Submission date: 27 February 2017

Abstract:

In unpublished notes, Pila discussed some theory surrounding the modular function j and its derivatives. A focal point of these notes was the statement of two conjectures regarding j, j' and j″: a Zilber-Pink type statement incorporating j, j' and j″, which was an extension of an apparently weaker conjecture of André-Oort type. In this paper, I cover some background regarding j, j' and j″ - mostly covering the work already done by Pila - and then prove Pila's “Modular André-Oort with Derivatives” conjecture. The proof uses ideas of o-minimality, adapting the usual Pila-Zannier strategy for diophantine problems of this type.

Mathematics Subject Classification:

Keywords and phrases:

Full text arXiv 1702.08403: pdf, ps.