The hyperbolic Ax-Lindemann-Weierstraß conjecture

Klingler, B.; Ullmo, E.; Yafaev, A.

Publications Mathématiques de l'IHÉS, Tome 123 (2016), pp. 333-360.

[1.] Ash, A.; Mumford, D.; Rapoport, M.; Tai, Y. Smooth Compactification of Locally Symmetric Varieties (1975) | Zbl

[2.] Ax, J. On Schanuel’s conjecture, Ann. Math., Volume 93 (1971), pp. 1-24 | DOI | MR | Zbl

[3.] Baily, W. L.; Borel, A. Compactification of arithmetic quotients of bounded symmetric domains, Ann. Math., Volume 84 (1966), pp. 442-528 | DOI | MR | Zbl

[4.] Borel, A. Introduction aux Groupes Arithmétiques (1969) | Zbl

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

[6.] Deligne, P. Variétés de Shimura: interprétation modulaire et techniques de construction de modèles canoniques, Automorphic Forms, Representations, and $L$-Functions, Part. 2 (1979), pp. 247-290 | DOI | Zbl

[7.] van den Dries, L. Tame Topology and o-Minimal Structures (1998) | DOI | MR | Zbl

[8.] van den Dries, L.; Miller, C. On the real exponential field with restricted analytic functions, Isr. J. Math., Volume 85 (1994), pp. 19-56 | DOI | MR | Zbl

[9.] Edixhoven, B.; Yafaev, A. Subvarieties of Shimura varieties, Ann. Math., Volume 157 (2003), pp. 621-645 | DOI | MR | Zbl

[10.] Fortuna, E.; Lojasiewicz, S. Sur l’algébricité des ensembles analytiques complexes, J. Reine Angew. Math., Volume 329 (1981), pp. 215-220 | MR | Zbl

[11.] Hwang, J. M.; To, W. K. Volumes of complex analytic subvarieties of Hermitian symmetric spaces, Am. J. Math., Volume 124 (2002), pp. 1221-1246 | DOI | MR | Zbl

[12.] Klingler, B.; Yafaev, A. The André-Oort conjecture, Ann. Math., Volume 180 (2014), pp. 867-925 | DOI | MR | Zbl

[13.] Lindemann, F. Über die Zahl $\pi$, Math. Ann., Volume 20 (1882), pp. 213-225 | DOI | JFM | MR

[14.] Margulis, G. A. Discrete Subgroups of Semisimple Lie Groups (1991) | DOI | MR | Zbl

[15.] Mok, N. Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds (1989) | DOI | MR | Zbl

[16.] Mok, N. On the Zariski closure of a germ of totally geodesic complex submanifold on a subvariety of a complex hyperbolic space form of finite volume, Complex Analysis (2010) | DOI | MR | Zbl

[17.] Mok, N. Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric, J. Eur. Math. Soc., Volume 14 (2012), pp. 1617-1656 | DOI | MR | Zbl

[18.] Moonen, B. Linearity properties of Shimura varieties. I, J. Algeb. Geom., Volume 7 (1998), pp. 539-567 | MR | Zbl

[19.] Mumford, D. Hirzebruch’s proportionality theorem in the non-cocompact case, Invent. Math., Volume 42 (1979), pp. 239-272 | DOI | MR | Zbl

[20.] Peterzil, Y.; Starchenko, S. Definability of restricted theta functions and families of Abelian varieties, Duke Math. J., Volume 162 (2013), pp. 731-765 | DOI | MR | Zbl

[21.] Peterzil, Y.; Starchenko, S. Tame complex analysis and o-minimality, Proceedings of the ICM (2010) (Available on first author’s web-page) | MR | Zbl

[22.] Pyateskii-Shapiro, I. I. Automorphic Functions and the Geometry of Classical Domains (1969) (translated from the Russian) | MR | Zbl

[23.] Pila, J. O-minimality and the Andre-Oort conjecture for ${\mathbf{C}}^n$, Ann. Math., Volume 173 (2011), pp. 1779-1840 | DOI | MR | Zbl

[24.] Pila, J.; Wilkie, A. The rational points on a definable set, Duke Math. J., Volume 133 (2006), pp. 591-616 | DOI | MR | Zbl

[25.] Pila, J.; Tsimerman, J. The André-Oort conjecture for the moduli space of abelian surfaces, Compos. Math., Volume 149 (2013), pp. 204-216 | DOI | MR | Zbl

[26.] Pila, J.; Tsimerman, J. Ax-Lindemann for ${\mathcal{A}}_g$, Ann. Math., Volume 179 (2014), pp. 659-681 | DOI | MR | Zbl

[27.] Pila, J.; Zannier, U. Rational points in periodic analytic sets and the Manin-Mumford conjecture, Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl., Volume 19 (2008), pp. 149-162 | DOI | MR | Zbl

[28.] Satake, I. Algebraic Structures of Symmetric Domains (1980) | MR | Zbl

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

[30.] Ullmo, E. Applications du théorème d’Ax-Lindemann hyperbolique, Compos. Math., Volume 150 (2014), pp. 175-190 | DOI | MR | Zbl

[31.] Ullmo, E.; Yafaev, A. Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture, Ann. Math., Volume 180 (2014), pp. 823-865 | DOI | MR | Zbl

[32.] Ullmo, E.; Yafaev, A. A characterisation of special subvarieties, Mathematika, Volume 57 (2011), pp. 263-273 | DOI | MR | Zbl

[33.] Ullmo, E.; Yafaev, A. Nombre de classes des tores de multiplication complexe et bornes inférieures pour orbites Galoisiennes de points spéciaux, Bull. Soc. Math. Fr., Volume 143 (2015), pp. 197-228 | DOI | MR | Zbl

[34.] Ullmo, E.; Yafaev, A. Hyperbolic Ax-Lindemann theorem in the cocompact case, Duke Math. J., Volume 163 (2014), pp. 433-463 | DOI | MR | Zbl

[35.] Tsimermann, J. Brauer-Siegel theorem for tori and lower bounds for Galois orbits of special points, J. Am. Math. Soc., Volume 25 (2012), pp. 1091-1117 | DOI | MR | Zbl

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

[37.] Wolf, J. A.; Korányi, A. Generalized Cailey transformations of bounded symmetric domains, Am. J. Math., Volume 87 (1965), pp. 899-939 | DOI | MR | Zbl