Rankin-Cohen brackets for Calabi-Yau modular forms

Abstract

For any positive integer n, we consider a modular vector field R on a moduli space 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 R, which are provided with definite weights, are called Calabi-Yau modular forms. Using R we introduce a derivation D and the Ramanujan-Serre type derivation ∂ on the space of Calabi-Yau modular forms. We show that they are degree 2 differential operators and there exists a proper subspace M 2 of the space of Calabi-Yau modular forms which is closed under ∂. Employing the derivation D, we define the Rankin-Cohen brackets for Calabi-Yau modular forms and prove that the subspace generated by the positive weight elements of M 2 is closed under the Rankin-Cohen brackets.

Ramanujan-type systems of nonlinear ODEs for $Γ_0(2)$ and $Γ_0(3)$

“…Indeed, in a geometric framework, the considered systems can be seen as vector fields, known as modular vector fields, in certain enhanced moduli spaces arising from the Dwork family. In general, solution components of modular vector fields generate the space of Calabi-Yau modular forms, which are interesting objects to study, see [Mov16,Nik19]. In lower dimensions 1 and 2, which are studied in this paper, these spaces of Calabi-Yau modular forms coincide with the spaces of classical quasi-modular forms on Γ 0 (3) and Γ 0 (2), respectively.…”

“…Part 3 of Theorem 1.1 and Theorem 1.2 provides the spaces M Γ 0 (2) and M Γ 0 (3) with an sl 2 (C)-module structure, respectively. This property, in general, is important on the one hand to assign correct weights to the Calabi-Yau modular forms, see [Nik20,Nik19], and on the other hand to study the dynamics of modular vector fields, see [Gui07,GR12].…”

Ramanujan systems of Rankin-Cohen type and hyperbolic triangles

“…The functions generated by the solutions of these systems were called Calabi-Yau (CY) modular forms. In subsequent works [24,23], the second author deepened the analogy between CY modular forms and quasimodular forms; in particular, he showed the existence of a natural Rankin-Cohen structure on the space of CY modular forms. Given a Fuchsian group Γ, the Rankin-Cohen brackets [ , ] n [27,5] on the space of modular forms M(Γ) are defined, for f ∈ M k (Γ) and g ∈ M l (Γ), by…”