A modular form on an even lattice M of signature (l, 2) is called reflective if it vanishes only on quadratic divisors orthogonal to the roots of M. We show that every reflective modular form on a lattice of type \(2U\hspace{1.111pt}{\oplus }\hspace{1.111pt}L\) induces a root system that satisfies certain constraints. As applications, (1) we prove that there is no lattice of signature (21, 2) with a reflective modular form and that \(2U\hspace{1.111pt}{\oplus }\hspace{1.111pt}D_{20}\) is the unique lattice of signature (22, 2) and type \(U\hspace{1.111pt}{\oplus }\hspace{1.111pt}K\) that has a reflective Borcherds product; (2) we give an automorphic proof of the theorem of Shvartsman and Vinberg, asserting that the algebra of modular forms for an arithmetic subgroup of \(\textrm{O}(l,2)\) can never be generated freely when \(l\geqslant 11\) . We also prove several results on the finiteness of lattices with reflective modular forms, and give an explicit example of hyperbolic reflection groups not coming from reflective modular forms.
