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2010 Calculation of Hilbert Borcherds Products
Sebastian Mayer
Experiment. Math. 19(2): 243-256 (2010).

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.

Citation

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Sebastian Mayer. "Calculation of Hilbert Borcherds Products." Experiment. Math. 19 (2) 243 - 256, 2010.

Information

Published: 2010
First available in Project Euclid: 17 June 2010

zbMATH: 1211.11061
MathSciNet: MR2676751

Subjects:
Primary: 11F41

Rights: Copyright © 2010 A K Peters, Ltd.

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Vol.19 • No. 2 • 2010
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