Submission date: 9 May 2018

Abstract:

Let (K ; +, ⠂, D) be a differentially closed field with constant field C. Let also E_j(x,y) be the differential equation of the the j-function. We prove a Zilber style classification result for strongly minimal sets in the reduct K_{E_j}:=(K ; +, ⠂, E_j) assuming an Existential Closedness (EC) conjecture for E_j. More precisely, assuming EC we show that in K_{E_j} all strongly minimal sets are geometrically trivial or non-orthogonal to C. The Ax-Schanuel inequality for the j-function and its adequacy play a crucial role in this classification.

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Full text arXiv 1805.03985: pdf, ps.