O-minimality and certain atypical intersections

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

A non-archimedean Ax-Lindemann theorem

We prove a statement of Ax-Lindemann type for the uniformization of products of Mumford curves whose associated fundamental groups are non-abelian Schottky subgroups of $\mathop{\rm PGL}(2,\bar{\mathbf Q_p})$ contained in $\mathop{\rm PGL}(2,\bar{\mathbf Q})$. In particular, we characterize bi-algebraic irreducible subvarieties of the uniformization.Comment: 31 pages; revised versio

Ax-Schanuel for Shimura varieties

We prove the Ax-Schanuel theorem for a general (pure) Shimura variety

Ax-Schanuel and strong minimality for the j-function

Let $\mathcal{K}:=(K;+,\cdot, D, 0, 1)$ be a differentially closed field of characteristic $0$ with field of constants $C$. In the first part of the paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation $E(x,y)$ and the geometry of the fibres $U_s:=\{ y:E(s,y) \wedge y \notin C \}$ where $s$ is a non-constant element. We show that certain types of predimension inequalities imply strong minimality and geometric triviality of $U_s$. Moreover, the induced structure on the Cartesian powers of $U_s$ is given by special subvarieties. In particular, since the $j$-function satisfies an Ax-Schanuel inequality of the required form (due to Pila and Tsimerman), applying our results to the $j$-function we recover a theorem of Freitag and Scanlon stating that the differential equation of $j$ defines a strongly minimal set with trivial geometry. In the second part of the paper we study strongly minimal sets in the $j$-reducts of differentially closed fields. Let $E_j(x,y)$ be the (two-variable) differential equation of the $j$-function. We prove a Zilber style classification result for strongly minimal sets in the reduct $\mathsf{K}:=(K;+, \cdot, E_j)$. More precisely, we show that in $\mathsf{K}$ all strongly minimal sets are geometrically trivial or non-orthogonal to $C$. Our proof is based on the Ax-Schanuel theorem and a matching Existential Closedness statement which asserts that systems of equations in terms of $E_j$ have solutions in $\mathsf{K}$ unless having a solution contradicts Ax-Schanuel.Comment: 27 pages. This is a combination of arXiv:1606.01778v3 and arXiv:1805.03985v1 (with substantial revisions