A generalization of reduced Arakelov divisors of a number field

TL;DR

This paper introduces strongly C-reduced divisors in number fields, generalizing reduced Arakelov divisors inspired by the LLL-algorithm, and proves they maintain key properties of the original divisors.

Contribution

It defines a new class of divisors called strongly C-reduced, extending the concept of reduced Arakelov divisors with preserved structural properties.

Findings

Strongly C-reduced divisors form a finite set.

They are regularly distributed in the Arakelov class group.

Properties of reduced divisors are retained in the generalized form.

Abstract

Let $C \geq 1$. Inspired by the LLL-algorithm, we define strongly $C$-reduced divisors of a number field $F$ which are generalized from the concept of reduced Arakelov divisors. Moreover, we prove that strongly $C$-reduced Arakelov divisors still retain outstanding properties of the reduced ones: they form a finite, regularly distributed set in the Arakelov class group and the oriented Arakelov class group of $F$.