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Galois groups of number fields generated by torsion points of elliptic curves

Published online by Cambridge University Press:  22 January 2016

Kay Wingberg
Kay Wingberg*
Affiliation:
NWF I—Mathematik der Universität Regensburg Universitätsstraße 31, D-84-00 Regensburg, F.R.G.
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Coates and Wiles [1] and B. Perrin-Riou (see [2]) study the arithmetic of an elliptic curve E defined over a number field F with complex multiplication by an imaginary quadratic field K by using p-adic techniques, which combine the classical descent of Mordell and Weil with ideas of Iwasawa’s theory of Zp-extensions of number fields. In a special case they consider a non-cyclotomic Zp-extension F∞ defined via torsion points of E and a certain Iwasawa module attached to E/F, which can be interpreted as an abelian Galois group of an extension of F∞. We are interested in the corresponding non-abelian Galois group and we want to show that the whole situation is quite analogous to the case of the cyclotomic Zp-extension (which is generated by torsion points of Gm).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 104 , December 1986 , pp. 43 - 53
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1986

References

[ 1 ] Coates, J. and Wiles, A., On the conjecture of Birch and Swinnerton-Dyer, Invent. Math., 39 (1977), 223–251.Google Scholar
[ 2 ] Coates, J. and Wiles, A., Infinite descent on elliptic curves with complex multiplication, Progress in Mathematics Vol. 35 (Birkhäuser) Arithmetic and Geometry I dedicated to I.R. Šafarevič Boston-Basel-Stuttgart 1983.Google Scholar
[ 3 ] Haberland, K., Galois cohomology of algebraic number fields, Deutscher Verlag der Wissenschaften, Berlin (1978).Google Scholar
[ 4 ] Iwasawa, K., On Zl-extensions of algebraic number fields, Ann. of Math., 98 (1973), 246–326.Google Scholar
[ 5 ] Jannsen, U., Über Galoisgruppen lokaler Körper, Invent. Math., 70 (1982), 53–69.Google Scholar
[ 6 ] Neukirch, J., Einbettungsprobleme mit lokaler Vorgabe und freie Produkte lokaler Galoisgruppen, J. reine angew. Math., 259 (1973), 1–47.Google Scholar
[ 7 ] Neumann, O., On p-closed number fields and an analogue of Riemann’s existence theorem, In A. Fröhlich, Algebraic number fields, Acad. Press, London (1977), 625–647.Google Scholar
[ 8 ] Wingberg, K., Duality theorems for Γ-extensions of algebraic number fields, Composite Math., 55 (1985), 333–381.Google Scholar
[ 9 ] Wingberg, K., Ein Analogon zur Fundamentalgruppe einer Riemann’schen Fläche im Zahlkörperfall, Invent. Math., 77 (1984), 557–584.CrossRefGoogle Scholar