Rankin-Cohen Bracket for Vector-Valued Modular Forms

TL;DR

The paper extends Rankin-Cohen brackets to the setting of vector-valued modular forms and establishes explicit Petersson-pairing formulas with these brackets, including the adjoint maps. It then links Jacobi forms to vector-valued modular forms via theta decompositions, showing that holomorphic and skew-holomorphic Rankin-Cohen brackets are compatible with the vector-valued theory through natural isomorphisms. Consequently, the authors derive corresponding adjoint maps and provide Jacobi-form analogues of the main results, offering a unified framework across vector-valued modular forms and Jacobi forms. This yields tools for constructing new vector-valued modular forms and transferring structural results between the vector-valued and Jacobi-form settings.

Abstract

In this paper, we explore the relationship between Rankin-Cohen brackets for vector-valued modular forms and Petersson's inner products, deriving an explicit description of the adjoint map for the bracket operator. The study extends to the cases of Jacobi forms and skew-holomorphic Jacobi forms, establishing connections between their respective Rankin-Cohen brackets and those defined for vector-valued modular forms through an isomorphism. Adjoint maps for these extended bracket operators are also examined.

Theorems & Definitions (32)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• Theorem 3.1
• proof
• Theorem 3.2
• proof