André–Oort conjecture

In mathematics, the André–Oort conjecture is an open problem in number theory that generalises the Manin–Mumford conjecture. A prototypical version of the conjecture was stated by Yves André in 1989 and a more general version was conjectured by Frans Oort in 1995. The modern version is a natural generalisation of these two conjectures. In mathematics, the André–Oort conjecture is an open problem in number theory that generalises the Manin–Mumford conjecture. A prototypical version of the conjecture was stated by Yves André in 1989 and a more general version was conjectured by Frans Oort in 1995. The modern version is a natural generalisation of these two conjectures. The conjecture in its modern form is as follows. Let S be a Shimura variety and let V be a set of special points in S. Then the irreducible components of the Zariski closure of V are special subvarieties. André's first version of the conjecture was just for one dimensional subvarieties of Shimura varieties, while Oort proposed that it should work with subvarieties of the moduli space of principally polarised Abelian varieties of dimension g. Various results have been established towards the full conjecture by Ben Moonen, Yves André, Andrei Yafaev, Bas Edixhoven, Laurent Clozel, and Emmanuel Ullmo, among others. Most of these results were conditional upon the generalized Riemann hypothesis being true. In 2009, Jonathan Pila used techniques from o-minimal geometry and transcendental number theory to prove the conjecture for arbitrary products of modular curves, a result which earned him the 2011 Clay Research Award. For the case of the Siegel modular variety, work by Pila and Jacob Tsimerman resulted in a proof of the André–Oort conjecture by reducing the problem to the averaged Colmez conjecture which was subsequently proved by Xinyi Yuan and Shou-Wu Zhang and independently by Andreatta, Goren, Howard and Madapusi-Pera Just as the André–Oort conjecture can be seen as a generalisation of the Manin–Mumford conjecture, so too the André–Oort conjecture can be generalised. The usual generalisation considered is the Zilber–Pink conjecture, an open problem which combines a generalisation of the André–Oort conjecture proposed by Richard Pink and conjectures put forth by Boris Zilber.