Wintenberger’s functor for abelian extensions

Abstract

Let k be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian p-adic Lie extensions E/F , where F is a local field with residue field k, and a category whose objects are pairs (K, A), where K ∼ = k((T )) and A is an abelian p-adic Lie subgroup of Aut k (K). In this paper we extend this equivalence to allow Gal(E/F ) and A to be arbitrary abelian pro-p groups.

Height of Some Automorphisms of Local Fields

“…Moreover, the canonical isomorphism from Gal(L/K) onto G preserves the ramification filtration [6,12]. This equivalence has been extended to allow Gal(L/K) and G to be arbitrary abelian pro-p groups by Keating [3]. In the following, we will simply say that G is corresponding to L/K if the extension L/K corresponds to (k((x)), G) by the equivalence of categories given by the field of norms functor.…”