The André–Oort conjecture is an open problem in number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

 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.

Statement

The conjecture in its modern form is as follows. Let S be a Shimura variety

Shimura variety

In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. The term "Shimura variety" applies to the higher-dimensional case, in the case of...

 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

Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...

 of dimension g.

Partial results

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

Generalized Riemann hypothesis

The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function...

 being true. The biggest unconditional results came in 2009 when Jonathan Pila

Jonathan Pila

Jonathan S. Pila is an Australian mathematician at the University of Oxford. He won the 2011 Clay Research Award for his resolution of the André–Oort conjecture in the case of products of modular curves. In June of 2011, he was awarded the Senior Whitehead Prize by the London Mathematical Society...

 used techniques from o-minimal

O-minimal theory

In mathematical logic, and more specifically in model theory, an infinite structure which is totally ordered by In mathematical logic, and more specifically in model theory, an infinite structure which is totally ordered by...

 geometry and transcendence theory

Transcendence theory

Transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...

 to prove the conjecture for arbitrary products of modular curve

Modular curve

In number theory and algebraic geometry, a modular curve Y is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL...

s, a result which earned him the 2011 Clay Research Award

Clay Research Award

The Clay Research Award is given annually by the Clay Mathematics Institute to mathematicians to recognize their achievement in mathematical research...

.

Generalisations

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.