A complete characterization of Galois subfields of the generalized Giulietti--Korchm\'aros function field

TL;DR

This paper provides a comprehensive classification of Galois subfields of generalized Giulietti--Korchmaros function fields, expanding the known list of genera of maximal function fields through explicit genus calculations.

Contribution

It offers a complete characterization of all Galois subfields for these function fields, including new genus values, advancing the understanding of their structure and properties.

Findings

Complete classification of Galois subfields for $ ext{GCK}$ function fields.

New genus values of maximal function fields identified.

Enhanced understanding of the structure of generalized Giulietti--Korchmaros fields.

Abstract

We give a complete characterization of all Galois subfields of the generalized Giulietti--Korchm\'aros function fields $\mathcal C_n / \fqn$ for $n\ge 5$. Calculating the genera of the corresponding fixed fields, we find new additions to the list of known genera of maximal function fields.