Computing Arakelov class groups

TL;DR

This paper explores the properties of Arakelov class groups and reduced divisors, providing a new perspective on algorithms for computing class and unit groups of algebraic number fields.

Contribution

It offers a detailed analysis of Arakelov class groups and adapts Buchmann's algorithm within this framework, enhancing understanding of computational methods in algebraic number theory.

Findings

Analysis of Arakelov class groups and reduced divisors

Reformulation of Buchmann's algorithm in the Arakelov context

Insights into the structure of class groups and units

Abstract

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.