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Global units and ideal class groups. (English) Zbl 0628.12007

Let K be a number field and F/K an abelian extension of K. The author shows how certain “special” global units of F can be used to define elements of \({\mathbb{Z}}[Gal(F/K)]\) which annihilate subquotients of the ideal class group of F. This result generalizes recent work of F. Thaine, who dealt with the case \(K={\mathbb{Q}}\) and F/\({\mathbb{Q}}^ a \)real abelian extension [“On the ideal class groups of real abelian number fields”, to appear].
The author shows that cyclotomic units are special, which allows him to recover Thaine’s results; and in a subsequent paper, he shows that the elliptic units in an imaginary quadratic field are special. This latter result forms an integral part of his spectacular proof that certain elliptic curves with complex multiplication have finite Tate-Shafarevich groups [Invent. Math. 89, 527-560 (1987; Zbl 0628.14018)].
Reviewer: J.H.Silverman

MSC:

11R27 Units and factorization
11R23 Iwasawa theory
11R18 Cyclotomic extensions
14H52 Elliptic curves
11R37 Class field theory

Citations:

Zbl 0628.14018

References:

[1] Atiyah, M., Wall, C.: Cohomology of groups. In: Algebraic Number Theory, Cassels, J.W.S., Fröhlich, A. (eds.) p.. 94-115, London: Academic Press 1967
[2] Eilenberg, S., Nakayama, T.: On the dimension of modules and algebras, II. Nagoya Math. J.9, 1-16 (1955) · Zbl 0068.26503
[3] Iwasawa, K.: OnZ l -extensions of algebraic nuber fields. Ann. Math.98, 246-326 (1973) · Zbl 0285.12008 · doi:10.2307/1970784
[4] Iwasawa, K.: On cohomology groups of units forZ p -extensions. Am. J. Math.105, 189-200 (1983) · Zbl 0525.12009 · doi:10.2307/2374385
[5] Kummer, E.: Über eine besondere Art, aus complexen Einheiten gebildeter Ausdrücke. J. Reine Angew. Math.50, 212-232 (1855) · ERAM 050.1333cj · doi:10.1515/crll.1855.50.212
[6] Mazur, B., Wiles, A.: Class fields of abelian extensions ofQ. Invent. Math.76, 179-330 (1984) · Zbl 0545.12005 · doi:10.1007/BF01388599
[7] Rubin, K.: Tate-Shafarevich groups andL-functions of elliptic curves with complex multiplication. Invent. Math.89, 527-560 (1987) · Zbl 0628.14018 · doi:10.1007/BF01388984
[8] Tate, J.: Les Conjectures de Stark sur les FonctionsL d’Artin ens=0. Prog. Math.47, Boston: Birkhäuser (1984)
[9] Thaine, F.: On the ideal class groups of real abelian extensions ofQ. (To appear) · Zbl 0665.12003
[10] Wintenberger, J.-P.: Structure galoisienne de limites projectives d’unités locales. Comp. Math.42, 89-103 (1981) · Zbl 0414.12008
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