19 problems

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…

Let $\mathbb{M}$ be the Monster finite simple group, and let $J(q)$ be the normalized Hauptmodul for $\operatorname{SL}(2,\mathbb{Z})$. Monstrous Moonshine conjecture. There exists…

Let $d4dc$ be the monster group, let … be an infinite-dimensional graded representation with finite-dimensional graded parts, and for each conjugacy class $[g]$ of $d4dc$ let … be…

Let $\tilde J=J-744$ be the normalised $J$-function, and let $\rho atural$ be a graded representation on an infinite-dimensional vector space … whose graded parts $V^ aturali$ are f…

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 $V^\natural$ 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 … 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…

Let $V=\bigoplus_{n=0}^\fty V_n$ be a graded infinite-dimensional $\mathbb C$-vector space with $\dim V_n<\infty$ and a natural algebraic structure whose automorphism group is the…

Let $\mathbb M$ denote the Monster group, and let $H^3(\mathbb M,\mathrm U(1))$ be its third cohomology group with coefficients in $\mathrm U(1)$. Moonshine anomaly cohomology conj…

Moonshine uniqueness conjecture. Then $V$ is isomorphic to the moonshine module vertex operator algebra.

Let $\Gamma_g$ be the group associated to $g\in\mathbb{M}$, and let $T^{(-1)}_{\Gamma_g}$ denote the normalized Rademacher sum attached to $\Gamma_g$. Duncan--Frenkel conjecture. T…

Let $\mathbb{M}$ be the monster, and for each $g\in\mathbb{M}$ let $T_g$ be its monstrous McKay--Thompson series. A normalized principal modulus is a generator of the function fiel…

Let $\mathbb{M}$ be the monster, let $g,h\in\mathbb{M}$ be commuting elements, and let $\mathfrak{H}$ denote the upper half-plane. A generalized character is a function $Z$ associa…

Let $\mathbb M$ be the Monster. For each $g\in\mathbb M$, let $\mathfrak m_g$ be the corresponding twisted monstrous Lie algebra and let $\mathcal I_g$ be its instanton space. Assu…

Let $\mathbb M$ be the Monster, let $g\in\mathbb M$, let $\Gamma_g$ be the associated group, and let $\Gamma_{\infty}$ denote the cusp subgroup. Let $T_{\Gamma}^{(1)}$ be the norma…