Weak Modular Zilber-Pink with Derivatives

TL;DR

This paper proves a weak form of the Modular Zilber-Pink with Derivatives conjecture for the modular j-function and its derivatives, using differential algebra, complex geometry, and Ax-Schanuel theorems.

Contribution

It introduces D-special varieties and establishes two differential analogues of the MZPD conjecture, including a functional Modular André-Oort with Derivatives statement.

Findings

Proves a weak version of MZPD for the j-function and derivatives.

Establishes functional analogues of Zilber-Pink conjectures.

Utilizes Ax-Schanuel theorem and differential algebra techniques.

Abstract

In unpublished notes Pila proposed a Modular Zilber-Pink with Derivatives (MZPD) conjecture, which is a Zilber-Pink type statement for the modular $j$-function and its derivatives. In this article we define D-special varieties, then state and prove two functional (differential) analogues of the MZPD conjecture for those varieties. In particular, we prove a weak version of MZPD. As a special case of our results, we obtain a functional Modular Andr\'e-Oort with Derivatives statement. The main tools used in the paper come from (model theoretic) differential algebra and complex analytic geometry, and the Ax-Schanuel theorem for the $j$-function and its derivatives (established by Pila and Tsimerman) plays a crucial role in our proofs. In the proof of the second Zilber-Pink type theorem we also use an Existential Closedness statement for the differential equation of the $j$-function.