Weak Modular Zilber-Pink with Derivatives
Abstract
In unpublished notes Pila proposed a Modular Zilber-Pink with Derivatives (MZPD) conjecture, which is a Zilber-Pink type statement for the modular -function and its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 -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 -function.
Cite
@article{arxiv.1803.05895,
title = {Weak Modular Zilber-Pink with Derivatives},
author = {Vahagn Aslanyan},
journal= {arXiv preprint arXiv:1803.05895},
year = {2021}
}
Comments
v5: substantial changes throughout the paper, both Zilber-Pink type theorems are now proven unconditionally, v6-v7: minor revisions, v8: minor changes as suggested by the reviewer