On Abelianized Absolute Galois Group of Global Function Fields

TL;DR

This paper characterizes the abelianized absolute Galois group of a global function field as a pro-finite group, linking it to the field's characteristic and class group, and showing the extent of its invariance.

Contribution

It provides a detailed description of the abelianized Galois group and establishes its connection to key invariants of the function field, with near-complete characterization.

Findings

The characteristic of the field is determined by the abelianized Galois group.

The non p-part of the class group is encoded in the Galois group.

Isomorphism type of the Galois group is almost determined by field invariants.

Abstract

The main purpose of this paper is to describe the abelian part $\mathcal G^{ab}_{K}$ of the absolute Galois group of a global function field $K$ as pro-finite group. We will show that the characteristic $p$ of $K$ and the non $p$-part of the class group of $K$ are determined by $\mathcal G^{ab}_{K}$. The converse is almost true: isomorphism type of $\mathcal G_K^{ab}$ as pro-finite group is determined by the invariant $d_K$ of the constant field $\mathbb F_q$ introduced in first section and the non $p$-part of the class group.