Cyclotomic System and their Arithmetic
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912159264407552 |
|---|---|
| author | Zhu, Li Liu, Jinle Wu, Hongfeng |
| author_facet | Zhu, Li Liu, Jinle Wu, Hongfeng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12455 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cyclotomic System and their Arithmetic Zhu, Li Liu, Jinle Wu, Hongfeng Number Theory 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. |
| title | Cyclotomic System and their Arithmetic |
| topic | Number Theory |
| url | https://arxiv.org/abs/2412.12455 |