Experimental Mathematics

Calculation of Hilbert Borcherds Products

Sebastian Mayer

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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.

Article information

Experiment. Math., Volume 19, Issue 2 (2010), 243-256.

First available in Project Euclid: 17 June 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20]

Hilbert modular forms Borcherds products


Mayer, Sebastian. Calculation of Hilbert Borcherds Products. Experiment. Math. 19 (2010), no. 2, 243--256. https://projecteuclid.org/euclid.em/1276784793

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