We prove that every vector of $\mathrm{II}_{8n+1,1}$, with $n\ge1$, is a sum of two roots if and only if $n=1$ or $2$. This settles the Borisov--Nuer conjecture and implies the existence of Ulrich line bundles for every very ample polarization on a complex unnodal Enriques surface. For vectors of negative norm, we expr...
The space $M(\rho;\mathcal{E})$ of weakly holomorphic modular forms for a dimension-$r$ complex representation $\rho$ of $\mathrm{SL}_2(\mathbb{Z})$ with growth prescribed by $\mathcal{E}$ at the cusp is a free module over $\mathbb{C}[E_4,E_6]$. We first prove that a $\mathbb{C}[E_4,E_6]$-linearly independent homogeneo...
After proving the moonshine conjecture, Borcherds asked in 1995 whether there exist only finitely many holomorphic automorphic products of singular weight. We show that, up to conjugation, there are exactly 27 such products on lattices of the form $2U\oplus L$. The classification proceeds by identifying each product at...
