Bounds for the relative class number problem for function fields

TL;DR

This paper derives bounds on finite separable extensions of function fields based on the relative class number, enabling finite classification, and solves the relative class number two problem for most base fields.

Contribution

It provides explicit bounds linking extension properties to the relative class number and completely solves the relative class number two case for certain base fields.

Findings

Established bounds on extensions via relative class number

Reduced classification to finite computation

Solved the relative class number two problem for base fields not equal to _2

Abstract

We establish bounds on a finite separable extension of function fields in terms of the relative class number, thus reducing the problem of classifying extensions with a fixed relative class number to a finite computation. We also solve the relative class number two problem in all cases where the base field has constant field not equal to $\mathbb{F}_2$.