Holomorphic Borcherds products of singular weight for simple lattices of arbitrary level

TL;DR

This paper classifies holomorphic Borcherds products of singular weight for simple lattices of signature (2,n), introduces new automorphic products of weight 1/2, and relates some to Siegel modular forms, providing explicit descriptions and coefficient estimates.

Contribution

It provides a complete classification of such products for all simple lattices of signature (2,n) and introduces new automorphic forms of weight 1/2, expanding the understanding of these structures.

Findings

Classified all holomorphic Borcherds products of singular weight for simple lattices.

Discovered new automorphic products of singular weight 1/2 for signature (2,3).

Derived estimates for Fourier coefficients of vector valued Eisenstein series.

Abstract

We classify the holomorphic Borcherds products of singular weight for all simple lattices of signature $(2,n)$ with $n \geq 3$. In addition to the automorphic products of singular weight for the simple lattices of square free level found by Dittmann, Hagemeier and the second author, we obtain several automorphic products of singular weight $1/2$ for simple lattices of signature $(2,3)$. We interpret them as Siegel modular forms of genus $2$ and explicitly describe them in terms of the ten even theta constants. In order to rule out further holomorphic Borcherds products of singular weight, we derive estimates for the Fourier coefficients of vector valued Eisenstein series, which are of independent interest.