## 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

Zbl 0768.11044
Full Text:

### 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 [3] Cohen, H.; Lenstra, H., Heuristics on class groups of number fields, Number theory, noordwijkerhout, 1983, Lecture notes in math., 1068, (1984), Springer-Verlag Berlin/New York, p. 33-62 · Zbl 0558.12002 [4] de Smit, B.; Lenstra, H., Explicit construction of universal deformation rings, (), 313-326 · Zbl 0907.13010 [5] J. Dixon, M. du Sautoy, A. Mann, D. Segal, Analytic Pro-p, Cambridge Univ. Press, Cambridge, UK [6] I. Fesenko, On just-infinite pro-p · Zbl 0997.11107 [7] Fontaine, J.-M.; Mazur, B., Geometric Galois representations, () · 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 ofpp, 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 [11] Itô, N., Note onp, Nagoya math. J., 1, 113-116, (1950) · Zbl 0038.16502 [12] Klaas, G.; Leedham-Green, C.R.; Plesken, W., Linear pro-p, Lecture notes in math., 1674, (1997), Springer-Verlag New York/Berlin · Zbl 0901.20013 [13] Koch, H.; Venkov, B., Über denp, Astérisque, 24, 57-67, (1975) · Zbl 0335.12021 [14] A. Lubotzky, Group presentation,pSL2{\bfC}, Ann. Math. 118, 115, 130 [15] Lubotzky, A.; Shalev, A., On someλp, 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) · JFM 61.0102.02 [17] Miyake, K., Somep, Tôhoku math. J., 44, 443-469, (1992) · Zbl 0763.20010 [18] Nomura, A., A remark on Boston’s question concerning the existence of unramifiedp, 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., Finitep, Finite and locally finite groups (Istanbul, 1994), (1995), Kluwer Academic Dordrecht, p. 401-450 · Zbl 0839.20028 [23] Wingberg, K., On the maximal unramifiedp, J. reine angew. math., 440, 129-156, (1993) · Zbl 0779.11054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.