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

Opitz, Sebastian; Schwagenscheidt, Markus

doi:10.1090/proc/14650

Holomorphic Borcherds products of singular weight for simple lattices of arbitrary level
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by Sebastian Opitz and Markus Schwagenscheidt PDF

Proc. Amer. Math. Soc. 147 (2019), 4639-4653 Request permission

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.

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