Non-isomorphic subfields of the BM and GGS maximal function fields

TL;DR

This paper constructs new subfields of the second generalized GK function field (BM) and compares them to previously known subfields of the first generalized GK function field (GGS), revealing non-isomorphism conditions based on automorphism groups.

Contribution

It introduces analogous subfields of the BM function field, analyzes their automorphism groups, and establishes criteria for non-isomorphism with GGS subfields.

Findings

Subfields of BM and GGS are not isomorphic unless specific divisibility conditions are met.

Automorphism groups serve as invariants to distinguish non-isomorphic subfields.

The difference between BM and GGS function fields is reflected at the subfield level.

Abstract

In 2016 Tafazolian et al. introduced new families of $\mathbb{F}_{q^{2n}}$-maximal function fields $\mathcal{Y}_{n,s}$ and $\mathcal{X}_{n,s,a,b}$ arising as subfields of the first generalized GK function field (GGS). In this way the authors found new examples of maximal function fields that are not isomorphic to subfields of the Hermitian function field. In this paper we construct analogous function fields $\tilde{\mathcal{Y}}_{n,s}$ and $\tilde{\mathcal{X}}_{n,s,a,b}$ as subfields of the second generalized GK function field (BM) and determine their automorphism groups. Using that the automorphism group is an invariant under isomorphism, we show that the function fields $\tilde{\mathcal{Y}}_{n,s}$ and ${\mathcal{Y}}_{n,s}$, as well as $\tilde{\mathcal{X}}_{n,s,a,b}$ and $\mathcal{X}_{n,s,a,b}$, are not isomorphic unless $m/s$ divides $q^2-q+1$ and $3$ divides $n$. In other words, the difference between the BM and GGS function fields can be found again at the level of the subfields that we consider.