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On just infinite pro-\(p\)-groups and arithmetically profinite extensions of local fields. (English) Zbl 0997.11107

Let \(F\) be a local field of characteristic \(p>0\) with perfect residue field \(k\). The wild group \(R_k= \operatorname{Aut}_1F\) is the group of wild continuous automorphisms \(\{\sigma: (\sigma-1){\mathcal O}_F\subset {\mathcal M}_F^2\}\) of the local field \(F\). A choice of a prime element \(t\) of a local field \(F\) determines an isomorphism of \(\operatorname{Aut}_1F\) and the group of formal power series \(f(t)= t+a_2t^2+\dots\) with coefficients from \(k\) with respect to the composition \((f\circ g)(t)= f(g(t))\).
The group \(R_k\) plays an important role in the theory of pro-\(p\)-groups. In particular it is a representative of so-called hereditarily just infinite pro-\(p\)-groups (in which every nontrivial closed subgroup of an open subgroup is open). The group \(R_{\mathbb{F}_p}\) has two generators.
In the first part of the paper Fontaine-Wintenberger’s theory of fields of norms [J.-M. Fontaine and J.-P. Wintenberger, C. R. Acad. Sci., Paris, Sér. A 288, 367-370 (1979; Zbl 0475.12020)] is applied to study the structure of the wild group \(R_k\). In particular a new short proof of R. Camina’s theorem [J. Algebra 196, 101-113 (1997; Zbl 0883.20015)] which says that every countably based pro-\(p\)-group (i.e. with countably many open subgroups) is isomorphic to a closed subgroup of \(F_{\mathbb{F}_p}\) is supplied. The proof is a corollary of the following proposition: every pro-\(p\)-group with countably many open subgroups is realizable as the Galois group of an arithmetically profinite extension of a local field of characteristic \(p\).
It is an open problem which finitely generated pro-\(p\)-groups are realizable as the Galois group of an arithmetically profinite extension of a local field of characteristic 0. Due to S. Sen [Invent. Math. 17, 44-50 (1972; Zbl 0242.12012)] \(p\)-adic Lie groups are realizable; another collection of realizable groups \(T[r]\) is provided in the second part of the paper.
For \(p^r\) define \(T[r]= \{\sum_{i\geq 0} a_it^{1+p^{r_i}}: a_0=1\), \(a_i\in \mathbb{F}_p\}\). These subgroups of \(R_{\mathbb{F}_p}\) have various bizarre properties: the commutator subgroup is unusually small and the abelian quotient is of exponent greater than \(p\) (which is important for number theory applications). Subgroups \(T[r]\) are torsion free and hereditarily just infinite, they don’t have infinite subquotients which are \(p\)-adic Lie groups.
Using these properties, it is proved in the paper that the group \(T[r]\) for \(r\geq 2\) can be realized as the Galois group of an arithmetically profinite extension of \({\mathfrak p}\)-adic fields. This provides in particular an affirmative answer to Coates-Greenberg’s problem on deeply ramified extensions of local fields stated in [J. Coates and R. Greenberg, Invent. Math. 124, 129-174 (1996; Zbl 0858.11032)].

MSC:

11S15 Ramification and extension theory
20E18 Limits, profinite groups
12F12 Inverse Galois theory
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