17 problems

Cheng–Duncan conjecture. For each conjugacy class $g$ of $M_{24}$, the product $\Phi_g$ defines a Siegel modular form with some multiplier on the congruence subgroup…

Let $M_{24}$ be the Mathieu group, and let $H^{(2)}(\tau)$ be the mock-modular form arising from the decomposition of the elliptic genus of a K3 surface into irreducible characters…

Siegel modularity conjecture. For every $g\in M_{24}$, the function $\Phi_g(\Omega)$ is a Siegel modular form, possibly with a nontrivial multiplier, for a subgroup of…

Let $V^{f\natural}$ be the holomorphic SCFT with automorphism group $\mathrm{Co}_1$, and let $\overline{\operatorname{Fer}(3)}$ be the antiholomorphic three-fermion theory. Set … T…

Let $\mathrm{Tcf}^\bullet$ be the spectrum of topological cusp forms, and let $[\overline{V^{f\natural}}]\nu$ denote the $\mathrm M_{24}$-equivariant class refining the nonequivari…

Let $\nu\in\mathrm{TMF}^{-3}(\mathrm{pt})$ be the class corresponding to $[\overline{\operatorname{Fer}(3)}]$, let $\{24\Delta\}\in\mathrm{TMF}^{-24}(\mathrm{pt})$ be the class cor…

Let $V^{f\natural}$ be the holomorphic SCFT with automorphism group containing $\mathrm M_{24}$, let $\overline{\operatorname{Fer}(3)}$ be the antiholomorphic theory of three antic…

Let $b^\beta$ be a particular linear combination of the twisted moduli, and let $V^{\mathbb{M}_{24}}_1$ denote the proposed degree-one Mathieu moonshine representation. Taormina–We…

Let $g\in M_{23}\subset M_{24}$, let $\alpha$ satisfy the cocycle conditions referred to in the source, and let $C^\alpha_{M_{24}}(g)$ be the corresponding central extension, with…

Wall-crossing interpretation conjecture. Under the stated supposition, this equation has a physical interpretation as a wall-crossing formula. The conjecture connects the factoriza…

All-commuting-pairs S-duality conjecture. The displayed S-duality relation should hold for all commuting pairs $(g,h)$, not only for the pairs of generators for which it was proved…

Cheng's automorphy conjecture. The function $\Phi_h$ is automorphic with respect to a subgroup $\Gamma_h^{(2)}$ of $Sp(4;\mathbb Z)$. This conjecture proposes an automorphic proper…

Untwisted Margolin representation conjecture. The group $\widetilde G$ acts without a twist on Margolin's representation on $V_{45}$.

Let $K=\bigoplus_{n=-1}^{\infty}K_n q^{n/8}$ be the virtual graded $M_{24}$-module determined by the McKay–Thompson series, with $K_n$ the representation whose character is given b…

Elliptic-cohomology realization conjecture. The elliptic cohomology of $K3$ surfaces furnishes this infinite-dimensional, $\mathbb{Z}$-graded $M_{24}$-supermodule, whose characters…

Twisted elliptic-genus module conjecture. For $k\equiv0,3\pmod4$ and $k\geq-1$, one has

Full $M_{24}$ symmetry conjecture. The full group $M_{24}$ acts on these moduli-invariant objects in $K3$-compactified string theories.