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Journal of the London Mathematical Society Advance Access originally published online on May 27, 2009
Journal of the London Mathematical Society 2009 80(1):191-211; doi:10.1112/jlms/jdp022
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© 2009 London Mathematical Society

The Nottingham group and local class field theory

François Laubie and Olivier Moioli

Département de Mathématiques
Xlim UMR 6172
Université de Limoges
123, Avenue Albert Thomas
87060 Limoges cedex
France
oliviermoioli@hotmail.fr

Let Formulaq be the algebraic closure of the finite field Fq. Let G be the group of continuous Formulaq-automorphisms {sigma} of the abelian closure of the series field Fq((t)) such that {sigma}(t) isin tFq[[t]]x. The set Formulaq [[t]]x is a group for the non-commutative Ore multiplication of the series (the ordinary multiplication twisted by the Frobenius map). In this paper, by means of the Koch–de Shalit reciprocity map, we construct a subgroup W of this group and an isomorphism {iota} of W onto G, which extends the Artin reciprocity map. Thus the Nottingham group can be described with Ore multiplication; this description gives rise to a larger interpretation of the Schmid local symbol and gives some information on its finite abelian subgroups. We study the behaviour of {iota} relative to ramification and norm mapping.

2000 Mathematics Subject Classification 11S31.

Received January 24, 2008; revised October 16, 2008; published online May 27, 2009.


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