A Modular Andre-Oort Statement with Derivatives
arXiv:1702.08403 · doi:10.1017/S001309151800041X
Abstract
In unpublished notes, Pila discussed some theory surrounding the modular function and its derivatives. A focal point of these notes was the statement of two conjectures regarding , and : a Zilber-Pink type statement incorporating , and , which was an extension of an apparently weaker conjecture of Andre-Oort type. In this paper, I first cover some background regarding , and , 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 Andre-Oort with Derivatives" conjecture. Under the assumption of a Schanuel-type conjecture, the central theorem of the paper implies Pila's conjecture in full generality, as well as a more precise statement on the same lines.
Version 3 fixes a few typos and adds a discussion of uniform versions of the main results. Version 2 implements significant changes. An error was found in version 1 which undermined the result. To amend the error required the introduction of a Schanuel-type conjecture. Version 2 also adds, under the Schanuel-type conjecture, a new and stronger version of the main theorem