Reflective modular forms: A Jacobi forms approach

TL;DR

This paper develops an explicit formula for the weight of 2-reflective modular forms, classifies such forms on large lattices, and proves non-existence results for certain signatures, advancing the understanding of reflective modular forms.

Contribution

It provides a new explicit formula for the weight of 2-reflective modular forms and classifies these forms on large lattices, including non-existence results.

Findings

No 2-reflective lattice of signature (2,n) for n≥15 and n≠19, except specific unimodular cases.

A simple proof that certain lattices are not 2-reflective for n>1.

Classification of reflective modular forms with simplest reflective divisors.

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 $n\geq 15$ and $n\neq 19$ 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 $2U\oplus 2E_8(-1)\oplus\langle -2n\rangle$ 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.