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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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Full Text: DOI arXiv

References:

[1] Boston, N., Some cases of the Fontaine-Mazur conjecture, J. Number Theory, 42, 285-291 (1992) · Zbl 0768.11044
[2] D. Cartwright, Queen Mary College, London; D. Cartwright, Queen Mary College, London
[3] Cohen, H.; Lenstra, H., Heuristics on class groups of number fields, Number Theory, Noordwijkerhout, 1983. Number Theory, Noordwijkerhout, 1983, Lecture Notes in Math., 1068 (1984), Springer-Verlag: Springer-Verlag Berlin/New York, p. 33-62 · Zbl 0558.12002
[4] de Smit, B.; Lenstra, H., Explicit construction of universal deformation rings, (Cornell, G.; Silverman, J.; Stevens, G., Modular Forms and Fermat’s Last Theorem (1997), Springer-Verlag: Springer-Verlag Berlin/New York), 313-326 · Zbl 0907.13010
[5] J. Dixon, M. du Sautoy, A. Mann, D. Segal, Analytic Pro-\(p\); J. Dixon, M. du Sautoy, A. Mann, D. Segal, Analytic Pro-\(p\)
[6] I. Fesenko, On just-infinite pro-\(p\); I. Fesenko, On just-infinite pro-\(p\) · Zbl 0997.11107
[7] Fontaine, J.-M.; Mazur, B., Geometric Galois representations, (Coates, J. H.; Yau, S. T., Elliptic Curves and Modular Forms. Elliptic Curves and Modular Forms, Proceedings, Conference in Hong Kong, December 18-21, 1993 (1993), International Press: International Press Cambridge) · Zbl 0839.14011
[8] Golod, E.; Shafarevich, I., On class field towers, Izv. Akad. Nauk SSSR, 28, 261-272 (1964) · Zbl 0136.02602
[9] Hajir, F., On the growth of \(pp\), J. Algebra, 188, 256-271 (1997) · Zbl 0879.11069
[10] H. Hasse, Bericht über neuere Untersuchungen und Probleme der Theorie der algebraischen Zahlkörper, Teil 1, Jber. dt. Matverein. 35; H. Hasse, Bericht über neuere Untersuchungen und Probleme der Theorie der algebraischen Zahlkörper, Teil 1, Jber. dt. Matverein. 35
[11] Itô, N., Note on \(p\), Nagoya Math. J., 1, 113-116 (1950) · Zbl 0038.16502
[12] Klaas, G.; Leedham-Green, C. R.; Plesken, W., Linear Pro-\(p\). Linear Pro-\(p\), Lecture Notes in Math., 1674 (1997), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0901.20013
[13] Koch, H.; Venkov, B., Über den \(p\), Astérisque, 24, 57-67 (1975) · Zbl 0335.12021
[14] A. Lubotzky, Group presentation,\(p SL_2\textbf{C} \); A. Lubotzky, Group presentation,\(p SL_2\textbf{C} \)
[15] Lubotzky, A.; Shalev, A., On some \(Λp\), Israel J. Math., 85, 307-337 (1994) · Zbl 0819.20030
[16] Magnus, W., Beziehung zwischen Gruppen und Idealen in einen speziellen Ring, Math. Ann., 111 (1935)
[17] Miyake, K., Some \(p\), Tôhoku Math. J., 44, 443-469 (1992) · Zbl 0763.20010
[18] Nomura, A., A remark on Boston’s question concerning the existence of unramified \(p\), J. Number Theory, 58, 66-70 (1996) · Zbl 0856.11052
[19] Scholz, A.; Taussky, O., Die Hauptideale der kubischen Klassenkörper imaginärquadratischer Zahlkörper; ihre rechnerische Bestimmung und ihr Einfluss auf den Klassenkörperturm, J. Reine Angew. Math., 171, 19-41 (1934) · Zbl 0009.10202
[20] Shafarevich, I., Extensions with prescribed ramification points, Inst. Hautes Études Sci. Publ. Math., 18, 71-95 (1964)
[21] Shalev, A., On almost fixed point free automorphisms, J. Algebra, 157, 271-282 (1993) · Zbl 0797.20023
[22] Shalev, A., Finite \(p\), Finite and Locally Finite Groups (Istanbul, 1994) (1995), Kluwer Academic: Kluwer Academic Dordrecht, p. 401-450 · Zbl 0839.20028
[23] Wingberg, K., On the maximal unramified \(p\), J. Reine Angew. Math., 440, 129-156 (1993) · Zbl 0779.11054
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