16 problems

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 $U(\tau)$ 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 $z_1,\ldots,z_n\in\mathbb{H}^{+}$, 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 $\vec v(q)$ 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…

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…

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 $d4dc$ 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 $d_1$ and $d_2$ be the discriminants under consideration, with $d1 e d2$ and $d1 e 0$, $d2 e 0$. Assume $d1 \equiv d2 \equiv 5\pmod 8$. 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…

Atkin's conjecture. For any prime $p\neq13$ such that $c_{p^m}(j)\neq0$ for all $m\geq1$,