Abstract
We give an unconditional proof of the André–Oort conjecture for Hilbert modular surfaces asserting that an algebraic curve contained in such a surface and containing an infinite set of special points, is special. The proof relies on a combination of Galois-theoretic techniques and results from the theory of o-minimal structures.
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References
Bombieri E., Gubler W.: Heights in Diophantine Goemetry. Cambridge University Press, Cambridge (2006)
Cox D.: Primes of the form x 2 + ny 2. Wiley, New York (1989)
Edixhoven, B.: The André–Oort conjecture for Hilbert modular surfaces. Moduli of abelian varieties (Texel Island, 1999). Progress in Mathematics, vol. 195, pp. 133–155. Birkhäuser, Basel (2001)
Edixhoven B., Yafaev A.: Subvarieties of Shimura varieties. Ann. Math. (2) 157(2), 621–645 (2003)
Freitag E.: Hilbert Modular Forms. Springer, Berlin (1990)
Klingler, B., Yafaev, A.: The André–Oort conjecture. Available at www.math.jussieu.fr/~klingler (2006) Preprint
Moonen B.: Linearity properties of Shimura varieties. I. J. Algebr. Geom. 7(3), 539–567 (1998)
Noot, R.: Correspondances de Hecke, action de Galois et la conjecture d’André–Oort (d’aprs̀ Edixhoven et Yafaev) Séminaire Bourbaki, vol. 2004/2005. Astérisque No. 307, Exp. No. 942, vii, 165–197 (2006)
Peterzil, Y., Starchenko, S.: Tame complex analysis and o-minimality. Proceedings of the ICM, Hyderabad, 2010. Available on first author’s web-page
Pila J., Wilkie A.J.: The rational points of a definable set. Duke Math. J. 133, 591–616 (2006)
Pila J.: Rational points of definable sets and results of André–Oort–Manin–Mumford type. IMRN (13), 2476–2507 (2009)
Pila J.: On algebraic points of a definable set. Sel. Math. New Ser. 15, 151–170 (2009)
Pila, J.: o-minimality and André–Oort conjecture for \({{\mathbb C}^n}\) . Ann. Math. (2010) Preprint
Ullmo, E., Yafaev, A.: A characterisation of special subvarieties. Mathematica. Available at www.ucl.ac.uk/~ucahaya (2011)
Ullmo, E., Yafaev, A.: Galois orbits and equidistribution: towards the André–Oort conjecture. Available at www.math.u-psud.fr/~ullmo (2006) Preprint
Van Der Geer G., Ueno K.: Families of abelian surfaces with real multiplications over Hilbert modular surfaces. Nagoya Math. J. 88, 17–53 (1982)
Van Der Geer G.: Hilbert Modular Surfaces. Ergeb. Math. Grenzeb. (3). Springer-Verlag, Berlin (1987)
Yafaev, A.: The André–Oort conjecture-a survey. L-functions and Galois representations. London Mathematical Society. Lecture Note Series, vol. 320, pp. 381-406. Cambridge University Press, Cambridge (2007)
Yafaev A.: Galois orbits and equidistribution: Manin–Mumford and André–Oort. J. Théorie de Nombres de Bordeaux 21(2), 493–502 (2009)
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Daw, C., Yafaev, A. An unconditional proof of the André–Oort conjecture for Hilbert modular surfaces. manuscripta math. 135, 263–271 (2011). https://doi.org/10.1007/s00229-011-0445-x
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DOI: https://doi.org/10.1007/s00229-011-0445-x