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Preprint Number 692
692. James Freitag and Thomas Scanlon
Strong minimality of the j-function
Submission date: 19 February 2014.
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
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