Igusa's $\chi_{10}$ and Borcherds products

Question

Igusa defined a genus 2 Siegel modular form $\chi_{10}$, which vanishes on the Humbert surface $G_{1}$ (the image of a "degenerate" Hilbert modular surface, the product of modular curves, inside the Siegel modular threefold, see for instance page 218 of van der Geer's Hilbert Modular Surfaces).

This should be related to Borcherds products, and in particular I have seen it mentioned (for instance in page 47 of this paper of Bruinier and Yang) that $\chi_{10}$ is up to a power of 2 the square of a Borcherds product. Is there a place where I can see this explicitly worked out? I would like to see, for example, the product expansion as well as which weakly holomorphic form this Borcherds product is lifted from, etc.

Answer 1

The product expansion and the weakly holomorphic (Jacobi) form it lifts from appear as Example 2.4 in Gritsenko, Nikulin - Automorphic Forms and Lorentzian Kac--Moody Algebras II, Internat. J. Math. 9(2) (1998), 201--275.

(arXiv).

The square root of $\chi_{10}$ is denoted there by $\Delta_5(Z)$.