Congruence Function Fields with Class Number One

TL;DR

This paper proves the uniqueness of a specific function field over a finite field with class number one and genus four, contributing to the classification of such fields.

Contribution

It establishes the existence and uniqueness of a function field over F_2 with class number one and genus four, completing part of the classification of congruence function fields.

Findings

Exactly one such function field exists over F_2 with genus four and class number one.

There are eight non-isomorphic congruence function fields with genus > 0 and class number one.

The result complements previous classifications by MacRae, Madan, Leitzel, Queen, and Stirpe.

Abstract

We prove that there exists, up to isomorphism, exactly one function field over the finite field of two elements of class number one and genus four. This result, together with the ones of MacRae, Madan, Leitzel, Queen and Stirpe, establishes that there exist eight non-isomorphic congruence function fields of genus larger than zero and class number one.