O-minimality and the Andre-Oort conjecture for Cn

• 成果类型: Article
• 署名作者: Pila, Jonathan
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.173.3.11
• 发表日期: 2011
• 页码: 1779-1840
• 关键词: manin-mumford conjecture; special points; rational-points; linearity properties; shimura varieties; elliptic-curves; real numbers; small height; equidistribution; INDEPENDENCE

摘要

We give an unconditional proof of the Andre-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 Manin Mumford 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 Andre-Oort-Manin-Mumford type by Pila