Algebra of Borcherds products

Ma, Shouhei

doi:10.1090/tran/8585

Algebra of Borcherds products
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by Shouhei Ma PDF

Trans. Amer. Math. Soc. 375 (2022), 4285-4305 Request permission

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.

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