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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| Médium: | Journal article |
| Jazyk: | English |
| Vydáno: |
2011
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| Shrnutí: | 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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