Ax-Schanuel and strong minimality for the j-function

Abstract

Let K:=(K;+,⋅,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 Us:={y:E(s,y)∧y∉C} where s is a non-constant element. We show that certain types of predimension inequalities imply strong minimality and geometric triviality of Us. Moreover, the induced structure on the Cartesian powers of Us 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 Ej(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 K:=(K;+,⋅,Ej). More precisely, we show that in 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 Ej have solutions in K unless having a solution contradicts Ax-Schanuel.