Submission date: 19 February 2014.

Abstract:

We show that the order three algebraic differential equation over Q satisfied by the analytic j-function defines a non-\aleph_0-categorical strongly minimal set with trivial forking geometry relative to the theory of differentially closed fields of characteristic zero answering a long-standing open problem about the existence of such sets. The theorem follows from Pila's modular Ax-Lindemann-Weierstrass with derivatives theorem using Seidenberg's embedding theorem and a theorem of Nishioka on the differential equations satisfied by automorphic functions. As a by-product of this analysis, we obtain a more general version of the modular Ax-Lindemann-Weierstrass theorem, which, in particular, applies to automorphic functions for arbitrary arithmetic subgroups of SL_2(Z).

Mathematics Subject Classification: 03C60, 11F03, 12H05

Keywords and phrases:

Full text arXiv 1402.4588: pdf, ps.