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

@article{Pila2012TheAC,
title={The Andr{\'e}–Oort conjecture for the moduli space of abelian surfaces},
author={Jonathan Pila and Jacob Tsimerman},
journal={Compositio Mathematica},
year={2012},
volume={149},
pages={204 - 216}
}
• Published 20 June 2011
• Mathematics
• Compositio Mathematica
Abstract 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.
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## References

SHOWING 1-10 OF 17 REFERENCES
Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture
• Mathematics
• 2014
In this paper we develop a strategy and some technical tools for proving the Andr e-Oort conjecture. We give lower bounds for the degrees of Galois orbits of geometric components of special
Definability of restricted theta functions and families of abelian varieties
• Mathematics
• 2011
We consider some classical maps from the theory of abelian varieties and their moduli spaces and prove their definability, on restricted domains, in the o-minimal structure $\Rae$. In particular, we
Symplectic Geometry
These are lecture notes for two courses, taught at the University of Toronto in Spring 1998 and in Fall 2000. Our main sources have been the books " Symplectic Techniques " by Guillemin-Sternberg and
Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces
• Mathematics
• 2007
We prove effective equidistribution, with polynomial rate, for large closed orbits of semisimple groups on homogeneous spaces, under certain technical restrictions (notably, the acting group should
Brauer-Siegel for arithmetic tori and lower bounds for Galois orbits of special points
In \cite{S}, Shyr derived an analogue of Dirichlet's class number formula for arithmetic Tori. We use this formula to derive a Brauer-Siegel formula for Tori, relating the Discriminant of a torus to
The rational points of a definable set
• Mathematics
• 2006
Let $X\R^n$ be a set that is definable in an o-minimal structure over $R$. This article shows that in a suitable sense, there are very few rational points of $X$ which do not lie on some connected
O-minimality and the André-Oort conjecture for $\mathbb{C}^{n}$
We give an unconditional proof of the Andre-Oort conjecture for arbitrary products of modular curves. We establish two generalizations. The first includes the Manin-Mumford conjecture for arbitrary
Rational points in periodic analytic sets and the Manin-Mumford conjecture
• Mathematics
• 2008
We present a new proof of the Manin-Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points
Tame Complex Analysis and o-minimality
• Mathematics
• 2011
We describe here a theory of holomorphic functions and analytic manifolds, restricted to the category of definable objects in an o-minimal structure which expands a real closed field R. In this