On some results of Bump-Choie and Choie-Kim

TL;DR

This paper explores the connections between results of Choie, Kim, Bump, and Choie on modular forms, providing new proofs and linking their conjectures to existing work by Boe, thereby deepening understanding of the representation-theoretic aspects.

Contribution

It demonstrates that Bump and Choie's conjecture follows from Boe's work and offers an alternative proof for the genus 2 case, extending the results to Jacobi forms.

Findings

Bump and Choie's conjecture follows from Boe's work.

Provided a second proof for the genus 2 case.

Extended results to Jacobi forms.

Abstract

This paper is motivated by a 2001 paper of Choie and Kim and a 2006 paper of Bump and Choie. The paper of Choie and Kim extends an earlier result of Bol for elliptic modular forms to the setting of Siegel and Jacobi forms. The paper of Bump and Choie provides a representation theoretic interpretation of the phenomenon, and shows how a natural generalization of Choie and Kim's result on Siegel modular forms follows from a natural conjecture regarding (g,K)-modules. In this paper, it is shown that the conjecture of Bump and Choie follows from work of Boe. A second proof which is along the lines of the proof given by Bump and Choie in the genus 2 case is also included, as is a similar treatment of the result of Choie and Kim on Jacobi forms.