Another product for a Borcherds form

TL;DR

This paper extends Borcherds' 1998 construction by providing a new product formula for Borcherds automorphic forms near 1-dimensional boundary components, enhancing understanding of their behavior on compactifications.

Contribution

It introduces an analogous product formula for Borcherds forms near 1-dimensional boundary components, expanding the analytic tools for studying these automorphic forms.

Findings

Derived a product formula near 1-dimensional boundary components.

Revealed the behavior of Borcherds forms on partial compactifications.

Added Fourier-Jacobi coefficient information to the analysis.

Abstract

In his celebrated 1998 Inventiones paper, Borcherds constructed meromorphic automorphic forms Psi(F) for arithmetic subgroups associated to even integral lattices M of signature (n,2). The input to his construction is a vector valued weakly holomorphic modular form F of weight 1 - n/2, and the resulting Borcherds form has an explicit divisor on the arithmetic quotient X = Gamma_M\ D. Most remarkably, in the neighborhood of each cusp (= rational point boundary component), there is a beautiful product formula for Psi(F), reminiscent of the classical product formula for the Dedekind eta-function. In this paper, we describe an analogous product formula for Psi(F) in the neighborhood of each 1-dimensional rational boundary component. This formula, which, like that of Borcherds, is obtained through the calculation of a regularized theta integral, reveals the behavior of Psi(F) on a (partial) smooth compactification of X. Information about Fourier-Jacobi coefficients is added to this revised version.