Monstrous Moonshine of higher weight

TL;DR

This paper characterizes the space of 1-point correlation functions for the Moonshine module as specific modular forms and extends the concept to Monster-equivariant analogues for various weights.

Contribution

It explicitly determines the modular forms associated with the Moonshine module and introduces Monster-equivariant analogues for different weights, generalizing classical Moonshine.

Findings

Identified the space of 1-point functions as certain holomorphic modular forms.

Established a correspondence between Monster elements and modular forms of fixed weight.

Extended Moonshine to higher weights with Monster-equivariant modular forms.

Abstract

We determine the space of 1-point correlation functions associated with the Moonshine module: they are precisely those modular forms of non-negative integral weight which are holomorphic in the upper half plane, have a pole of order at most 1 at infinity, and whose Fourier expansion has constant 0. There are Monster-equivariant analogues in which one naturally associates to each element in the Monster a modular form of fixed weight k, the case k=0 corresponding to the original ``Moonshine'' of Conway and Norton.