The Andre-Oort conjecture, in Diophantine geometry, concerns the characterization of the Zariski closure of sets of special points in Shimura varieties, and is regarded as a non-abelian analogue of the Manin-Mumford conjecture. A special case was stated by Yves Andre in 1989, and a more general statement was conjectured by Frans Oort in 1995; the modern version generalizes both.
Facts
StatementThe Andre-Oort conjecture states that each irreducible component of the Zariski closure of a set of special points in a Shimura variety is itself a special subvariety. A restricted special case was first proposed by Yves Andre in 1989, and Frans Oort conjectured the more general statement, with a restriction on the type of Shimura variety, in 1995. 1 Proposed Year Prize StatusNo prize has been offered for the conjecture as a whole, but Jonathan Pila received the 2011 Clay Research Award for proving an unconditional special case, the case of arbitrary products of modular curves. 1 Progress Toward ResolutionThe conjecture is now considered proven in general. Jonathan Pila, Ananth Shankar and Jacob Tsimerman posted a paper in September 2021 proving the Binyamini-Schmidt-Yafaev height conjecture, thereby completing a proof of the Andre-Oort conjecture using the Pila-Zannier strategy. 1 Prize Status
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Source Andre-Oort Conjecture (Wikipedia)
Sources
1. Andre-Oort Conjecture (Wikipedia)
Statement section
each irreducible component of the Zariski closure of a set of special points in a Shimura variety is a special subvariety
History section, Oort statement
a more general statement (albeit with a restriction on the type of the Shimura variety) was conjectured by Frans Oort in 1995
History section, Pila award
a result which earned him the 2011 Clay Research Award
History section, 2021 proof
thus completing the proof of the Andre-Oort conjecture using the Pila-Zannier strategy
- Lead section
- In Branch: Number Theory, Lead sentence
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