Calculation of Hilbert Borcherds Products

Abstract

In Brunier and Bundschuh, “On Borcherds Products Associated with Lattices of Prime Discriminant.” Ramanujan Journal 7 (2003), 49–61, the authors use Borcherds lifts to obtain Hilbert modular forms. Another approach is to calculate Hilbert modular forms using the Jacquet--Langlands correspondence, which was implemented by Lassina Dembele in "Magma". In Mayer, "Rings of Hilbert Modular Forms for the Fields $\Q(\sqrt{13})$ and $\Q(\sqrt{17})$,'' To appear, 2009, we use Brunier and Bundschuh to determine the rings of Hilbert modular forms for $\{Q}(\sqrt{13})$ and $\{Q}(\sqrt{17})$. In the present note we give the major calculational details and present some results for $\{K}=\{Q}(\sqrt{5})$, $\{K}=\{Q}(\sqrt{13})$, and $\K=\{Q}(\sqrt{17})$. For calculations in the ring $\{o}$ of integers of $\{K}$ we order $\{o}$ by the norm of its elements and get for fixed norm, modulo multiplication by $\pm \varepsilon_0^{2\Z}$, a finite set. We use this decomposition to describe Weyl chambers and their boundaries, to determine the Weyl vector of Borcherds products, and hence to calculate Borcherds products. As a further example we calculate Fourier expansions of Eisenstein series.