Computing dimensions of spaces of Arakelov divisors of number fields

TL;DR

This paper introduces a computational method for determining the $h^0$ function of number fields with small unit rank, leveraging lattice reduction, the jump algorithm, and Poisson summation, even for fields with large discriminants.

Contribution

It presents a novel approach combining lattice reduction, the jump algorithm, and Poisson summation to compute $h^0$ for number fields with unit rank up to 2.

Findings

Effective computation of $h^0$ for large discriminant fields

Method applicable to fields with unit rank ≤ 2

Utilizes LLL-reduced bases and Poisson summation

Abstract

The function $h^0$ for a number field is analogous to the dimension of the Riemann-Roch spaces at divisors on an algebraic curve. We provide a method to compute this function for number fields with unit group of rank at most 2, even with large discriminant. This method is based on using LLL-reduced bases, the "jump algorithm" and Poisson summation formula.