Reflective modular forms: A Jacobi forms approach

We give an explicit formula to express the weight of $2$-reflective modular forms. We prove that there is no $2$-reflective lattice of signature $(2,n)$ when $n\ge 15$ and $n\mathrm{\ne }19$ except the even unimodular lattices of signature $(2,18)$ and $(2,26)$. As applications, we give a simple proof of Looijenga’s theorem that the lattice $2U\oplus 2E_8(-1)\oplus ⟨-2n⟩$ is not $2$-reflective if $n>1$. We also classify reflective modular forms on lattices of large rank and the modular forms with the simplest reflective divisors.