André–Oort conjecture

In mathematics, the André–Oort conjecture is an open problem in Diophantine geometry, a branch of number theory, that builds on the ideas found in the Manin–Mumford conjecture, which is now a theorem. A prototypical version of the conjecture was stated by Yves André in 1989[1] and a more general version was conjectured by Frans Oort in 1995.[2] The modern version is a natural generalisation of these two conjectures.

Statement

The conjecture in its modern form is as follows. Each irreducible component of the Zariski closure of a set of special points in a Shimura variety is a special subvariety.

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.

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 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,[3][4] a result which earned him the 2011 Clay Research Award.[5]

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.[6]

Coleman–Oort conjecture

A related conjecture that has two forms, equivalent if the André–Oort conjecture is assumed, is the Coleman–Oort conjecture. Robert Coleman conjectured that for sufficiently large g, there are only finitely many smooth projective curves C of genus g, such that the Jacobian variety J(C) is an abelian variety of CM-type. Oort then conjectured that the Torelli locus – of the moduli space of abelian varieties of dimension g – has for sufficiently large g no special subvariety of dimension > 0 that intersects the image of the Torelli mapping in a dense open subset.[7]

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[8] and conjectures put forth by Boris Zilber.[9][10]

References

1. André, Yves (1989), G-functions and geometry, Aspects of Mathematics, E13, Vieweg.
2. Oort, Frans (1997), "Canonical liftings and dense sets of CM points", in Fabrizio Catanese (ed.), Arithmetic Geometry, Cambridge: Cambridge University Press.
3. Pila, Jonathan (2009), "Rational points of definable sets and results of André–Oort–Manin–Mumford type", Int. Math. Res. Not. IMRN (13): 2476–2507.
4. Pila, Jonathan (2011), "O-minimality and the André–Oort conjecture for Cⁿ", Annals of Mathematics, 173 (3): 1779–1840, doi:10.4007/annals.2011.173.3.11.
5. Clay Research Award website Archived 2011-06-26 at the Wayback Machine
6. "February 2018". Notices of the American Mathematical Society. 65 (2): 191. 2018. ISSN 1088-9477.
7. Carlson, James; Müller-Stach, Stefan; Peters, Chris (2017). Period Mappings and Period Domains. Cambridge University Press. p. 285. ISBN 9781108422628.
8. Pink, Richard (2005), "A combination of the conjectures of Mordell–Lang and André–Oort", Geometric methods in algebra and number theory, Progress in Mathematics, 253, Birkhauser, pp. 251–282.
9. Zilber, Boris (2002), "Exponential sums equations and the Schanuel conjecture", J. London Math. Soc., 65 (2): 27–44, doi:10.1112/S0024610701002861.
10. Rémond, Gaël (2009), "Autour de la conjecture de Zilber–Pink", J. Théor. Nombres Bordeaux (in French), 21 (2): 405–414, doi:10.5802/jtnb.677.

Further reading

• Zannier, Umberto (2012). "About the André–Oort conjecture". Some Problems of Unlikely Intersections in Arithmetic and Geometry. Princeton: Princeton University Press. pp. 96–127. ISBN 978-0-691-15370-4.