The relative class number one problem for function fields, III

TL;DR

This paper completes the classification of function fields with relative class number one over finite fields, focusing on genus 6 and 7 over _2, by identifying specific Weil polynomials and verifying unramified quadratic extensions.

Contribution

It finalizes the solution to the relative class number one problem for certain function fields, using explicit enumeration and geometric stratification techniques.

Findings

Identified three function fields with the desired properties.

Reduced the problem to finding specific Weil polynomials.

Verified the existence of unramified quadratic extensions for these fields.

Abstract

We complete the solution of the relative class number one problem for function fields of curves over finite fields. Using work from two earlier papers, this reduces to finding all function fields of genus 6 or 7 over $\mathbb{F}_2$ with one of 40 prescribed Weil polynomials; one may then verify directly that three of these fields admit an everywhere unramified quadratic extension with trivial relative class group. The search is carried out by carefully enumerating curves based on the Brill--Noether stratification of the moduli spaces of curves in these genera, and particularly Mukai's descriptions of the open strata.