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arXiv · 2610.26768

Meromorphic modular differential equations for the elliptic genus of Calabi--Yau varieties

Dmitrii Adler·Valery Gritsenko
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Abstract

We study differential equations -- not necessarily monic -- for Jacobi modular forms. As an application, we derive a second-order differential equation for the elliptic genus of a $K3$ surface involving the logarithmic derivative of the modular invariant $j(τ)$, and we explicitly find a third-order equation with a leading coefficient of weight 12 for the elliptic genus of any four-dimensional Calabi--Yau variety. Furthermore, we propose a simple recursive algorithm for investigating second-order differential equations with leading coefficient $E_4(τ)$ or $E_6(τ)$.

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Dmitrii Adler, Valery Gritsenko. 2026-10-19. Meromorphic modular differential equations for the elliptic genus of Calabi--Yau varieties. https://arxiv.org/abs/2610.26768

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