Notes on Explicit Constructions of Arithmetically Equivalent Global Function Fields via Torsion Points On Drinfeld Modules

TL;DR

This paper presents explicit examples of non-isomorphic global function fields that are arithmetically equivalent, constructed using torsion points of Drinfeld Modules, and provides code for verification and further exploration.

Contribution

It introduces a method to explicitly construct arithmetically equivalent global function fields using torsion points of Drinfeld Modules, with accompanying computational tools.

Findings

Examples of arithmetically equivalent non-isomorphic function fields

Explicit construction method via Drinfeld Modules torsion points

Magma scripts for verification and further examples

Abstract

In this short note we provide a few examples of non-isomorphic arithmetically equivalent global function fields. These examples are obtained via well-known technique of adjoining the torsion points of various Drinfeld Modules to realise the $Gl_n(\mathbb F_q)$ as a Galois group of extensions of global function fields. Furthermore we afford the code of the Magma scripts to verify the results and construct more examples in similar fashion.