Construction of Jacobi forms using adjoint of Jacobi-Serre derivative

TL;DR

This work shows that the Oberdieck derivative $\\mathcal{O}$ preserves Jacobi forms and constructs its adjoint with respect to the Petersson product, enabling the adjoint of the Jacobi-Serre derivative $\\partial^{J}$ to be obtained as well. It provides explicit Fourier-coefficient formulas for the adjoints $\\mathcal{L}_{k,m}^{*}$ and $\\mathcal{O}^{*}$, expressed via Dirichlet sums and hypergeometric-type sums, using Rankin unfolding with Jacobi-Poincaré series. The main contributions are the precise descriptions of $\\partial^{J*}$ and the resulting coefficient relations among Jacobi forms, along with a Key Ingredient that ensures convergence and computability of the adjoint operators. These results deepen understanding of differential-operator calculus on Jacobi forms and yield concrete relations among Fourier coefficients with potential applications in the study of Jacobi and Siegel modular forms.

Abstract

In the article, we study the Oberdieck derivative defined on the space of weak Jacobi forms. We prove that the Oberdieck derivative maps a Jacobi form to a Jacobi form. Moreover, we study the adjoint of Oberdieck derivative of a Jacobi cusp form with respect to Petersson scalar product defined on the space of Jacobi forms. As a consequence, we also obtain the adjoint of Jacobi-Serre derivative (defined in an unpublished work of Oberdieck). As an application, we obtain certain relations among the Fourier coefficients of Jacobi forms.

Theorems & Definitions (19)

• Definition 2.1
• Example 2.2
• Example 2.3
• Lemma 2.4
• Lemma 2.5: Choie, Kohnen choie5
• Theorem 2.6
• Theorem 2.7
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3