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Determinants of Laplacians on Hilbert modular surfaces. (English) Zbl 1440.11172

Summary: We study regularized determinants of Laplacians acting on the space of Hilbert-Maass forms for the Hilbert modular group of a real quadratic field. We show that these determinants are described by Selberg type zeta functions introduced in [Y. Gon, Proc. Japan Acad., Ser. A 88, No. 9, 145–148 (2012; Zbl 1305.11077); J. Number Theory 147, 396–453 (2015; Zbl 1380.11073); ].

MSC:

11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
58J52 Determinants and determinant bundles, analytic torsion
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