8 problems

Convergence conjecture. For any positive integer $k$, we have

The characterization of Eisenstein-type forms. The subspace of $M_{a,b}(\operatorname{SO}(\Lambda),\vartheta_d)$ spanned by forms of Eisenstein type is the span of constant functio…

Let $G_p$ be the genus of lattices of rank $6$ and discriminant $D=p$. Kernel-dimension conjecture. The kernel of $\theta^{(2)}$ on $G_p$ has dimension equal to the number of class…

Let $G_p$ be the genus of lattices of rank $6$ and discriminant $p$, and let $k_p$ be the number of isomorphism classes in $G_p$ of lattices with no automorphism of determinant…

Freeness conjecture. If $M_*(\Gamma')$ is a free algebra, then the smaller algebra $M_*(\Gamma)$ is also free.

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 a positive-definite lattice such that $2U\oplus L(-1)$ is a $2$-reflective lattice of type (b), meaning that every lattice in the genus of $L$ has no $2$-root. Leech-lat…

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…