Cyclotomic System and their Arithmetic

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 $\mathcal{P}\mathcal{C}(\ell ,q,n)$ with base module $n$ and the total $q$-cyclotomic system ${\mathcal{P}\mathcal{C}}_q$, 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.

31 pages