Ax-Schanuel and strong minimality for the $j$-function

Abstract

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.