Rankin–Cohen brackets for Calabi–Yau modular forms

Abstract

For any positive integer $n$, we consider a modular vector field $\textsf{R}$ on a moduli space $\textsf{T}$ of Calabi-Yau $n$-folds arising from the Dwork family enhanced with a certain basis of the $n$-th algebraic de Rham cohomology. The components of a particular solution of $\textsf{R}$, which are provided with definite weights, are called Calabi-Yau modular forms. Using $\textsf{R}$ we introduce a derivation $\mathcal{D}$ and the Ramanujan-Serre type derivation $\partial$ on the space of Calabi-Yau modular forms. We show that they are degree $2$ differential operators and there exists a proper subspace $\mathcal{M}^2$ of the space of Calabi-Yau modular forms which is closed under $\partial$. Employing the derivation $\mathcal{D}$, we define the Rankin-Cohen brackets for Calabi-Yau modular forms and prove that the subspace generated by the positive weight elements of $\mathcal{M}^2$ is closed under the Rankin-Cohen brackets.

Locations

• Communications in Number Theory and Physics - View
• arXiv (Cornell University) - View - PDF