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 >= 15 and n not equal 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 circle plus 2E(8)(-1) circle plus <-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.

Automated summary

The paper provides an explicit formula to express the weight of 2-reflective modular forms and proves the non-existence of 2-reflective lattices of signature (2, n) when n is greater than or equal to 15 and not equal to 19. Applications of the results include a simple proof of Looijenga's theorem and classification of reflective modular forms on lattices of large rank.