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The André-Oort conjecture for products of modular curves. (English) Zbl 1254.11062
Darmon, Henri (ed.) et al., Arithmetic geometry. Clay Mathematics Institute Summer School Arithmetic Geometry, Göttingen, Germany, July 17–August 11, 2006. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute (ISBN 978-0-8218-4476-2/pbk). Clay Mathematics Proceedings 8, 431-439 (2009).
The André-Oort conjecture concerns the distribution (for the Zariski topology) of the special points in a Shimura variety \(S\). It asserts that an irreducible component of the Zariski closure of any set of special points of \(S\) is a special subvariety of \(S\). A proof of the conjecture in full generality has been announced under the Generalised Riemann Hypothesis in work of Klingler, Ullmo, and Yafaev. In this paper the ideas and strategy underlying the general proof are illustrated in the particular case that \(S\) is an arbitrary product of modular curves. In this case a proof (also under GRH but with different methods that do not seem to generalise to the general case) was given by Edixhoven; an unconditional proof was recently given by the reviewer. For a product of 2 modular curves the conjecture was affirmed unconditionally by André, while an effective proof in this case was given very recently by L. Kühne [Ann. Math. (2) 176, No. 1, 651–671 (2012; Zbl 1341.11035)].
For the entire collection see [Zbl 1175.14002].

11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
14L05 Formal groups, \(p\)-divisible groups