The relative class number one problem for function fields, II

TL;DR

This paper proves that certain finite extensions of function fields with trivial relative class group are Galois and cyclic, using Weil polynomial classification and Jacobian analysis, advancing understanding of class number problems in algebraic geometry.

Contribution

It establishes that such extensions are Galois and cyclic, building on a classification of Weil polynomials and detailed analysis of Jacobian structures.

Findings

Extensions are Galois and cyclic under given conditions

Classification of Weil polynomials constrains possible extensions

Analysis of Jacobians and polarizations informs extension properties

Abstract

We establish that any finite extension of function fields of genus greater than 1 whose relative class group is trivial is Galois and cyclic. This depends on a result from a preceding paper which establishes a finite list of possible Weil polynomials for both fields. Given this list, we analyze most cases by computing options for the splittings of low-degree places in the extension, then consider the effect of these options on the Weil polynomials of certain isogeny factors of the Jacobian of the Galois closure. In one case, we use instead an analysis based on principal polarizations, modeled on an argument of Howe.