Maass-Jacobi Poincar\'e series and Mathieu Moonshine

TL;DR

This paper introduces semi-holomorphic Maass-Jacobi forms related to Mathieu moonshine, proving convergence of certain Poincaré series and characterizing forms associated with the Mathieu group.

Contribution

It presents a novel modification of Mathieu moonshine by replacing weak Jacobi forms with semi-holomorphic Maass-Jacobi forms and establishes their convergence and characterization.

Findings

Proved convergence of Maass-Jacobi Poincaré series of weight one.

Characterized semi-holomorphic Maass-Jacobi forms from the Mathieu group.

Extended Mathieu moonshine framework with new Maass-Jacobi forms.

Abstract

Mathieu moonshine attaches a weak Jacobi form of weight zero and index one to each conjugacy class of the largest sporadic simple group of Mathieu. We introduce a modification of this assignment, whereby weak Jacobi forms are replaced by semi-holomorphic Maass-Jacobi forms of weight one and index two. We prove the convergence of some Maass-Jacobi Poincar\'e series of weight one, and then use these to characterize the semi-holomorphic Maass-Jacobi forms arising from the largest Mathieu group.