On the proof of Humbert's volume formula
Let \(K\) be the imaginary quadratic number field of discriminant \(D<0\) and \({\mathfrak v}\) its ring of integers. The group \(\Gamma= SL_2({\mathfrak v})\) acts discontinuously on the three-dimensional hyperbolic space \(\mathbb{H}\), and the hyperbolic volume of the orbit space \(\Gamma\setminus \mathbb{H}\) is given by Humbert's volume formula \[ \text{vol} (\Gamma\setminus \mathbb{H})= \frac{| D|^{3/2}} {4\pi^2} \zeta_K(2), \] where \(\zeta_K\) denotes the Dedekind zeta function of \(K\). Various modern proofs of this formula are known, whereas Humbert's original proof [C. R. 169, 448-454 (1991; JFM 47.0138.01)] contains a gap. The authors close this gap and give a rigorous proof of a crucial lemma.
scientific article; zbMATH DE number 1158110