Rubin, Karl Global units and ideal class groups. (English) Zbl 0628.12007 Invent. Math. 89, 511-526 (1987). 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 Cited in 11 ReviewsCited in 25 Documents MSC: 11R27 Units and factorization 11R23 Iwasawa theory 11R18 Cyclotomic extensions 14H52 Elliptic curves 11R37 Class field theory Keywords:class field theory; Iwasawa theory; global units; ideal class group; cyclotomic units; elliptic units Citations:Zbl 0628.14018 × Cite Format Result Cite Review PDF Full Text: DOI EuDML 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 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.