O-minimality and the Andre-Oort conjecture for C-n

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 po...

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Hovedforfatter: Pila, J
Format: Journal article
Sprog:English
Udgivet: 2011
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author Pila, J
author_facet Pila, J
author_sort Pila, J
collection OXFORD
description 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.
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spelling oxford-uuid:40bb8a20-5dc4-4edc-8f99-f222cd54b0562022-03-26T14:39:30ZO-minimality and the Andre-Oort conjecture for C-nJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:40bb8a20-5dc4-4edc-8f99-f222cd54b056EnglishSymplectic Elements at Oxford2011Pila, JWe 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.
spellingShingle Pila, J
O-minimality and the Andre-Oort conjecture for C-n
title O-minimality and the Andre-Oort conjecture for C-n
title_full O-minimality and the Andre-Oort conjecture for C-n
title_fullStr O-minimality and the Andre-Oort conjecture for C-n
title_full_unstemmed O-minimality and the Andre-Oort conjecture for C-n
title_short O-minimality and the Andre-Oort conjecture for C-n
title_sort o minimality and the andre oort conjecture for c n
work_keys_str_mv AT pilaj ominimalityandtheandreoortconjectureforcn