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Zbl 1214.11130
Keating, Kevin
Wintenberger's functor for abelian extensions. (English)
[J] J. Théor. Nombres Bordx. 21, No. 3, 665-678 (2009). ISSN 1246-7405

Fix a finite field $k$ of characteristic $p$. Let $\mathcal{A}$ be the category whose objects are totally ramified abelian extensions $E/F$ where $F$ is a local field with residue field $k$, and $E/F$ is infinite if $\text {char}F=0$; a morphism from $E/F$ to $E'/F'$ is a continuous embedding $E\to E'$ satisfying several conditions. Let $\mathcal{B}$ be the category whose objects are pairs $(K,A)$ where $K$ is a local field of characteristic $p$ and residue field $k$, and $A$ is a closed abelian subgroup of $\Aut_k(K)$; a morphism $(K,A)\to (K',A')$ is a continuous embedding $K\to K'$ satisfying several conditions. \par {\it J.-P. Wintenberger} [C. R. Acad. Sci., Paris, Sér. A 290, 201--203 (1980; Zbl 0428.12012)] used the field of norms construction to give an equivalence $\mathcal{F}$ of the categories $\mathcal{A}_{Lie}$ and $\mathcal{B}_{Lie}$, obtained respectively by restricting to abelian $p$-adic Lie extensions $E/F$ and abelian $p$-adic subgroups $A$. The paper under review extends this equivalence to the case of arbitrary abelian groups, i.e. to the categories $\mathcal{A}$ and $\mathcal{B}$. \par Here is a sketch of the proof of the essential surjectivity to indicate the main ideas. The closed subgroup $A$ of $\Aut_k(K)$ is a $p$-adic Lie group if and only if it is (topologically) finitely generated, so we may suppose that it is not. Let $l_0<l_1<\dots$ be the positive lower ramification breaks of $A$, and $A_0\le A_1\le $ an increasing tower of finitely generated subgroups of $A$ satisfying $A[l_n]A_n=A$ for all $n$. Then each $A_n$ is a $p$-adic Lie group, so (by Wintenberger's result) is isomorphic to $\mathcal{F}(L_n/F_n)$ for some $L_n/F_n\in\mathcal{A}$. The fact that $A$ is not finitely generated, combined with Wintenberger's result, implies that $\Aut_k(F_n)$ is not finitely generated, whence $\text{char}\,F_n=p$; now we may identify $F_n$ with $k((T))$ and thus view each $L_n$ as an extension of the same field. Let $E_n$ be the fixed field of $\text {Gal}(L_n/k((T)))[l_n]$ and set $E=\bigcup_{n\ge 0} E_n$; then $(E/k((T)))\in\mathcal{A}$ and in fact $(K,A)\cong\mathcal{F}(E/k((T)))$.
[Matthew Morrow (Chicago)]

MSC 2000:: *11S15 Ramification and extension theory
11S20 Galois theory for local fields

Keywords: ramification; field of norms; extensions of local fields; automorphisms of local fields.

Citations: Zbl 0428.12012

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