On reduced Arakelov divisors of real quadratic fields

TL;DR

This paper introduces a generalized concept of reduced Arakelov divisors for real quadratic fields, defines $C$-reduced divisors, and presents a polynomial-time algorithm to test their reduction status, with limitations shown through a cubic field example.

Contribution

It extends the theory of reduced Arakelov divisors by defining $C$-reduced divisors and provides an efficient algorithm for testing their properties in real quadratic fields.

Findings

The algorithm runs in polynomial time relative to the logarithm of the discriminant.

$C$-reduced divisors share key properties with traditional reduced divisors.

An example cubic field demonstrates the algorithm's limitations.

Abstract

We generalize the concept of reduced Arakelov divisors and define $C$-reduced divisors for a given number $C \geq 1$. These $C$-reduced divisors have remarkable properties which are similar to the properties of reduced ones. In this paper, we describe an algorithm to test whether an Arakelov divisor of a real quadratic field $F$ is $C$-reduced in time polynomial in $\log|\Delta_F|$ with $\Delta_F$ the discriminant of $F$. Moreover, we give an example of a cubic field for which our algorithm does not work.