Generalised Kawada–Satake method for Mackey functors in class field theory

Abstract

We propose and study a generalised Kawada-Satake method for Mackey functors in the class field theory of positive characteristic. The root of this method is in the use of explicit pairings, such as the Artin-Schreier-Witt pairing, for groups describing abelian extensions. We separate and simplify the algebraic component of the method and discuss a relation between the existence theorem in class field theory and topological reflexivity with respect to the explicit pairing. We apply this method to derive higher local class field theory of positive characteristic, using advanced properties of topological Milnor K-groups of such fields. CONTENTS

Class field theory, its three main generalisations, and applications

“…For the p-primary part of CFT in characteristic p one cannot use Kummer theory. However, the Kawada-Satake method and its generalisations [13], using Witt theory, makes the p-primary part the easiest part of CFT in characteristic p. The nature of existence theorem in positive characteristic is clarified in [13]: it corresponds to topological reflexivity of (generally non-locally compact) groups with respect to a generalised explicit pairing. An explicit approach to higher CFT in positive characteristic by Parshin, [62,63], is a higher local fields extension of the Kawada-Satake method, [36], several errors and gaps in [62,63] were fixed in [13].…”

“…Unusually for CFT, this approach does not (yet) have local and local-to-global parts. 13 Remark. Unlike the classical one-dimensional case, 2dCFT is somehow separated from geometry.…”

“…In various generalisations of the classical class field theory such as class field theory for local fields with quasi-finite or perfect residue field,[5], the topology associated with norm subgroups is strictly weaker than the discrete valuation topology on the multiplicative group 3. See also[13] which connects aspects of existence theorem with topological reflexivity with respect to a related explicit pairing.…”