Borcherds Products Everywhere

Abstract

We prove the Borcherds Products Everywhere Theorem, Theorem 6.6, that constructs holomorphic Borcherds Products from certain Jacobi forms that are theta blocks without theta denominator. The proof uses generalized valuations from formal series to partially ordered abelian semigroups of closed convex sets. We present nine infinite families of paramodular Borcherds Products that are simultaneously Gritsenko lifts. This is the first appearance of infinite families with this property in the literature.