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arXiv · 1409.0771

O-minimality and certain atypical intersections

Philipp Habegger·Jonathan Pila
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Abstract

We show that the strategy of point counting in o-minimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of Manin-Mumford and Andr\'e-Oort. We verify the so-called Zilber-Pink Conjecture in a product of modular curves on assuming a lower bound for Galois orbits and a sufficiently strong modular Ax-Schanuel Conjecture. In the context of abelian varieties we obtain the Zilber-Pink Conjecture for curves unconditionally when everything is defined over a number field. For higher dimensional subvarieties of abelian varieties we obtain some weaker results and some conditional results.

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Philipp Habegger, Jonathan Pila. 2014-09-02. O-minimality and certain atypical intersections. https://arxiv.org/abs/1409.0771

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