Maass–Jacobi Poincaré series and Mathieu Moonshine

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 Poincaré series of weight one, and then use these to characterize the semi-holomorphic Maass–Jacobi forms arising from the largest Mathieu group.