On Sp_4 modularity of Picard--Fuchs differential equations for Calabi--Yau threefolds (with an appendix by Vicentiu Pasol)

TL;DR

This paper explores the modularity properties of Picard--Fuchs differential equations of order 4 and 5 related to Calabi--Yau threefolds, extending classical modular function concepts to higher-order cases.

Contribution

It introduces a new framework for understanding Sp_4 modularity in Picard--Fuchs equations of Calabi--Yau threefolds, expanding the scope beyond classical cases.

Findings

Identifies Sp_4 modularity in higher-order Picard--Fuchs equations

Establishes connections between modular functions and Calabi--Yau threefolds

Provides a foundation for further exploration of modularity in complex differential equations

Abstract

Motivated by the relationship of classical modular functions and Picard--Fuchs linear differential equations of order 2 and 3, we present an analogous concept for equations of order 4 and 5.