15 problems

Let the pure theta block $\Theta_f$ be a holomorphic Jacobi form of weight $k$ and index $m$ with vanishing order $1$ in $q$, where $k,m\in\mathbb Z_{>0}$. Define the nearly holomo…

Cheng–Duncan conjecture. For each conjugacy class $g$ of $M_{24}$, the product $\Phi_g$ defines a Siegel modular form with some multiplier on the congruence subgroup…

Let $F$ be the real quadratic field and let $K$ be a CM extension of $F$ as in the preceding construction. Write $\mathcal{O}_F$ for the ring of integers of $F$, let $\mathfrak l$…

Let $g$ be an automorphism used to twist the fake monster algebra, and let its twisted denominator function be the associated denominator function. An automorphic form has singular…

A generalized Kac–Moody algebra $\mathfrak{g}$ whose denominator $\Phi_\mathfrak{g}$ is a singular Borcherds product is expected to arise from the BRST cohomology of a suitable ver…

Let $\Theta_f$ be a theta block of $q$-order one that is a holomorphic Jacobi form of integral weight $k$ and integral index $N$ for $A_1$. Let $\operatorname{Grit}(\Theta_f)$ deno…

Let $f:\mathbb N\to\mathbb N$ have finite support, and let … be the associated pure theta block, where $\eta$ is the Dedekind eta function and $\vartheta$ is the odd Jacobi theta f…

Let $d$ be the discriminant parameter of the imaginary quadratic field underlying the unitary group, let $L$ be a Hermitian lattice of signature $(n,1)$, and let…

Let $\Lambda$ be the Leech lattice, and let $g$ range over conjugacy classes of its automorphism group with non-trivial fixed-point lattice. Borcherds's conjecture. Each such conju…

Let $F$ be a Borcherds product of singular weight for an even lattice $M$ of signature $(2,n)$ with $n\geq 3$. Singular-weight Borcherds products conjecture. There exists an even l…

Let $L$ be the lattice indexing the Jacobi-form space $J_{k,L,1}$, and let $\phi\in J_{k,L,1}$. Let $T_{-}(2)$ denote the index-raising Hecke operator, and let…

Let $L$ be an even lattice of signature $(2,n)$ with $n\geq 3$. A holomorphic Borcherds product for $L$ has singular weight $\frac{n}{2}-1$, the smallest possible weight of a nonco…

Let $p$ be a prime number, and let a modular form have odd weight and come from a Borcherds product. Odd-weight Borcherds-product conjecture. Any modular form of odd weight coming…

Let $L$ be an even lattice of signature $(2,n)$, and let a lattice be simple if the space of cusp forms of weight $1+n/2$ for the Weil representation associated with the discrimina…

Let $J_{k,t}$ denote the space of Jacobi forms of weight $k$ and index $t$. Let $\theta$ be a theta block without theta denominator and with vanishing order one in $q=e(\tau)$, and…