28 problems

For each of the 23 umbral cases, let $G^X$ be the corresponding finite group, let $m$ be its associated positive integer, and let $I^X\subset\{1,2,\dots,m-1\}$ be the specified sub…

For each $g\in M_{24}$, let $n_g$ be the order of $g$, let $h_g$ be the length of a minimal cycle in its induced permutation, and let $H_g$ denote the corresponding McKay–Thompson…

Let $M_{24}$ denote the largest Mathieu group, and let the $23$ Niemeier lattices with root systems of full rank be the even unimodular positive-definite lattices of rank $24$ havi…

Let $X$ be a Niemeier root system and let $g\in G^X$. Let $\check{H}^X_g$ denote the vector formed by the components of $H^X_g$ indexed by $0<r<m$, and let…

Umbral Moonshine module conjecture. There exists such a super-module for $G^X$ whose graded super-trace at $g\in G^X$ is recovered from the vector-valued mock modular form $H^X_g$…

Let $N$ be a Niemeier lattice with non-trivial root system, let $g\in G_N$ be an element, and let $\phi_g^N$ be the weight-zero, index-one Jacobi form constructed from the umbral m…

The generalised umbral moonshine phenomenon consists of finite umbral groups, twisted-twined functions, and associated mock modular forms. A deformed Drinfel'd double is the quasi-…

Consider the unique umbral moonshine case with $n=3$, and let $z$ be the relevant central element. For $g\in G^{(\ell)}$ commuting with $z$ and with order $n_g$ not divisible by…

Let $G^{(\ell)}$ be an umbral group with $n=2$, let $z\in n$ be its nontrivial central element, and let $m$ be the index of the lambency. For $g\in G^{(\ell)}$, let $\psi_{(z,g)}$…

The 23 umbral cases are associated with umbral finite groups and their mock modular forms; a deformed Drinfel'd double is the quasi-triangular algebra $D^{\omega}(G)$ determined by…

Umbral moonshine module conjecture. There is a naturally defined module $\check{K}^X$ such that, for $g\in G^X$ and $r\in I^X$,

Let $X$ be a Niemeier root system, let $G^X$ be its umbral group, let $I^X$ and $m$ be the associated index set and Coxeter-number parameter, and let $H^X_g=(H^X_{g,r})$ be the vec…

Let $L^X$ be a Niemeier lattice whose root system has only A-type components, equivalently $m-1$ divides $24$, and let $K^X_r$ be the $G^X$-module underlying the component…

For a Niemeier lattice $L^X$, let $G^X$ be its associated finite group and let $H^X_g$ be the vector-valued mock modular form of weight $1/2$ for $\Gamma_0(n_g)$ with specified sha…

For each Niemeier root system $X$ and each element $g\in G^X$, let $H_g^X$ be the associated vector-valued mock modular form of weight $1/2$, and let $S_g^X$ be a vector-valued cus…

For each of the nine Niemeier root systems … assume the associated umbral modules and McKay--Thompson series are as in the preceding conjectures. All-doublets conjecture. Because t…

Let $X$ be a Niemeier root system with Coxeter number $m$. A representation is a doublet if it is the direct sum of two copies of one representation, and a sextet if it is the dire…

Let $X$ be a Niemeier root system with Coxeter number $m$, and let $D$ be a discriminant of $H^X$. Dual-pair conjecture. If … for some integer $\lambda$, then the representation…

Let $X$ be a Niemeier root system, let $m$ be its Coxeter number, and let $H^X_g$ be the associated vector-valued mock modular form with multiplier $\nu^X_g$ and shadow $S^X_g$. A…

Let $X$ be a Niemeier root system and $g\in G^X$. Let $\nu^X_g$ be the multiplier system of $H^X_g$, and let $S^X$ be the untwisted umbral shadow. Multiplier conjecture. The multip…

Let $X$ be a Niemeier root system with Coxeter number $m$, let $G^X$ be its associated group, and let $S^X_g$ be the attached cusp form for $g\in G^X$. For $g\in G^X$, let $n_g$ be…

Factoring-through conjecture. The $\mathbb{Q}$-graded $G^X$-module $K^X_r$ is a faithful representation of $G^X$ when $r\equiv0\pmod c$, and factors through $\bar G^X$ otherwise. T…

Let $X$ be a Niemeier root system with Coxeter number $m$, and let $G^X$ be its associated finite group. A $\mathbb{Z}/2m\rrbracket\times\mathbb{Q}$-graded super-module is a…

For $\ell\in\{2,3,4,5,7,13\}$ and $g\in G^{(\ell)}$, let $H^{(\ell)}_g$ be the umbral McKay–Thompson series, and let $R^{[\mu]}_{\Gamma_0(n_g),\psi^{(\ell)}\rho^{(\ell)}_g,1/2}$ de…

Umbral McKay–Thompson modularity conjecture. For every $g\in G^{(\ell)}$, the vector $H^{(\ell)}_g=(H^{(\ell)}_{g,r})$ transforms as a vector-valued mock modular form with specifie…