Weak modular Zilber–Pink with derivatives

In unpublished notes Pila proposed a Modular Zilber–Pink with derivatives (MZPD) conjecture, which is a Zilber–Pink type statement for the modular j-function and its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 André–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.