33 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…

Let $z_1,\ldots,z_n\in\mathbb{H}^{+}$, write $\mathbf{z}=(z_1,\ldots,z_n)$, and let $j$ be the modular function. Let $\Sigma$ be the set of special points, and let…

Let $f=q^{-1}+\sum_{n=1}^\infty a_nq^n$ be a formal $q$-series with rational coefficients that is replicable. Norton's conjecture. The series $f$ is either a Hauptmodul for a genus…

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…

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

Let $K$ be an imaginary quadratic field of discriminant $d_K$ other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, let $\mathfrak{n}$ be a proper nontrivial ideal of its…

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

Kaneko's interlacing conjecture. The function $\mathrm{Re}(\mathrm{val})$ has an “interlacing property” on the Markov tree; equivalently,…

Let $U(\tau)$ be the modular function considered above, and for an odd prime $p\neq 5$ let $U_p(X,Y)=0$ denote its modular equation of level $p$. Kronecker congruence conjecture. F…

Let $p_d(x)$ be the minimal polynomial of $\beta(w/3)$, where … according as $-d \equiv 1 \pmod{3}$ is even or odd, and where $9 \mid N(w)$. Conjecture. The irreducible factors of…

Let $w$ be a Markov irrationality, and consider the Markov–Hurwitz tree of Markov irrationalities. The two Markov irrationalities lying above $w$ on the tree determine two correspo…

The $p$-adic singular-moduli approximation conjecture. Every algebraic number is badly approximable in $\mathbb{C}_p$ by the singular moduli of the $j$-invariant.

Let $\vec v(q)$ be a vector-valued modular function with components $v_i(q)$, and write the Fourier expansion of $v_i(q)$ with leading exponent $m_0^{(i)}$ and Fourier coefficients…

Degree-seven Rogers–Ramanujan relation. The function satisfies

Let $J:\operatorname{\mathbb{H}}\to\operatorname{\mathbb{C}}^3$ be defined by … and extend it coordinatewise to $\operatorname{\mathbb{H}}^n$, with graph…

Let $\operatorname{\mathbb{H}}$ be the complex upper half-plane, let $j:\operatorname{\mathbb{H}}\to\operatorname{\mathbb{C}}$ be the modular $j$-function, and let…

Composite-order graded Brauer character conjecture. Each of these series is a linear combination of Hauptmoduls. This predicts a uniform Hauptmodul description for the graded Braue…

Suppose $\rho$ is a tree monodromy representation giving rise to Shabat–Blaschke products of degree $n$ with exactly two critical values in $\mathbb{D}$. For each fixed…

Effective André–Oort-type conjecture. There are effectively computable constants $c_i=c_i(V)$ such that whenever

Collinear triples conjecture. There are only finitely many triples with the $P_i$ pairwise distinct and lying on a straight line which is neither horizontal, vertical nor the diago…

Let $D$ range over fundamental discriminants. For the values of the Klein modular invariant $j$ at real quadratic irrationalities, define … where the sum is over classes of indefin…

Let $d_1$ and $d_2$ be the discriminants under consideration, with $d1 e d2$ and $d1 e 0$, $d2 e 0$. Assume $d1 equiv d2 equiv 5pmod 8$. For each $t=frac{m+sqrt D}{2}inmathcal OF$,…

Let $\mathfrak{L}_{j}$ denote the language of the reduct associated with the differential equation of the modular $j$-function, and let $\emph{EC}$ denote the existential-closednes…

Let $E_i=\mathbb{Q}(\sqrt{d_i})$ be imaginary quadratic fields of fundamental discriminants $d_i$ with $(d_1,d_2)=1$, let $F=\mathbb{Q}(\sqrt D)$ with $D=d_1d_2$, and let…

Let $\chi^*$ be the level $1$ almost holomorphic modular function defined by … where $E_4$ and $E_6$ are the usual Eisenstein series and…