On the non-existence of singular Borcherds products

TL;DR

This paper proves that certain reflective modular forms with specific properties are highly restricted, showing that non-trivial singular Borcherds products cannot exist beyond known cases, thus answering a long-standing question by Borcherds.

Contribution

It establishes the non-existence of singular Borcherds products for most cases, confirming that only known reflective forms occur in specific dimensions, and resolves a question posed by Borcherds in 1995.

Findings

Reflective modular forms have only simple zeros.

Such forms are anti-invariant under certain reflections.

Existence of these forms is limited to specific dimensions, notably 20 and 26.

Abstract

Let $l\geq 3$ and $F$ be a modular form of weight $l/2-1$ on $\mathrm{O}(l,2)$ which vanishes only on rational quadratic divisors. We prove that $F$ has only simple zeros and that $F$ is anti-invariant under every reflection fixing a quadratic divisor in the zeros of $F$. In particular, $F$ is a reflective modular form. As a corollary, the existence of $F$ leads to $l\leq 20$ or $l=26$, in which case $F$ equals the Borcherds form on $\mathrm{II}_{26,2}$. This answers a question posed by Borcherds in 1995.