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A Modular André–Oort Statement with Derivatives

Part of: Model theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  16 November 2018

Haden Spence
Haden Spence*
Affiliation:
Mathematical Institute, Andrew Wiles Building, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK ([email protected])
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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In unpublished notes, Pila discussed some theory surrounding the modular function j and its derivatives.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 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.

Keywords

definable setrational pointAndré–Oort conjecturederivatives of jPila–Wilkie theorem

MSC classification

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties
Secondary: 03C64: Model theory of ordered structures; o-minimality
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 2 , May 2019 , pp. 323 - 365
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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