Projects per year
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In unpublished notes Pila proposed a Modular Zilber–Pink with derivatives (MZPD) conjecture, which is a Zilber–Pink type statement for the modular j-function and its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.
Original language | English |
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Pages (from-to) | 433–474 |
Number of pages | 42 |
Journal | Mathematische Annalen |
Volume | 383 |
Issue number | 1-2 |
Early online date | 24 May 2021 |
DOIs | |
Publication status | Published - Jun 2022 |
Projects
- 1 Finished
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Exponentially Algebraically closed fields.
Engineering and Physical Sciences Research Council
1/09/19 → 31/08/22
Project: Research