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Some cases of the Fontaine-Mazur conjecture. II. (English) Zbl 0928.11050

The Fontaine-Mazur conjecture can be formulated as follows: There do not exist a number field \(K\) and an infinite everywhere unramified Galois pro-\(p\)-extension \(L\) such that \(\text{Gal}(L/K)\) is uniform.
In his earlier paper about this conjecture [J. Number Theory 42, 285-291 (1992; Zbl 0768.11044)] the author proved it in the case where \(K\) is a cyclic extension of prime degree \(l\neq p\) of a number field \(F\) such that \(p\nmid h(F)\).
In the paper at hand, the author defines the notion of a self-similar group as a generalization of the notion of uniform group. He proves the following generalization of his normal theorem above: Suppose \(K\) is a number field containing a subfield \(F\) such that \(K/F\) is cyclic of degree \(n\) prime to \(p\), and such that \(p\) does not divide the class number \(h(F)\) of \(F\). Then, there is no everywhere unramified pro-\(p\)-extension \(L\) of \(K\), Galois over \(F\), with self-similar Galois group \(\text{Gal}(L/K)=G\) such that \(H(G,n)\) holds. Here \(H(G,n)\) means that there is a function of \(n\) which is an upper bound for the derived length of every quotient of \(G\) that admits a fixed-point-free automorphism of order \(n\). \(H(G,n)\) holds for \(n\) a prime and for \(n=4\).
Reviewer: H.Koch (Berlin)

MSC:

11R32 Galois theory
11S20 Galois theory
11S15 Ramification and extension theory

Citations:

Zbl 0768.11044
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References:

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