Weak modular Zilber–Pink with derivatives

Aslanyan, Vahagn (2022) Weak modular Zilber–Pink with derivatives. Mathematische Annalen, 383 (1-2). 433–474. 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In unpublished notes Pila proposed a Modular Zilber–Pink with derivatives (MZPD) conjecture, which is a Zilber–Pink type statement for the modular j-function and its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
Additional Information:	Acknowledgements: This work was done while I was a postdoctoral associate at Carnegie Mellon University, and some revisions were made at the University of East Anglia. Partially supported by EPSRC grant EP/S017313/1.
Uncontrolled Keywords:	mathematics(all) ,/dk/atira/pure/subjectarea/asjc/2600
Faculty \ School:	Faculty of Science > School of Mathematics (former - to 2024)
UEA Research Groups:	Faculty of Science > Research Groups > Logic (former - to 2024)
Related URLs:	http://www.scopus.com/inward/record.url?...
Depositing User:	LivePure Connector
Date Deposited:	27 May 2021 00:10
Last Modified:	04 Jul 2025 00:01
URI:	https://ueaeprints.uea.ac.uk/id/eprint/80132
DOI:	10.1007/s00208-021-02213-7

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