Algebra of Borcherds products

Shouhei Ma

Published 2020-07-20Version 1

Borcherds lift for an even lattice L of signature (p,q) is a lifting from weakly holomorphic modular forms of weight (p-q)/2 for the Weil representation of L. We introduce a product operation on the space of such modular forms, depending on the choice of a maximal isotropic sublattice of L, which makes this space a finitely generated filtered associative algebra, without unit element in general. This algebra structure is functorial with respect to embedding of lattices by the quasi-pullback map. We study the basic properties, prove for example that the algebra is commutative if and only if L is unimodular. When L is unimodular with p=2, the multiplicative group of Borcherds products of integral weight forms a subring.