Computing Arakelov class groups

Summary

Shanks’s infrastructure algorithm and Buchmann’s algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class groups and of the set of reduced Arakelov divisors. As an application we describe Buchmann’s algorithm in this context.

Daniel Shanks [1972] observed that the forms in the principal cycle of reduced binary quadratic forms of positive discriminant exhibit a group-like behavior. This was a surprising phenomenon, because the principal cycle itself constitutes the trivial class of the class group. Shanks called this group-like structure ‘inside’ the neutral element of the class group the infrastructure . He exploited it by designing an efficient algorithm to compute the regulator of a real quadratic number field. Later, H. W. Lenstra [1982] (see also [Schoof 1982]) made Shanks’ observations more precise, introducing a certain topological group and providing a satisfactory framework for Shanks’s algorithm. Both Shanks [1976, Section 1; 1979, 4.4], and Lenstra [1982, section 15] indicated that the infrastructure ideas could be generalized to arbitrary number fields. This was done first by H.Williams and his students [Williams et al. 1983] for complex cubic fields, then by J. Buchmann [1987a; 1987b; 1987c] and by Buchmann and Williams [1989]. Finally Buchmann [1990; 1991] described an algorithm for computing the class group and regulator of an arbitrary number field that, under reasonable assumptions, has a subexponential running time. It has been implemented in the computer algebra packages LiDIA, MAGMA and PARI.