Number theory - Rational points in periodic analytic sets and the Manin-Mumford conjecture
We present a new proof of the Manin-Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points as rational points on a complex torus and then compare (i) upper bounds for the number of rational poi...
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| Format: | Journal article |
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2008
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| _version_ | 1797073234892423168 |
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| author | Pila, J Zannier, U |
| author_facet | Pila, J Zannier, U |
| author_sort | Pila, J |
| collection | OXFORD |
| description | We present a new proof of the Manin-Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points as rational points on a complex torus and then compare (i) upper bounds for the number of rational points on a transcendental analytic variety (Bombieri-Pila-Wilkie) and (ii) lower bounds for the degree of a torsion point (Masser), after taking conjugates. In order to be able to deal with (i), we discuss (Thm. 2.1) the semialgebraic curves contained in an analytic variety supposed invariant under translations by a full lattice, which is a topic with some independent motivation. |
| first_indexed | 2024-03-06T23:19:12Z |
| format | Journal article |
| id | oxford-uuid:682a7a47-0781-41f7-a554-67908a68529a |
| institution | University of Oxford |
| last_indexed | 2024-03-06T23:19:12Z |
| publishDate | 2008 |
| record_format | dspace |
| spelling | oxford-uuid:682a7a47-0781-41f7-a554-67908a68529a2022-03-26T18:43:06ZNumber theory - Rational points in periodic analytic sets and the Manin-Mumford conjectureJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:682a7a47-0781-41f7-a554-67908a68529aSymplectic Elements at Oxford2008Pila, JZannier, UWe present a new proof of the Manin-Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points as rational points on a complex torus and then compare (i) upper bounds for the number of rational points on a transcendental analytic variety (Bombieri-Pila-Wilkie) and (ii) lower bounds for the degree of a torsion point (Masser), after taking conjugates. In order to be able to deal with (i), we discuss (Thm. 2.1) the semialgebraic curves contained in an analytic variety supposed invariant under translations by a full lattice, which is a topic with some independent motivation. |
| spellingShingle | Pila, J Zannier, U Number theory - Rational points in periodic analytic sets and the Manin-Mumford conjecture |
| title | Number theory - Rational points in periodic analytic sets and the Manin-Mumford conjecture |
| title_full | Number theory - Rational points in periodic analytic sets and the Manin-Mumford conjecture |
| title_fullStr | Number theory - Rational points in periodic analytic sets and the Manin-Mumford conjecture |
| title_full_unstemmed | Number theory - Rational points in periodic analytic sets and the Manin-Mumford conjecture |
| title_short | Number theory - Rational points in periodic analytic sets and the Manin-Mumford conjecture |
| title_sort | number theory rational points in periodic analytic sets and the manin mumford conjecture |
| work_keys_str_mv | AT pilaj numbertheoryrationalpointsinperiodicanalyticsetsandthemaninmumfordconjecture AT zannieru numbertheoryrationalpointsinperiodicanalyticsetsandthemaninmumfordconjecture |