A Modular André–Oort Statement with Derivatives

Abstract

AbstractIn unpublished notes, Pila discussed some theory surrounding the modular function j and its derivatives.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 first cover some background regarding j, j′ and j″, mostly covering the work already done by Pila. Then I use a seemingly novel adaptation of the o-minimal Pila–Zannier strategy to prove a weakened version of Pila's ‘Modular André–Oort with Derivatives’ conjecture. Under the assumption of a certain number-theoretic conjecture, the central theorem of the paper implies Pila's conjecture in full generality, as well as a more precise statement along the same lines.