2009/06/24 by Gunther Cornelissen, Aristides Kontogeorgis, Cornelissen, Gunther +3
Mathematics · #11G09 #11M38 #11R58 #14H05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.0906.4424
openalex publication_date 2009/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.