Harvey–Rayhaun's Thompson Moonshine conjecture

Let $Th$ be the Thompson sporadic simple group. A graded $Th$-supermodule is a decomposition

$$W=\bigoplus_{\substack{m=-3 \\ m\equiv 0,1\ \left(\operatorname{mod} 4\right)}}^\infty W_m.$$

For $m\geq 0$, suppose that $W_m$ has vanishing odd part when $m\equiv 0\pmod 4$ and vanishing even part when $m\equiv 1\pmod 4$. For $g\in Th$, define its McKay–Thompson series by

$$\mathcal{T}_{[g]}(\tau):=\sum_{\substack{m=-3 \\ m\equiv 0,1\ \left(\operatorname{mod} 4\right)}}^\infty \operatorname{strace}_{W_m}(g)q^m.$$

Harvey–Rayhaun's Thompson Moonshine conjecture. There exists such a graded $Th$-supermodule $W$ for which, for every $g\in Th$, the series $\mathcal{T}_{[g]}(\tau)$ is a specifically given weakly holomorphic modular form of weight $\tfrac12$ in the Kohnen plus space. This conjecture concerns a half-integral-weight Moonshine phenomenon for the Thompson group; the paper's abstract proof establishes the asserted module and modular-form identities.