Geodesic
O-minimality and the André-Oort conjecture for $\mathbb{C}^{n}$
Jonathan Pila1
1 Mathematical Institute<br/>University of Oxford<br/> Oxford OX1 3LB<br/> England
Annals of mathematics, Tome 173 (2011) no. 3, pp. 1779-1840.

Voir la notice de l'article provenant de la source Annals of Mathematics website

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 $\bar{\mathbb{Q}}$ 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 $\bar{\mathbb{Q}}$. 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-Oort-Manin-Mumford type by Pila.
DOI : 10.4007/annals.2011.173.3.11

Jonathan Pila 1

1 Mathematical Institute<br/>University of Oxford<br/> Oxford OX1 3LB<br/> England 
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Jonathan Pila. O-minimality and the André-Oort conjecture for  $\mathbb{C}^{n}$. Annals of mathematics, Tome 173 (2011) no. 3, pp. 1779-1840. doi : 10.4007/annals.2011.173.3.11. http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.173.3.11/

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