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A proof of the André-Oort conjecture via mathematical logic. (English) Zbl 1271.14030
Séminaire Bourbaki. Volume 2010/2011. Exposés 1027–1042. Avec table par noms d’auteurs de 1948/49 à 2009/10. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-351-5/pbk). Astérisque 348, 299-315 (2012).
In [Ann. Math. (2), 173, No. 3, 1779–1840 (2011; Zbl 1243.14022)], J. Pila gave an unconditional proof of the André-Oort conjecture for products of modular curves, using methods from model theory, a subdomain of logic. The article under review is a nicely written survey of that proof, which needs little prior knowledge. In particular, it explains the André-Oort conjecture in this particular case and it contains an introduction to the model theory needed in the proof. Here’s a survey of the survey.
For simplicity, the only modular curve considered is $$Y := \text{SL}_2(\mathbb Z) \backslash \mathfrak h$$, where $$\mathfrak h$$ is the upper complex half plane and $$\text{SL}_2(\mathbb Z)$$ acts by Möbius transformations. This quotient parametrizes elliptic curves and there exists a holomorphic map $$j: \mathfrak h \to \mathbb C$$ (the “modular function”) inducing a bijection $$Y \to \mathbb C$$.
A point in $$\mathbb C^n$$ is called “special” if each of its coordinates corresponds to an elliptic curve with complex multiplication. There is also a notion of “special subvarieties” in $$\mathbb C^n$$. In each special subvariety, the special points are dense. The claim of the André-Oort conjecture is the converse, namely if an irreducible subvariety $$X \subseteq \mathbb C^n$$ contains a dense set of special points, then it is already a special subvariety.
The main novelty in Pila’s proof is a new way to obtain an upper bound for the number of special points in $$X$$ if it contains no special subvariety of positive dimension. The central ingredient from model theory is a result by J. Pila and A. J. Wilkie [Duke Math. J. 133, No. 3, 591–616 (2006; Zbl 1217.11066)] giving upper bounds for the number of rational points of subsets of $$\mathbb R^m$$ that are “definable in an o-minimal structure”. If $$D$$ is a fundamental domain for the action of $$\text{SL}_2(\mathbb Z)$$ on $$\mathfrak h$$, then $$\mathfrak X := j^{-1}(X) \cap D^n$$ is such a set, when considered as a subset of $$\mathbb R^{2n}$$, and a bound on the number of rational points in $$\mathfrak X$$ essentially yields a bound on the number of special points in $$X$$. To be able to apply the Pila-Wilkie theorem to $$\mathfrak X$$, one has to show that $$\mathfrak X$$ contains no infinite semi-algebraic set; an important part of the proof consists in showing that this follows from the assumption that $$X$$ contains no special subvariety of positive dimension.
For the entire collection see [Zbl 1257.00012].

##### MSC:
 14G35 Modular and Shimura varieties 11Gxx Arithmetic algebraic geometry (Diophantine geometry) 03C64 Model theory of ordered structures; o-minimality