The relative class number one problem for function fields, III

TL;DR

The paper completes the relative class number one classification for function fields by reducing to genus $6$ and $7$ curves over $\bF_2$ whose Weil polynomials lie in a fixed set of $40$ entries, then identifying three curves that admit an everywhere unramified quadratic extension with trivial relative class group. It blends Mukai’s Brill–Noether descriptions of genus $6$ and $7$ canonical models with a geometry-guided Brill–Noether stratification to enumerate candidate curves, and introduces an orbit lookup-tree technique to manage $G$-orbits efficiently in the search. The main contributions include explicit realizations of two genus $6$ curves and one genus $7$ curve with the desired properties, a detailed computational framework (SageMath and Magma) to perform the necessary checks, and a discussion of strategies toward a full census of genus $6$ and $7$ curves over $\bF_2$. The work provides a rigorous, geometry-informed path to classify relative class number one extensions in these genera and offers methodological tools applicable to higher genera and related arithmetic-geometric counting problems.

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.

Theorems & Definitions (22)

• Theorem 1.1: Solution of the relative class number one problem
• Theorem 1.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Remark 3.4
• Lemma 4.1
• proof