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The hyperbolic Ax-Lindemann-Weierstraß conjecture

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References

  1. A. Ash, D. Mumford, M. Rapoport and Y. Tai, Smooth Compactification of Locally Symmetric Varieties, Lie Groups: History Frontiers and Applications, vol. 4, Math. Sci. Press, Brookline, 1975.

    MATH  Google Scholar 

  2. J. Ax, On Schanuel’s conjecture, Ann. Math., 93 (1971), 1–24.

    Article  MathSciNet  MATH  Google Scholar 

  3. W. L. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. Math., 84 (1966), 442–528.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Borel, Introduction aux Groupes Arithmétiques, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, vol. 1341, Hermann, Paris, 1969.

    MATH  Google Scholar 

  5. C. Daw and M. Orr, Heights of pre-special points of Shimura varieties, Math. Ann., to appear, arXiv:1502.00822.

  6. P. Deligne, Variétés de Shimura: interprétation modulaire et techniques de construction de modèles canoniques, in A. Borel and W. Casselman (eds.) Automorphic Forms, Representations, and \(L\) -Functions, Part. 2, Proc. of Symp. in Pure Math., vol. 33, pp. 247–290, Am. Math. Soc., Providence, 1979.

    Chapter  Google Scholar 

  7. L. van den Dries, Tame Topology and o-Minimal Structures, LMS Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.

    Book  MATH  Google Scholar 

  8. L. van den Dries and C. Miller, On the real exponential field with restricted analytic functions, Isr. J. Math., 85 (1994), 19–56.

    Article  MathSciNet  MATH  Google Scholar 

  9. B. Edixhoven and A. Yafaev, Subvarieties of Shimura varieties, Ann. Math., 157 (2003), 621–645.

    Article  MathSciNet  MATH  Google Scholar 

  10. E. Fortuna and S. Lojasiewicz, Sur l’algébricité des ensembles analytiques complexes, J. Reine Angew. Math., 329 (1981), 215–220.

    MathSciNet  MATH  Google Scholar 

  11. J. M. Hwang and W. K. To, Volumes of complex analytic subvarieties of Hermitian symmetric spaces, Am. J. Math., 124 (2002), 1221–1246.

    Article  MathSciNet  MATH  Google Scholar 

  12. B. Klingler and A. Yafaev, The André-Oort conjecture, Ann. Math., 180 (2014), 867–925.

    Article  MathSciNet  MATH  Google Scholar 

  13. F. Lindemann, Über die Zahl \(\pi\), Math. Ann., 20 (1882), 213–225.

    Article  MathSciNet  Google Scholar 

  14. G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grengebiete, vol. 17, Springer, Berlin, 1991.

    Book  MATH  Google Scholar 

  15. N. Mok, Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds, Series in Pure Math., vol. 6, World Scientific, Singapore, 1989.

    MATH  Google Scholar 

  16. N. Mok, On the Zariski closure of a germ of totally geodesic complex submanifold on a subvariety of a complex hyperbolic space form of finite volume, in Complex Analysis, Trends Math., vol. 2-79-300, Springer, Berlin, 2010.

    Google Scholar 

  17. N. Mok, Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric, J. Eur. Math. Soc., 14 (2012), 1617–1656.

    Article  MathSciNet  MATH  Google Scholar 

  18. B. Moonen, Linearity properties of Shimura varieties. I, J. Algeb. Geom., 7 (1998), 539–567.

    MathSciNet  MATH  Google Scholar 

  19. D. Mumford, Hirzebruch’s proportionality theorem in the non-cocompact case, Invent. Math., 42 (1979), 239–272.

    Article  MathSciNet  MATH  Google Scholar 

  20. Y. Peterzil and S. Starchenko, Definability of restricted theta functions and families of Abelian varieties, Duke Math. J., 162 (2013), 731–765.

    Article  MathSciNet  MATH  Google Scholar 

  21. Y. Peterzil and S. Starchenko, Tame complex analysis and o-minimality, in Proceedings of the ICM, Hyderabad, 2010. Available on first author’s web-page.

    Google Scholar 

  22. I. I. Pyateskii-Shapiro, Automorphic Functions and the Geometry of Classical Domains, Mathematics and Its Applications, vol. 8, Gordon & Breach, New York, 1969, translated from the Russian.

    Google Scholar 

  23. J. Pila, O-minimality and the Andre-Oort conjecture for \({ \mathbf{C}}^{n}\), Ann. Math., 173 (2011), 1779–1840.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Pila and A. Wilkie, The rational points on a definable set, Duke Math. J., 133 (2006), 591–616.

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Pila and J. Tsimerman, The André-Oort conjecture for the moduli space of abelian surfaces, Compos. Math., 149 (2013), 204–216.

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Pila and J. Tsimerman, Ax-Lindemann for \({\mathcal{A}}_{g}\), Ann. Math., 179 (2014), 659–681.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Pila and U. Zannier, Rational points in periodic analytic sets and the Manin-Mumford conjecture, Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl., 19 (2008), 149–162.

    Article  MathSciNet  MATH  Google Scholar 

  28. I. Satake, Algebraic Structures of Symmetric Domains, Kanu Memorial Lectures, vol. 4, Iwanami Shoten and Princeton University Press, Tokyo and Princeton, 1980.

    MATH  Google Scholar 

  29. T. Scanlon, O-minimality as an approach to the André-Oort conjecture, preprint (2012). Available on author’s web-page.

  30. E. Ullmo, Applications du théorème d’Ax-Lindemann hyperbolique, Compos. Math., 150 (2014), 175–190.

    Article  MathSciNet  MATH  Google Scholar 

  31. E. Ullmo and A. Yafaev, Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture, Ann. Math., 180 (2014), 823–865.

    Article  MathSciNet  MATH  Google Scholar 

  32. E. Ullmo and A. Yafaev, A characterisation of special subvarieties, Mathematika, 57 (2011), 263–273.

    Article  MathSciNet  MATH  Google Scholar 

  33. E. Ullmo and A. Yafaev, Nombre de classes des tores de multiplication complexe et bornes inférieures pour orbites Galoisiennes de points spéciaux, Bull. Soc. Math. Fr., 143 (2015), 197–228.

    MathSciNet  Google Scholar 

  34. E. Ullmo and A. Yafaev, Hyperbolic Ax-Lindemann theorem in the cocompact case, Duke Math. J., 163 (2014), 433–463.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Tsimermann, Brauer-Siegel theorem for tori and lower bounds for Galois orbits of special points, J. Am. Math. Soc., 25 (2012), 1091–1117.

    Article  Google Scholar 

  36. K. Weierstraß, Zu Lindemanns Abhandlung: “Über die Ludolph’sche Zahl”, Berl. Ber. (1885), 1067–1086.

  37. J. A. Wolf and A. Korányi, Generalized Cailey transformations of bounded symmetric domains, Am. J. Math., 87 (1965), 899–939.

    Article  MATH  Google Scholar 

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Klingler, B., Ullmo, E. & Yafaev, A. The hyperbolic Ax-Lindemann-Weierstraß conjecture. Publ.math.IHES 123, 333–360 (2016). https://doi.org/10.1007/s10240-015-0078-9

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