Arithmetic equivalence for function fields, the Goss zeta function and a generalization

TL;DR

This paper explores the relationship between the Goss zeta function and Gassmann-equivalence in function fields, revealing conditions under which they coincide and introducing a Teichmueller lift to generalize the equivalence.

Contribution

It establishes a connection between the Goss zeta function and Gassmann-equivalence, and introduces a Teichmueller lift to extend the correspondence.

Findings

Equality of Goss zeta functions implies Gassmann-equivalence when characteristic exceeds degree.

The equivalence fails when degree exceeds characteristic.

The Teichmueller lift ensures Gassmann-equivalence corresponds to zeta function equality.

Abstract

A theorem of Tate and Turner says that global function fields have the same zeta function if and only if the Jacobians of the corresponding curves are isogenous. In this note, we investigate what happens if we replace the usual (characteristic zero) zeta function by the positive characteristic zeta function introduced by Goss. We prove that for function fields whose characteristic exceeds their degree, equality of the Goss zeta function is the same as Gassmann-equivalence (a purely group theoretical property), but this statement fails if the degree exceeds the characteristic. We introduce a `Teichmueller lift' of the Goss zeta function and show that equality of such is always the same as Gassmann equivalence.