38 problems

Let $\mathcal M$ be the Monster group, let $g\in\mathcal M$, and let $T_g(q)$ be the McKay–Thompson series associated with $g$. Conway–Norton monstrous moonshine conjecture. The se…

Norton's generalized moonshine conjecture. For every commuting pair $(g,h)$ in $\mathbb M$, there exists such a genus-zero modular function $Z(g,h,\tau)$; explicitly, if…

Let $V_n$ be the graded pieces of the Monster module and, for an element $g$ of the Monster group, let … be the associated McKay–Thompson series. A hauptmodul is a generator of the…

Let $\mathbb{M}$ be the Monster group, let $g\in\mathbb{M}$, and let $[g]$ denote its conjugacy class. Let $T_g(\tau)$, with $\tau\in\mathbb{H}$, be the modular function associated…

Let $J(q)=\sum_{n=-1}^{\infty}c(n)q^n$ be the normalized elliptic modular invariant, where $q=e^{2\pi i\tau}$ for $\tau\in\mathbb{C}$ with $\operatorname{Im}(\tau)>0$. McKay–Thomps…

Monstrous moonshine conjecture. The McKay–Thompson series for the Monster group are Hauptmoduls.

Let $M$ be the Monster simple group. The group $(M\times M)\rtimes\mathbb{Z}/2$ is proposed as a quotient of an orbifold fundamental group associated with a complex thirteen-dimens…

Monstrous Moonshine conjecture. There exists a natural infinite-dimensional $\mathbb{Z}$-graded $\mathbb{M}$-module $W$ such that

For each conjugacy class $[g]\in Co_0$, let $\pi_g$ be its Frame shape, let $\eta_g(\tau)$ be the associated eta quotient, and let $\Theta_g(\tau)$ be the $g$-twined Leech-lattice…

Let $M$ denote the Monster, and let the phenomena of Generalised Moonshine be the occurrence of $24$th roots of unity and certain central extensions of centraliser subgroups of $M$…

Let $g$ be an element of the Monster group, and let $j_g(q)$ be its McKay–Thompson series. For a prime level $N$, let $\Gamma_0(N)$ be the congruence subgroup of the modular group…

Quasi-replicable power series conjecture. Every quasi-replicable power series is either a Hauptmodul or a finite Laurent polynomial in $q$.

Monstrous Moonshine for group rings. For every ring homomorphism $\phi:\operatorname{Rep}^\natural_R(G)\to\mathbb{C}$, the generalized McKay–Thompson series

Let $p$ be a prime and let $f$ be a modular function. Say that $f$ is $p$-adically annihilated if the sequence $f\mid U_p^n$ uniformly converges to $0$ in the $p$-adic limit as…

Let $V^\natural_{\mathbb{Z}}$ be a self-dual integral form of $V^\natural$ with Monster symmetry, and let $W_\infty M$ denote the big Witt vectors of a graded module $M$.…

Let $\mathbb{M}$ be the Monster group, let $g\in\mathbb{M}$ be non-Fricke, and let ${}^g\hat V$ be the module assigned to $g$ in the proposed unified Moonshine construction. Non-Fr…

Let $\mathbb{M}$ be the Monster group. For each $g\in\mathbb{M}$, seek a $\frac{1}{|g|}\mathbb{Z}$-graded super-module ${}^g\hat V$ over $\mathbb{Z}[e^{2\pi i/|g|}]$, equipped with…

Let $\mathbb{M}$ be the Monster group, let $V^\natural_{\mathbb{Z}}$ be a self-dual integral form of the Moonshine vertex operator algebra, and let $g\in\mathbb{M}$ be a Fricke ele…

Let $V^\natural$ be the Moonshine vertex operator algebra, let $\mathbb{M}$ be the Monster group, and let $\alpha^\natural\in H^3(\mathbb{M},U(1))$ be the Moonshine anomaly. Write…

Let $\mathbb{M}$ be the Monster group, and let $\alpha^\natural\in H^3(\mathbb{M},U(1))$ be the Moonshine anomaly controlling the constants in Generalized Moonshine functions…

Conway–Norton–Thompson moonshine conjectures. There exists such a graded vector space with

Moonshine-module heterotic theory conjecture. Such a theory can be found by making this replacement.

Let … be the indicated completely replicable function. Non-moonshine-likeness conjecture for $f_{9a}$. The function $f_{9a}$ is non-moonshine-like. Establishing this would remove t…

Let $f$ be a moonshine-like function, meaning a completely replicable function given by the graded trace of a finite-order automorphism of a $C_2$-cofinite, holomorphic vertex oper…

The Monster CFT is the conformal field theory associated with the Monster module, and $\mathrm{AdS}_3$ denotes three-dimensional anti-de Sitter space. Monster CFT holographic duali…