2-Reflective Lattices of Signature (n, 2) with n ≥ 8

Abstract

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 2017, Ma [ 25] 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 $n\geq 8$.