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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.
J. Ax, On Schanuel’s conjecture, Ann. Math., 93 (1971), 1–24.
W. L. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. Math., 84 (1966), 442–528.
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.
C. Daw and M. Orr, Heights of pre-special points of Shimura varieties, Math. Ann., to appear, arXiv:1502.00822.
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.
L. van den Dries, Tame Topology and o-Minimal Structures, LMS Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.
L. van den Dries and C. Miller, On the real exponential field with restricted analytic functions, Isr. J. Math., 85 (1994), 19–56.
B. Edixhoven and A. Yafaev, Subvarieties of Shimura varieties, Ann. Math., 157 (2003), 621–645.
E. Fortuna and S. Lojasiewicz, Sur l’algébricité des ensembles analytiques complexes, J. Reine Angew. Math., 329 (1981), 215–220.
J. M. Hwang and W. K. To, Volumes of complex analytic subvarieties of Hermitian symmetric spaces, Am. J. Math., 124 (2002), 1221–1246.
B. Klingler and A. Yafaev, The André-Oort conjecture, Ann. Math., 180 (2014), 867–925.
F. Lindemann, Über die Zahl \(\pi\), Math. Ann., 20 (1882), 213–225.
G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grengebiete, vol. 17, Springer, Berlin, 1991.
N. Mok, Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds, Series in Pure Math., vol. 6, World Scientific, Singapore, 1989.
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.
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.
B. Moonen, Linearity properties of Shimura varieties. I, J. Algeb. Geom., 7 (1998), 539–567.
D. Mumford, Hirzebruch’s proportionality theorem in the non-cocompact case, Invent. Math., 42 (1979), 239–272.
Y. Peterzil and S. Starchenko, Definability of restricted theta functions and families of Abelian varieties, Duke Math. J., 162 (2013), 731–765.
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.
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.
J. Pila, O-minimality and the Andre-Oort conjecture for \({ \mathbf{C}}^{n}\), Ann. Math., 173 (2011), 1779–1840.
J. Pila and A. Wilkie, The rational points on a definable set, Duke Math. J., 133 (2006), 591–616.
J. Pila and J. Tsimerman, The André-Oort conjecture for the moduli space of abelian surfaces, Compos. Math., 149 (2013), 204–216.
J. Pila and J. Tsimerman, Ax-Lindemann for \({\mathcal{A}}_{g}\), Ann. Math., 179 (2014), 659–681.
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.
I. Satake, Algebraic Structures of Symmetric Domains, Kanu Memorial Lectures, vol. 4, Iwanami Shoten and Princeton University Press, Tokyo and Princeton, 1980.
T. Scanlon, O-minimality as an approach to the André-Oort conjecture, preprint (2012). Available on author’s web-page.
E. Ullmo, Applications du théorème d’Ax-Lindemann hyperbolique, Compos. Math., 150 (2014), 175–190.
E. Ullmo and A. Yafaev, Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture, Ann. Math., 180 (2014), 823–865.
E. Ullmo and A. Yafaev, A characterisation of special subvarieties, Mathematika, 57 (2011), 263–273.
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.
E. Ullmo and A. Yafaev, Hyperbolic Ax-Lindemann theorem in the cocompact case, Duke Math. J., 163 (2014), 433–463.
J. Tsimermann, Brauer-Siegel theorem for tori and lower bounds for Galois orbits of special points, J. Am. Math. Soc., 25 (2012), 1091–1117.
K. Weierstraß, Zu Lindemanns Abhandlung: “Über die Ludolph’sche Zahl”, Berl. Ber. (1885), 1067–1086.
J. A. Wolf and A. Korányi, Generalized Cailey transformations of bounded symmetric domains, Am. J. Math., 87 (1965), 899–939.
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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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DOI: https://doi.org/10.1007/s10240-015-0078-9