Algebra of Borcherds products

TL;DR

This paper introduces a new algebraic product on weakly holomorphic modular forms associated with Borcherds lift, creating a finitely generated filtered algebra structure that is functorial and has rational closure properties.

Contribution

It develops a novel product operation on modular forms for Borcherds lift, establishing a new algebraic framework and exploring its properties and examples.

Findings

The space of modular forms becomes a finitely generated filtered algebra.

The algebra is functorial under lattice embeddings.

Rational forms with rational principal parts are closed under the product.

Abstract

Borcherds lift for an even lattice of signature (p,q) is a lifting from weakly holomorphic modular forms of weight (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, the multiplicative group of Borcherds products of integral weight forms a subring.