbaki, Exposé 389, Fevrier 1971 (Lecture Notes in Mathematics 244), Springer (Berlin, 1971), 123–165.Google Scholar
[4]Deligne, P., La conjecture de Weil pour les surfaces K3. Invent. Math. 15 (1972), 206–226.CrossRefGoogle Scholar
[5]Deligne, P., Variétés de Shimura: interprétation modulaire et techniques de construction de modèles canoniques, dans. In Automorphic Forms, Representations, and L-functions Part. 2 (Proceedings of Symposia in Pure Mathematics 33) (eds Borel, A. and Casselman, W.), American Mathematical Society (Providence, RI, 1979), 247–290.CrossRefGoogle Scholar
[6]Edixhoven, B., Special points on products of two modular curves. Compositio Math. 114(3) (1998), 315–328.CrossRefGoogle Scholar
[7]Edixhoven, B., The André–Oort conjecture for Hilbert modular surfaces. In Moduli of Abelian Varieties (Texel Island, 1999) (Progress in Mathematics 195), Birkhäuser (Basel, 2001), 133–155.CrossRefGoogle Scholar
[8]Edixhoven, B. and Yafaev, A., Subvarieties of Shimura varieties. Ann. of Math. (2) 157(2) (2003), 621–645.CrossRefGoogle Scholar
[9]Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces (Pure and Applied Mathematics 80), Academic Press (New York, 1978).Google Scholar
[10]Klingler, B. and Yafaev, A., The André–Oort conjecture. Preprint, available athttp://people.math.jussieu.fr/∼klingler/papers.html.Google Scholar
[11]Korányi, A. and Wolf, J. A., Generalized Cayley transformation of bounded symmetric domains. Amer. J. Math. 87 (1965), 899–939.Google Scholar
[12]Milne, J., Introduction to Shimura varieties. In Harmonic Analysis, the Trace Formula, and Shimura Varieties (Clay Mathematics Proceedings 4), American Mathematical Society (Providence, RI, 2005), 265–378.Google Scholar
[13]Milne, J. S., Canonical models of (mixed) Shimura varieties and automorphic vector bundles. In Automorphic Forms, Shimura Varieties, and L-functions (Ann Arbor, MI, 1988), Vol. I (Perspectives in Mathematics 10), Academic press (Boston, MA, 1990), 283–414.Google Scholar
[14]Mok, N., Metric Rigidity Theorems on Hermitian Locally Symmetric Submanifolds (Series in Pure Mathematics 6), World Scientific (Singapore, 1989).CrossRefGoogle Scholar
[15]Moonen, B., Linearity properties of Shimura varieties I. J. Algebraic Geom. 7(3) (1998), 539–567.Google Scholar
[16]Moonen, B., Models of Shimura varieties in mixed characteristics. In Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996) (London Mathematical Society Lecture Note Series 254), Cambridge University Press (Cambridge, 1998), 267–350.CrossRefGoogle Scholar
[17]Piatetskii-Shapiro, I. I., Geometry of Classical Domains and Theory of Automorphic Functions, Fizmatgiz (Moscow, 1961), (in Russian); Dunot (Paris, 1966) (French translation); Gordon and Breach (New York, 1969) (English translation).Google Scholar
[18]Pila, J., Rational points of definable sets and results of André–Oort–Manin–Mumford type. Int. Math. Res. Not. IMRN 13 (2009), 2476–2507.Google Scholar
[19]Pila, J., O-minimality and the Andre–Oort conjecture for ℂn. Preprint, 2010.Google Scholar
[20]Satake, I., Algebraic Structures of Bounded Symmetric Spaces, Iwanami Shoten and Princeton University Press (Tokyo and Princeton, NJ, 1980).Google Scholar
[21]Shiga, H. and Wolfart, J., Criteria for complex multiplication and transcendence properties of automorphic functions. J. Reine Angew. Math. 463 (1995), 1–25.Google Scholar
[22]Ullmo, E. and Yafaev, A., Galois orbits and equidistribution: towards the André–Oort conjecture. Preprint, available at http://www.math.u-psud.fr/∼ullmo/.Google Scholar
[23]Voisin, C., Théorie de Hodge et géométrie algébrique complexe (Cours Spécialisés 10), Société Mathématique de France (Paris, 2002).Google Scholar