Simple lattices and free algebras of modular forms

We study the algebras of modular forms on type IV symmetric domains for simple lattices; that is, lattices for which every Heegner divisor occurs as the divisor of a Borcherds product. For every simple lattice L of signature (n, 2) with 3 < n < 10, we prove that the graded algebra of modular forms for the maximal reflection subgroup of the orthogonal group of L is freely generated. We also show that, with five exceptions, the graded algebra of modular forms for the maximal reflection subgroup of the discriminant kernel of L is also freely generated.(c) 2022 Elsevier Inc. All rights reserved.