On the classification of reflective modular forms

TL;DR

This paper studies reflective modular forms on lattices, showing their connection to root systems, proving uniqueness and non-existence results for certain signatures, and providing an automorphic proof related to modular form algebras.

Contribution

It establishes a link between reflective modular forms and root systems on specific lattices, proves uniqueness and non-existence results, and offers an automorphic proof of a theorem on modular form algebras.

Findings

No lattice of signature (21,2) admits a reflective modular form.

The lattice 2U⊕D20 is the unique (22,2) lattice with a reflective Borcherds product.

The algebra of modular forms for certain groups is not freely generated when l≥11.

Abstract

A modular form on an even lattice $M$ of signature $(l,2)$ is called reflective if it vanishes only on quadratic divisors orthogonal to roots of $M$. In this paper we show that every reflective modular form on a lattice of type $2U\oplus L$ induces a root system satisfying certain constrains. As applications, (1) we prove that there is no lattice of signature $(21,2)$ with a reflective modular form and that $2U\oplus D_{20}$ is the unique lattice of signature $(22,2)$ and type $U\oplus K$ which has a reflective Borcherds product; (2) we give an automorphic proof of Shvartsman and Vinberg's theorem, asserting that the algebra of modular forms for an arithmetic subgroup of $\mathrm{O}(l,2)$ is never freely generated when $l\geq 11$. We also prove several results on the finiteness of lattices with reflective modular forms.