Galois groups of number fields generated by torsion points of elliptic curves. (English) Zbl 0621.12011

Wingberg, Kay

Nagoya Math. J. 104, 43-53 (1986).

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

Keywords:

torsion points of elliptic curves; Iwasawa theory; Galois cohomology; maximal p-extension; \({bbfZ}_ p\)-extension; Galois groups; free pro-p- groups

References:

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[2]	On p-closed number fields and an analogue of Riemann’s existence theorem pp 625– (1977)
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