O-minimality and the André-Oort conjecture for C^n

Abstract

We give an unconditional proof of the André-Oort conjecture for arbitrary products of modular curves. We establish two generalizations. The first includes the Manin-Mumford conjecture for arbitrary products of elliptic curves defined over Q as well as Lang's conjecture for torsion points in powers of the multiplicative group. The second includes the ManinMumford conjecture for abelian varieties defined over Q. Our approach uses the theory of o-minimal structures, a part of Model Theory, and follows a strategy proposed by Zannier and implemented in three recent papers: a new proof of the Manin-Mumford conjecture by Pila-Zannier; a proof of a special (but new) case of Pink's relative Manin-Mumford conjecture by Masser-Zannier; and new proofs of certain known results of André-OortManin-Mumford type by Pila.

Heights of pre-special points of Shimura varieties

“…This was first proved for products of modular curves by Pila in his seminal work [Pil11], then for compact Shimura varieties by Ullmo and Yafaev in [UY14b] and for A g by Pila and Tsimerman in [PT14]. The case of a general Shimura variety has recently been demonstrated by Klingler, Ullmo and Yafaev (see [KUY16]).…”

Non-Archimedean Yomdin–gromov Parametrizations and Points of Bounded Height

“…In [35], Pila and Wilkie proved a general estimate for the number of rational points on the transcendental part of sets definable in an o-minimal structure; this has been used in a spectacular way by Pila to provide an unconditional proof of some cases of the André-Oort conjecture [34] (see also [38][39][40]44] for surveys on applications in diophantine geometry of the Pila-Wilkie Theorem). Lying at the heart of Pila and Wilkie's approach is the possibility of having uniform (in terms of number of parametrizations and in terms of bounds on the partial derivatives) C k -parametrizations.…”

Counting special points: Logic, diophantine geometry, and transcendence theory

“…The paper in which the proof of Theorem 5.4 appears [30] includes proofs of theorems in the direction of the Pink-Zilber conjectures. On the other hand, while many parts of this argument succeed when applied to other Shimura varieties, some steps are incomplete.…”

“…The reader interested in an exposition of Pila's proof of the André-Oort conjecture pitched to the general mathematician should consult my notes for the Bourbaki seminar [36], while the reader who wishes to read a detailed précis of these proofs should read my notes for the Luminy lectures [37]. Even better, because the original papers [26] and [30] are well written and contain extensive introductions, the reader should go straight to the source.…”

“…In the paper [30], Pila presents an unconditional proof of a version of the socalled André-Oort conjecture about algebraic relations amongst the j-invariants of elliptic curves with complex multiplication using a novel technique that comes from model theory. This proof of the André-Oort conjecture comes on the heels of reproofs of the Manin-Mumford conjecture (or, Raynaud's theorem [34]) by Pila and Zannier [31] (and then extended by Peterzil and Starchenko [24]) and a proof of a remarkable theorem due to Masser and Zannier [21] about simultaneous torsion for pairs of points on families of elliptic curves, all of which employ Pila's method.…”