Dedekind sums and quadratic residue symbols of imaginary quadratic fields (Q1179717)

Let \(K\) be an imaginary quadratic field, \({\mathcal O}_ K\) its ring of integers and \(\Gamma(8)\) the principal congruence subgroup of level 8 of \(SL(2,{\mathcal O}_ K)\). The main result of this paper is the ``lifting'' of the Kubota homomorphism \(\chi:\Gamma(8)\to\mathbb{Z}/2\mathbb{Z}\) to a homomorphism \(\Phi:\Gamma(8)\to\mathbb{Z}\). This is accomplished by considering the generalized Dedekind sums of \textit{R. Sczech} [Invent. Math. 76, 523-551 (1984; Zbl 0521.10021)] and recognizing a) that these sums are homomorphisms from \(\Gamma(8)\) to the additive group of \(\mathbb{C}\), b) certain linear combinations of these Dedekind sums/homomorphisms provide the necessary lift.