O-minimality and the Andre-Oort conjecture for C-n - Pila, J (2011)
We give an unconditional proof of the Andŕe-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 Q̄ as well as Lang's conjecture for torsion points in powers of the multiplicative group. The second includes the Manin-Mumford conjecture for abelian varieties defined over Q̄. 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 Andŕe-Oort-Manin-Mumford type by Pila.