Higher $K$-Groups of Smooth Projective Curves Over Finite Fields

TL;DR

This paper investigates the structure of higher $K$-groups of smooth projective curves over finite fields, establishing analogues of Iwasawa theory and applying results to elliptic cryptography.

Contribution

It introduces an Iwasawa-theoretic framework for $K$-groups over finite fields and determines the structure of these groups for elliptic curves.

Findings

Established an analogue of Iwasawa theorem for $K$-groups.

Determined the structure of $K_n$ for elliptic curves.

Applied results to construct efficient elliptic cryptography systems.

Abstract

Let $X$ be a smooth projective curve over a finite field $\mathbb{F}$ with $q$ elements. For $m\geq 1,$ let $X_m$ be the curve $X$ over the finite field $\mathbb{F}_m$, the $m$-th extension of $\mathbb{F}.$ Let $K_n(m)$ be the $K$-group $K_n(X_m)$ of the smooth projective curve $X_m.$ In this paper, we study the structure of the groups $K_n(m).$ If $l$ is a prime, we establish an analogue of Iwasawa theorem in algebraic number theory for the orders of the $l$-primary part $K_n(l^m)\{l\}$ of $K_n(l^m)$. In particular, when $X$ is an elliptic curve $E$ defined over $\mathbb{F},$ our method determines the structure of $K_n(E).$ Our results can be applied to construct an efficient {\bf DL} system in elliptic cryptography.