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On the theta constant of genus 8 and Hilbert modular groups over certain cyclic biquadratic fields. (English) Zbl 0635.10023

Let \(F\) be a real cyclic biquadratic extension of \(\mathbb Q\) and \(\mathfrak o\) its ring of integers. Under the assumption that the discriminant of the unique quadratic subfield of \(F\) is even and using the arithmetic of \(F\) [H. Hasse, Abh. Deutsch. Akad. Wiss. Berlin, Math.-Nat. Kl. 2 (1950; Zbl 0035.305)], the author constructs an embedding of the Hilbert modular group \(\Gamma = \mathrm{SL}_2(\mathfrak o)\) into the unitary group \(\mathrm{SU}_4(\mathbb Z[i])\). \(\mathrm{SU}_4(\mathbb Z[i])\) can be embedded, after G. Shimura [Ann. Math. (2) 107, 569–605 (1978; Zbl 0409.10016)], into the symplectic group \(\mathrm{Sp}_8(\mathbb Z)\) and so the composite supplies an embedding of \(\Gamma\) into \(\mathrm{Sp}_8(\mathbb Z)\). At the end a theta function of \(\mathrm{Sp}_8(\mathbb Z)\) is used to get a Hilbert modular form of weight 2 with respect to \(\Gamma\).

MSC:

11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F06 Structure of modular groups and generalizations; arithmetic groups
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References:

[1] Hasse, H.: Arithmetische Bestimmung von Grundeinheit und Klassenzahl in zyklischen, kubischen und biquadratischen Zahlkörpern. Abh. Deutsch. Akad. Wiss. Berlin Math. Nat. Wiss. Kl.2 (1950) · Zbl 0035.30502
[2] Naganuma, H.: Remarks on the modular imbedding of Hammond. Jap. J. Math.10, 379-387 (1984) · Zbl 0588.10030
[3] Shimura, G.: The arithmetic of automorphic forms with respect to a unitary group. Ann. Math.107, 569-605 (1978) · Zbl 0409.10016 · doi:10.2307/1971129
[4] Vaserstein, L.N.: On the groupSL 2 over Dedekind ring of arithmetic type. Mat. Sb.89, 313-322 (1972)
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