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The André–Oort conjecture for the moduli space of abelian surfaces

Part of: Model theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  11 December 2012

Jonathan Pila  and
Jacob Tsimerman
Jonathan Pila
Affiliation:
Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK (email: Jonathan.Pila@maths.ox.ac.uk)
Jacob Tsimerman
Affiliation:
Mathematics Department, Harvard University, One Oxford Street, Cambridge, MA 02138, USA (email: jacobt@math.harvard.edu)
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Abstract

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We provide an unconditional proof of the André–Oort conjecture for the coarse moduli space 𝒜2,1 of principally polarized abelian surfaces, following the strategy outlined by Pila–Zannier.

Keywords

André–Oort conjecturemoduli of abelian surfaces

MSC classification

Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties 03C64: Model theory of ordered structures; o-minimality
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 2 , February 2013 , pp. 204 - 216
Copyright
© The Author(s) 2012

References

[Del79]Deligne, P., Variétés de Shimura: interpretation modulaire, et techniques de construction de modèles canoniques, Proc. Symp. Pure Math. 33(part 2) (1979), 247–290.Google Scholar
[EMV09]Einsiedler, M., Margulis, G. A. and Venkatesh, A., Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces, Invent. Math. 177 (2009), 137–212.Google Scholar
[GV95]Galligo, A. and Vorobjov, N., Complexity of finding irreducible components of a semialgebraic set, J. Complexity 11 (1995), 174–193.Google Scholar
[PS11a]Peterzil, Y. and Starchenko, S., Tame complex analysis and o-minimality, in Proceedings of The International Congress of Mathematicians 2010 (ICM 2010): Vol. 2, Hyderabad, India (World Scientific, 2011).Google Scholar
[PS11b]Peterzil, Y. and Starchenko, S., Definability of restricted theta functions and families of abelian varieties, Duke Math. J., to appear, http://arxiv.org/abs/1103.3110.Google Scholar
[Pil11]Pila, J., O-minimality and the André–Oort conjecture for ℂn, Annals Math. 173 (2011), 1779–1840.Google Scholar
[PW06]Pila, J. and Wilkie, A., The rational points of a definable set, Duke Math. J. 133 (2006), 591–616.Google Scholar
[PZ08]Pila, J. and Zannier, U., Rational points in periodic analytic sets and the Manin–Mumford conjecture, Rend. Lincei Mat. Appl. 19 (2008), 149–162.Google Scholar
[Sie64]Siegel, C-L., Symplectic geometry (Academic Press, New York–London, 1964).Google Scholar
[Tsi12]Tsimerman, J., Brauer–Siegel for arithmetic tori and lower bounds for Galois orbits of special points, J. Amer Math. Soc. 25 (2012), 1091–1117.CrossRefGoogle Scholar
[UY]Ullmo, E. and Yafaev, A., Galois orbits and equidistribution of special subvarieties: towards the André–Oort conjecture, http://www.math.u-psud.fr/∼ullmo/.Google Scholar
[UY11]Ullmo, E. and Yafaev, A., A characterization of special subvarieties, Mathematika 57 (2011), 263–273.Google Scholar
[vdDM94]van den Dries, L. and Miller, C., On the real exponential field with restricted analytic functions, Israel J. Math. 85 (1994), 19–56.Google Scholar