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Cyclotomic units and class groups in p-extensions of real abelian fields

Published online by Cambridge University Press:  16 June 2009

FILIPPO ALBERTO EDOARDO NUCCIO
FILIPPO ALBERTO EDOARDO NUCCIO*
Affiliation:
Università “La Sapienza”, P.le Aldo Moro, 5-00186, Rome, Italy. e-mail: nuccio@mat.uniroma1.it
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Abstract

For a real abelian number field F and for a prime p we study the relation between the p-parts of the class groups and of the quotients of global units modulo cyclotomic units along the cyclotomic p-extension of F. Assuming Greenberg's conjecture about the vanishing of the λ-invariant of the extension, a map between these groups has been constructed by several authors, and shown to be an isomorphism if p does not split in F. We focus in the split case, showing that there are, in general, non-trivial kernels and cokernels.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 1 , January 2010 , pp. 93 - 106
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

[BNQD01]Belliard, J.-R. and Do, T. Nguyen QuangFormules de classes pour les corps abéliens réels. Ann. Inst. Fourier (Grenoble) 51 (2001), no. 4, 903–937.CrossRefGoogle Scholar
[CF86]Cassels, J. W. S. and Fröhlich, A. (eds.). Algebraic number theory. (Academic Press Inc. [Harcourt Brace Jovanovich Publishers], 1986.) Reprint of the 1967 original.Google Scholar
[FW79]Ferrero, B. and Washington, L. C.The Iwasawa invariant μp vanishes for abelian number fields. Ann. of Math. (2) 109 (1979), no. 2, 377–395.CrossRefGoogle Scholar
[Gil79a]Gillard, R.Remarques sur les unités cyclotomiques et les unités elliptiques. J. Number Theory 11 (1979), no. 1, 21–48.CrossRefGoogle Scholar
[Gil79b]Gillard, R.Unités cyclotomiques, unités semi-locales et Zl-extensions, II. Ann. Inst. Fourier (Grenoble) 29 (1979), no. 4, viii, 1–15.CrossRefGoogle Scholar
[Gre76]Greenberg, R.On the Iwasawa invariants of totally real number fields. Amer. J. Math. 98 (1976), no. 1, 263–284.CrossRefGoogle Scholar
[Gre77]Greenberg, R.On p-adic L-functions and cyclotomic fields, II. Nagoya Math. J. 67 (1977), 139–158.CrossRefGoogle Scholar
[Iwa60]Iwasawa, K.On local cyclotomic fields. J. Math. Soc. Japan 12 (1960), 16–21.CrossRefGoogle Scholar
[Iwa73]Iwasawa, K.On Zl-extensions of algebraic number fields. Ann. of Math. (2) 98 (1973), 246–326.CrossRefGoogle Scholar
[KS95]Kraft, J. S. and Schoof, R.Computing Iwasawa modules of real quadratic number fields. Compositio Math. 97 (1995), no. 1-2, 135–155, Special issue in honour of Frans Oort.Google Scholar
[Kuz96]Kuz′min, L. V.On formulas for the class number of real abelian fields. Izv. Ross. Akad. Nauk Ser. Mat. 60 (1996), no. 4, 43–110.Google Scholar
[Lan90]Lang, S.Cyclotomic fields I and II, second ed., Graduate Texts in Mathematics, vol. 121. (Springer-Verlag, 1990) With an appendix by Karl Rubin.CrossRefGoogle Scholar
[NSW00]Neukirch, J., Schmidt, A. and Wingberg, K.Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, (Springer-Verlag, 2000).Google Scholar
[Oza97]Ozaki, M.On the cyclotomic unit group and the ideal class group of a real abelian number field, I, II. J. Number Theory 64 (1997), no. 2, 211–222, 223–232.CrossRefGoogle Scholar
[Sch88]Schoof, R.Cohomology of class groups of cyclotomic fields: an application to Morse-Smale diffeomorphisms. J. Pure Appl. Algebra 53 (1988), no. 1-2, 125–137. MR MR955614 (89j:11111)CrossRefGoogle Scholar
[Sin81]Sinnott, W.On the Stickelberger ideal and the circular units of an abelian field. Invent. Math. 62 (1980/81), no. 2, 181–234.CrossRefGoogle Scholar
[Was97]Washington, L. C.Introduction to cyclotomic fields, second ed. Graduate Texts in Mathematics, vol. 83 (Springer-Verlag, 1997).CrossRefGoogle Scholar