Algebra of Borcherds products

Abstract

Borcherds lift for an even lattice of signature ( p , q ) (p, q) is a lifting from weakly holomorphic modular forms of weight ( p − q ) / 2 (p-q)/2 for the Weil representation. We introduce a new product operation on the space of such modular forms and develop a basic theory. The product makes this space a finitely generated filtered associative algebra, without unit element and noncommutative in general. This is functorial with respect to embedding of lattices by the quasi-pullback. Moreover, the rational space of modular forms with rational principal part is closed under this product. In some examples with p = 2 p=2 , the multiplicative group of Borcherds products of integral weight forms a subring.