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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.

MSC:
11Y40Algebraic number theory computations
14G40Arithmetic varieties and schemes; Arakelov theory; heights