O-minimality and certain atypical intersections

TL;DR

This paper applies o-minimal point counting techniques to unlikely intersection problems, verifying the Zilber-Pink Conjecture in certain cases and establishing new results for abelian varieties.

Contribution

It extends o-minimal methods to unlikely intersections, proving cases of the Zilber-Pink Conjecture unconditionally and conditionally for various algebraic structures.

Findings

Zilber-Pink Conjecture verified for products of modular curves under certain assumptions

Unconditional proof of Zilber-Pink for curves in abelian varieties over number fields

Conditional results for higher-dimensional subvarieties of abelian varieties

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.