Cyclotomic System and their Arithmetic

TL;DR

This paper introduces $ ext{q}$-cyclotomic systems based on projective limits of cyclotomic cosets, providing detailed classifications and algorithms for their representatives and sizes, with applications in number theory.

Contribution

It defines and analyzes $ ext{q}$-cyclotomic systems and offers algorithms for classifying and computing their cosets, advancing understanding of cyclotomic structures.

Findings

Characterization of $ ext{q}$-cyclotomic systems for odd $ ext{l}$ and $ ext{l}=2$

Classification of compatible sequences of cyclotomic cosets

Algorithm for determining coset representatives and sizes

Abstract

Let $q=p^{e}$ be a prime power, $\ell$ be a prime number different from $p$, and $n$ be a positive integer divisible by neither $p$ nor $\ell$. In this paper we define the $\ell$-adic $q$-cyclotomic system $\mathcal{PC}(\ell,q,n)$ with base module $n$ and the total $q$-cyclotomic system $\mathcal{PC}_{q}$, which are projective limits of certain spaces of $q$-cyclotomic cosets. Comparing to $q$-cyclotomic cosets modulo a fixed integer, the compatible sequences of $q$-cyclotomic cosets lying in these systems can be characterized and classified in a natural way. We give a detailedd description of the $\ell$-adic $q$-cyclotomic system in the cases where $\ell$ is an odd prime and where $\ell=2$ respectively. As an application, we represent an algorithm to determine a full set of representatives and the sizes of the cosets with any given parameters.