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].

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].

Reviewer: Immanuel Halupczok (Münster)