An ordinary cyclotomic function field

TL;DR

This paper determines the precise conditions under which cyclotomic function fields are ordinary by analyzing the p-rank of their Jacobians, contributing to the understanding of their algebraic structure.

Contribution

It provides a complete characterization of when cyclotomic function fields are ordinary based on the p-rank of their Jacobians.

Findings

Derived necessary and sufficient conditions for ordinarity

Characterized the p-rank of Jacobians in cyclotomic function fields

Enhanced understanding of algebraic properties of cyclotomic function fields

Abstract

In this paper, we investigate the $p$-rank of Jacobian of cyclotomic function field $K_m$. Our goal in this paper is to give the necessary and sufficient condition $m$ such that $K_m$ is ordinary.