On the theta constant of genus 8 and Hilbert modular groups over certain cyclic biquadratic fields (Q1097292)

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 \(F\) [\textit{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 \textit{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 \(\Gamma\).