The Infrastructure of a Global Field of Arbitrary Unit Rank

TL;DR

This paper introduces a generalized framework for understanding the infrastructure of global fields with arbitrary unit rank, connecting it to divisor class groups and extending computational methods like baby-step giant-step.

Contribution

It generalizes existing concepts of infrastructure, giant steps, and f-representations to arbitrary unit rank global fields, linking them to divisor class groups and extending computational techniques.

Findings

Established a general interpretation of infrastructure for arbitrary unit rank fields

Connected infrastructure concepts to divisor class groups of global fields

Demonstrated effective implementation and generalization of baby-step giant-step method

Abstract

In this paper, we show a general way to interpret the infrastructure of a global field of arbitrary unit rank. This interpretation generalizes the prior concepts of the giant step operation and f-representations, and makes it possible to relate the infrastructure to the (Arakelov) divisor class group of the global field. In the case of global function fields, we present results that establish that effective implementation of the presented methods is indeed possible, and we show how Shanks' baby-step giant-step method can be generalized to this situation.