Weak modular Zilber–Pink with derivatives

Aslanyan, Vahagn (2021) Weak modular Zilber–Pink with derivatives. Mathematische Annalen. ISSN 0025-5831

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In unpublished notes Pila proposed a Modular Zilber–Pink with derivatives (MZPD) conjecture, which is a Zilber–Pink type statement for the modular j-function and its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

Item Type: Article
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: LivePure Connector
Date Deposited: 27 May 2021 00:10
Last Modified: 04 Jun 2021 00:04
URI: https://ueaeprints.uea.ac.uk/id/eprint/80132
DOI: 10.1007/s00208-021-02213-7

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