Schoof, RenĂ© Computing Arakelov class groups. (English) Zbl 1188.11076 Buhler, J. P. (ed.) et al., Algorithmic number theory. Lattices, number fields, curves and cryptography. Cambridge: Cambridge University Press (ISBN 978-0-521-80854-5/hbk). Mathematical Sciences Research Institute Publications 44, 447-495 (2008). The author studies Arakelov divisors and Arakelov class groups in number fields. He demonstrates that this concept is a natural generalization of the much earlier “infrastructure” theory of Shanks for quadratic number fields. He also shows that Buchmann’s algorithm for calculating the class group and the regulator of a number field (which has subexponential running time under reasonable assumptions) can be deduced from his Arakelov point of view. A major part of the article treats the development of a “geometry of numbers” for Arakelov divisors leading to the concept of reduction. In the last three sections computational aspects are considered and a deterministic algorithm for the calculation of the Arakelov class group of a number field is presented.For the entire collection see [Zbl 1154.11002]. Reviewer: Michael Pohst (Berlin) Cited in 2 ReviewsCited in 16 Documents MSC: 11Y40 Algebraic number theory computations 14G40 Arithmetic varieties and schemes; Arakelov theory; heights Keywords:Arakelov class groups; class group computation PDFBibTeX XMLCite \textit{R. Schoof}, Math. Sci. Res. Inst. Publ. 44, 447--495 (2008; Zbl 1188.11076) Full Text: arXiv