The Kubota symbol for \(\mathrm{Sp}(2n,\mathbb Q(i))\) (Q1892568)

From the introduction: Following work of Kubota, \textit{H. Bass} et al. [Publ. Math., Inst. Hautes Étud. Sci. 33, 59--137 (1968; Zbl 0174.05203)] constructed characters of congruence subgroups of the modular subgroups of \(\mathrm{SL}(n)\) and \(\mathrm{Sp}(2n)\) over a totally complex number field that contains the \(r\)-th roots of unity, which are related to the \(r\)-th power residue symbol. In 1990, \textit{D. Bump} et al. [Nagoya Math. J. 119, 173--188 (1990; Zbl 0686.10020)] gave a construction and a precise formula for the Kubota symbol for \(\mathrm{Sp}(4,\mathbb Q(i))\) independent of the work of Bass et al. in the case \(r=2\). The main result of this paper is a new proof and a generalization to \(\mathrm{Sp}(2n,\mathbb Q(i))\) of the result of Bump et al. For this purpose a new theta function \(\theta (Z)\), which transforms under a subgroup \(\Gamma_\theta\) of the complex symplectic group \(\mathrm{Sp}(2n,\mathbb Z[i])\), is defined. \(\mathrm{Sp}(2n,\mathbb C)\) acts on the corresponding symmetric space, the quaternionic upper half space \[ H_n= \{Z= X+ Y{\mathbf k}\in M(n,{\mathbf H}) \mid X,\;Y\in M(n, \mathbb C), {}^t X= X,\;{}^t\overline {Y}= Y>0\}. \] It is shown that for \(M\in \Gamma_\theta\), the new theta function satisfies the invariance property \[ \theta (MZ)= \psi (M) \theta (Z), \tag \(*\) \] where \(\psi (M)\) is an eighth root of unity. Since there are no choices of square root in this transformation formula, in fact the map \(\psi: \Gamma_\theta\to \{\pm 1\}\) is a homomorphism and generalizes the homomorphisms of Kubota and Bump, Friedberg and Hoffstein. For \(n=1\) this method was used by \textit{S. Friedberg} and \textit{S.-T. Wong} [Math. Ann. 290, 183--207 (1991; Zbl 0726.11031)].