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

作者：Cornelissen Gunther^*; Kontogeorgis Aristides; van der Zalm Lotte

来源：Journal of Number Theory, 2010, 130(4): 1000-1012.

DOI：10.1016/j.jnt.2009.08.002

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 'Teichmuller lift' of the Goss zeta function and show that equality of such is always the same as Gassmann equivalence.

• 出版日期2010-4