Traces of Singular Moduli and Moonshine for the Thompson Group

TL;DR

This paper explores a novel connection between the Thompson sporadic group and a specific modular form, providing evidence for a new moonshine phenomenon and conjecturing an associated infinite-dimensional module.

Contribution

It introduces a conjecture about an infinite-dimensional graded module for the Thompson group linked to modular forms and constructs McKay--Thompson series supporting this conjecture.

Findings

Construction of McKay--Thompson series matching modular forms

Evidence supporting the existence of a Thompson group moonshine

Observation of a discriminant property related to Umbral Moonshine

Abstract

We describe a relationship between the representation theory of the Thompson sporadic group and a weakly holomorphic modular form of weight one-half that appears in work of Borcherds and Zagier on Borcherds products and traces of singular moduli. We conjecture the existence of an infinite dimensional graded module for the Thompson group and provide evidence for our conjecture by constructing McKay--Thompson series for each conjugacy class of the Thompson group that coincide with weight one-half modular forms of higher level. We also observe a discriminant property in this moonshine for the Thompson group that is closely related to the discriminant property conjectured to exist in Umbral Moonshine.