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Cyclicity of the unramified Iwasawa module

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Abstract

For a number field k and a prime number p, let k ∞ be the cyclotomic Z p -extension of k with finite layers k n . We study the finiteness of the Galois group X ∞ over k ∞ of the maximal abelian unramified p-extension of k ∞ when it is assumed to be cyclic. We then focus our attention to the case where p = 2 and k is a real quadratic field and give the rank of the 2-primary part of the class group of k n . As a consequence, we determine the complete list of real quadratic number fields for which X ∞ is cyclic non trivial. We then apply these results to the study of Greenberg’s conjecture for infinite families of real quadratic fields thus generalizing previous results obtained by Ozaki and Taya.

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References

  1. Azizi A., Mouhib A.: Sur le rang du 2-groupe de classes de \({{\bf Q}(\sqrt m,\sqrt d)}\) où m = 2 ou un premier p ≡ 1 (mod 4). Trans. Am. Math. Soc. 353(7), 2741–2752 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ferrero B.: The cyclotomic Z 2-extension of imaginary quadratic fields. Am. J. Math. 102(3), 447–459 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ferrero B., Washington L.C.: The Iwasawa invariant μ p vanishes for abelian number fields. Ann. Math. 109(2), 377–395 (1979)

    Article  MathSciNet  Google Scholar 

  4. Fröhlich, A.: Central Extensions, Galois Groups, and Ideal Class Groups of Number Fields. Contemporary Mathematics, vol. 24. American Mathematical Society, Providence (1983)

  5. Fukuda T.: Remarks on Z p -extensions of number fields. Proc. Jpn. Acad. A 70, 264–266 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  6. Greenberg R.: On the Iwasawa invariants of totally real number fields. Am. J. Math. 98(1), 263–284 (1976)

    Article  MATH  Google Scholar 

  7. Iwasawa K.: On Z l -extensions of algebraic number fields. Ann. Math. 98(2), 246–326 (1973)

    Article  MathSciNet  Google Scholar 

  8. Kisilevsky H.: Number fields with class number congruent to 4 mod 8 and Hilbert’s theorem 94. J. Number Theory 8(3), 271–279 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kučera R.: On the parity of the class number of a biquadratic field. J. Number Theory 52(2), 457–472 (1995)

    Google Scholar 

  10. Mouhib A., Movahhedi A.: p-class tower of a Z p -extension. Tokyo J. Math. 31, 321–332 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Nguyen Quang Do T., Lescop M.: Iwasawa descent and co-descent for units modulo circular units. Pure Appl. Math. Quart. 2(2), 465–496 (2006)

    MATH  MathSciNet  Google Scholar 

  12. Ozaki M., Taya H.: On the Iwasawa λ2-invariants of certain families of real quadratic fields. Manuscr. Math. 94(4), 437–444 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculté Polydisciplinaire de Taza, Université de Fez, B.P 1223, Taza Gare, Morocco

    Ali Mouhib

  2. Mathématiques et Informatique, XLIM UMR 6172 CNRS, Université de Limoges, 123, Avenue A. Thomas, 87060, Limoges, France

    Abbas Movahhedi

Authors
  1. Ali Mouhib
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  2. Abbas Movahhedi
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Correspondence to Abbas Movahhedi.

Additional information

This work was partially financed as part of project no 18607 of the CNRS/CNRST cooperation.

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Mouhib, A., Movahhedi, A. Cyclicity of the unramified Iwasawa module. manuscripta math. 135, 91–106 (2011). https://doi.org/10.1007/s00229-010-0407-8

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  • DOI: https://doi.org/10.1007/s00229-010-0407-8

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