$2$-reflective lattices of signature $(n,2)$ with $n\geq 8$

Abstract

An even lattice $M$ of signature $(n,2)$ is called $2$-reflective if there is a non-constant modular form for the orthogonal group of $M$ which vanishes only on quadratic divisors orthogonal to $2$-roots of $M$. In [Amer. J. Math. 2017] Shouhei Ma proved that there are only finitely many $2$-reflective lattices of signature $(n,2)$ with $n\geq 7$. In this paper we extend the finiteness result of Ma to $n\geq 5$ and show that there are exactly forty-two $2$-reflective lattices of signature $(n,2)$ with $n\geq 8$.