# A Modular Andre-Oort Statement with Derivatives

## TL;DR

This paper advances the understanding of the modular function j and its derivatives by proving a weakened form of a conjecture using o-minimal techniques, and discusses implications under a Schanuel-type conjecture.

## Contribution

It introduces a novel adaptation of the Pila-Zannier strategy to prove a weakened modular Andre-Oort conjecture involving derivatives.

## Key findings

- Proves a weakened version of Pila's Modular Andre-Oort with Derivatives conjecture.
- Under a Schanuel-type conjecture, the full conjecture is implied.
- Provides background and builds on Pila's prior work on j, j', and j".

## 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In unpublished notes, Pila discussed some theory surrounding the modular function $j$ and its derivatives.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 Andre-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 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.