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Galois groups of number fields generated by torsion points of elliptic curves. (English) Zbl 0621.12011
Let \(F\) be an algebraic number field. Let \(p\neq 2\) be a prime, \(S_ p\) the set of all primes in \(F\) above \(p\), \(S\) a finite set of primes in \(F\), \(F_ S\) the maximal p-extension of \(F\) unramified outside \(S\), and \(F_{\infty}^ a\) \({\mathbb{Z}}_ p\)-extension in \(F_ S\). The main result in this article is a general theorem giving sufficient conditions for the Galois groups \(G(F_{S_ p}/F_{\infty})\), \(G(F_{S\cap S_ p}/F_{\infty})\), \(G(F_ S/F_{\infty})\), and \(G(F_{S\cup S_ p}/F_{\infty})\) to be free pro-p-groups. If \(T\supseteq S\), sufficient conditions are given for \(G(F_ T/F_ S)\) to be a free pro-p-product of certain inertia groups. This generalizes results of J. Neukirch [J. Reine Angew. Math. 259, 1-47 (1973; Zbl 0263.12006)] and O. Neumann [in Algebr. Number Fields, Proc. Symp. London math. Soc., Univ. Durham 1975, 625-647 (1977; Zbl 0393.12015)].
Reviewer: L.Olson

MSC:
11R32 Galois theory
11R18 Cyclotomic extensions
14H52 Elliptic curves
11R34 Galois cohomology
11S15 Ramification and extension theory
14G25 Global ground fields in algebraic geometry
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References:
[1] Composite Math 55 pp 333– (1985)
[2] On p-closed number fields and an analogue of Riemann’s existence theorem pp 625– (1977)
[3] J. reine angew. Math 259 pp 1– (1973)
[4] DOI: 10.1007/BF01393198 · Zbl 0534.12009 · doi:10.1007/BF01393198
[5] DOI: 10.1007/BF01388840 · Zbl 0563.12008 · doi:10.1007/BF01388840
[6] Galois cohomology of algebraic number fields (1978)
[7] Progress in Mathematics Vol. 35 (Birkhäuser) Arithmetic and Geometry I dedicated to I.R (1983)
[8] DOI: 10.1007/BF01402975 · Zbl 0359.14009 · doi:10.1007/BF01402975
[9] DOI: 10.2307/1970784 · Zbl 0285.12008 · doi:10.2307/1970784
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