Reflective modular forms: A Jacobi forms approach

Abstract

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 $n\ne 19$ 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.

20 pages, revised according to the referees' comments, to appear in Int. Math. Res. Not. IMRN

References in corpus (4)

• On the Kodaira dimension of orthogonal modular varieties
• Lorentzian Kac-Moody algebras with Weyl groups of 2-reflections
• Reflective modular forms in algebraic geometry
• The Singular Theta Correspondence, Lorentzian Lattices and Borcherds-Kac-Moody Algebras

Cited by in corpus (1)

• The classification of 2-reflective modular forms