Jacobi Forms of Affine Weight in Higher Cogenus and Nearly Holomorphic Functions

TL;DR

This paper extends the theory of Jacobi forms of vector-valued weights to higher cogenus using differential geometric methods, providing a structure theorem for nearly holomorphic functions without explicit calculations.

Contribution

It generalizes previous results by Ibukiyama and Kyomura to arbitrary cogenus and introduces a geometric approach to study these forms.

Findings

Isomorphisms between Jacobi forms and classical forms via covariant differential operators

A new structure theorem for nearly holomorphic functions on the Jacobi upper half space

Avoidance of explicit calculations through geometric arguments

Abstract

We describe Jacobi forms of vector-valued weights in terms of classical ones, extending previous results by Ibukiyama and Kyomura to the case of arbitrary cogenus. As in their result, our isomorphisms are given by holomorphic covariant differential operators. In contrast to previous work, however, we avoid explicit calculations, which we replace by general differential geometric arguments. In the process, we obtain a structure theorem on nearly holomorphic functions on the Jacobi upper half space.