On the proof of Humbert's volume formula
type: Item
instance of: scholarly article
title: On the proof of Humbert's volume formula
zbMATH Open document ID: 0897.11013
author: Stefan Kühnlein
author: Fritz Grunewald
published in: Manuscripta Mathematica
publication date: 1998-10-19T00:00:00Z
full work available at URL: https://eudml.org/doc/156386
review text: 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.
Mathematics Subject Classification ID: 11F06
Mathematics Subject Classification ID: 11H06
Mathematics Subject Classification ID: 11S40
zbMATH DE Number: 1158110
zbMATH Keywords: volume of fundamental domains
zbMATH Keywords: JFM 47.0138.01
zbMATH Keywords: hyperbolic volume
zbMATH Keywords: Humbert's volume formula
zbMATH Keywords: Dedekind zeta function
reviewed by: Jürgen Elstrodt
MaRDI profile type: Publication
OpenAlex ID: W2006506406
DOI: 10.1007/S002290050039
description: scientific article; zbMATH DE number 1158110