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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Format: | Journal article |
Sprog: | English |
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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. |
first_indexed | 2024-03-06T21:18:47Z |
format | Journal article |
id | oxford-uuid:40bb8a20-5dc4-4edc-8f99-f222cd54b056 |
institution | University of Oxford |
language | English |
last_indexed | 2024-03-06T21:18:47Z |
publishDate | 2011 |
record_format | dspace |
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 |