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[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. 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.

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. The biggest unconditional results came in 2009 when 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]

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[6] and conjectures put forth by Boris Zilber.[7][8]

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, 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: 1779–1840, doi:10.4007/annals.2011.173.3.11.
  5. ↑ Clay Research Award website
  6. ↑ 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.
  7. ↑ Zilber, Boris (2002), "Exponential sums equations and the Schanuel conjecture", J. London Math. Soc. 65 (2): 27–44, doi:10.1112/S0024610701002861.
  8. ↑ 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.
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