The classification of 2-reflective modular forms

Abstract

The classification of reflective modular forms is an important problem in the theory of automorphic forms on orthogonal groups. In this paper, we develop an approach based on the theory of Jacobi forms to give a full classification of 2-reflective modular forms. We prove that there are only 3 lattices of signature $(2,n)$ having 2-reflective modular forms when $n\geq 14$. We show that there are exactly 51 lattices of type $2U\oplus L(-1)$ which admit 2-reflective modular forms and satisfy that $L$ has 2-roots. We further determine all 2-reflective modular forms giving arithmetic hyperbolic 2-reflection groups. This is the first attempt to classify reflective modular forms on lattices of arbitrary level.