Arithmetic equivalence for function fields, the Goss zeta function and a generalisation |
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Authors: | Gunther Cornelissen Aristides Kontogeorgis |
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Affiliation: | a Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, The Netherlands b Department of Mathematics, University of the Aegean, Karlovasi, Samos, Greece |
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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 can fail if the degree exceeds the characteristic. We introduce a ‘Teichmüller lift’ of the Goss zeta function and show that equality of such is always the same as Gaßmann equivalence. |
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Keywords: | 11G09 11M38 11R58 14H05 |
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