O-minimality and certain atypical intersections

Journal article

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.

Authors: Habegger, P; Pila, J

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Société Mathématique de France
• Journal: Annales scientifiques de l'ENS
• Volume: 49
• Issue: 4
• Pages: 813-858
• Publication date: 2016-07-01
• Acceptance date: 2015-05-25
• ISSN: 0012-9593
• Keywords: math.LO; math.NT; SBTMR