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Zbl 1123.11038
Boston, Nigel
Galois groups of tamely ramified $p$-extensions. (English)
[J] J. Théor. Nombres Bordx. 19, No. 1, 59-70 (2007). ISSN 1246-7405

For a prime $p$ and a number field $K,$ let us denote $G_{K,S}(p)$ the Galois group of the maximal pro-$p$-extension of $K$ unramified outside a finite set of primes $S.$ The cohomological properties of $G_{K,S}(p)$ have been much studied in the case when $S$ contains all $p$-primes (see e.g. the book by [{\it H. Koch}, Galois theory of $p$-extensions. Springer Monographs in Mathematics. Berlin: Springer (2002; Zbl 1023.11002)]. On the contrary, not much is known in the case (let us call it ``case $(T)$'') when $S$ contains no $p$-prime. In this paper, the author describes some methods to compute $G_{K,S}(p)$ in case $(T)$ when it is finite and conjectural properties of it when it is infinite. Let us for example review the developments around the Fontaine-Mazur conjecture. This famous conjecture asserts that under $(T),$ $G_{K,S}(p)$ has no infinite analytic quotient. The author proposes a stronger refinement, called $(VGS)$ : in case $(T),$ any infinite quotient of $G_{K,S}(p)$ has an open subgroup which is Golod-Shafarevich (a pro-$p$-group $G$ is called Golod-Shafarevich if $\dim\ H^2(G, {\Bbb F}_p) \leq {1 \over 4}(\dim\ H^1(G, {\Bbb F}_p))^2).$ Together with the so-called ``conjecture $\varepsilon$'', namely that non abelian free pro-$p$-groups are not linear, conjecture $VGS$ implies the Fontaine-Mazur conjecture under $(T).$ More precisely, it implies that under $(T)$ any just-infinite quotient of $G_{K,S}(p)$ is a branch group. Here, a pro-$p$-group is called just-infinite if its only infinite quotient is itself; a branch pro-$p$-group is a certain kind of subgroup of the automorphisms of a locally finite, rooted tree. This implies that there should exist actions with infinite image of $G_{K,S}(p)$ on such trees. The author expresses the hope that there should be a theory of ``arboreal'' Galois representations which would help in investigating $G_{K, S}(p)$ in case $(T)$ (contrary to the established theory of $p$-adic Galois representations, as predicted by Fontaine-Mazur).
[Thong Nguyen Quang Do (Besançon)]

MSC 2000:: *11R34 Galois cohomology for global fields

Keywords: tame ramification

Citations: Zbl 1023.11002

Cited in: Zbl 1163.00002 Zbl 1155.11028

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