Arithmetic equivalence for non-geometric extensions of global function fields

TL;DR

This paper investigates arithmetically equivalent non-geometric extensions of function fields over finite fields, extending previous results and providing explicit examples that differentiate these extensions over larger constant fields.

Contribution

It extends the theory of arithmetic equivalence to non-geometric extensions and constructs explicit examples illustrating their properties and differences over various constant fields.

Findings

Extended arithmetic equivalence results to non-geometric extensions.

Constructed explicit examples of non-isomorphic, arithmetically equivalent extensions.

Solved a specific Inverse Galois Problem for function fields.

Abstract

In this paper we study couples of finite separable extensions of the function field $\mathbb{F}_q(T)$ which are arithmetically equivalent, i.e. such that prime ideals of $\mathbb{F}_q[T]$ decompose with the same inertia degrees in the two fields, up to finitely many exceptions. In the first part of this work, we extend previous results by Cornelissen, Kontogeorgis and Van der Zalm to the case of non-geometric extensions of $\mathbb{F}_q(T)$, which are fields such that their field of constants may be bigger than $\mathbb{F}_q$. In the second part, we explicitly produce examples of non-geometric extensions of $\mathbb{F}_2(T)$ which are equivalent and non-isomorphic over $\mathbb{F}_2(T)$ and non-equivalent over $\mathbb{F}_4(T)$, solving a particular Inverse Galois Problem.