Some families of non-isomorphic maximal function fields

TL;DR

This paper studies a specific family of maximal function fields derived from Hermitian fields, analyzing their automorphisms and isomorphism classes, revealing instances of non-isomorphic fields sharing key properties.

Contribution

It introduces a detailed analysis of Galois subfields of Hermitian function fields, including automorphism groups and isomorphism classifications, highlighting non-isomorphic fields with identical invariants.

Findings

Identified families of maximal function fields with the same genus and automorphism group

Demonstrated existence of non-isomorphic fields sharing key invariants

Computed automorphism groups and Weierstrass semigroups for these fields

Abstract

The problem of understanding whether two given function fields are isomorphic is well-known to be difficult, particularly when the aim is to prove that an isomorphism does not exist. In this paper we investigate a family of maximal function fields that arise as Galois subfields of the Hermitian function field. We compute the automorphism group, the Weierstrass semigroup at some special rational places and the isomorphism classes of such function fields. In this way, we show that often these function fields provide in fact examples of maximal function fields with the same genus, the same automorphism group, but that are not isomorphic.