Weber’s Class Number One Problem

  • Takashi Fukuda
  • Keiichi Komatsu
  • Takayuki Morisawa
Conference paper
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 7)

Abstract

Let p and be prime numbers. In this paper, we consider the -indivisibility of the class numbers of the intermediate fields in the cyclotomic Z p -extension of Q. Moreover, we study the class numbers of the intermediate fields in the composite of such extensions.

Keywords

Cond 

Notes

Acknowledgements

The authors thank the organizers of Iwasawa 2012 for giving us the opportunity to talk. The authors also thank the referee for reading this paper carefully and offering several invaluable suggestions.

References

  1. Aoki, M., Fukuda, T.: An algorithm for computing p-class groups of abelian number fields.In: Hess, F., Pauli, S., Pohst, M. (eds.) Algorithmic Number Theory. Lecture Notes in Computer Science, vol. 4076, pp. 56–71. Springer, Berlin/Heidelberg (2006)Google Scholar
  2. Bauer, H.: Numerische Bestimmung von Klassenzahlen reeller zyklischer Zahlkörper. J. Number Theory 1, 161–162 (1969)MathSciNetCrossRefMATHGoogle Scholar
  3. Coates, J.: The enigmatic Tate-Shafarevich group. In: Fifth International Congress of Chinese Mathematicians. Part 1, 2, AMS/IP Studies in Advanced Mathematics, vol. 51, Part. 1, 2, pp. 43–50. American Mathematical Society, Providence (2012)Google Scholar
  4. Cohn, H.: A numerical study of Weber’s real class number calculation I. Numer. Math. 2, 347–362 (1960)MathSciNetCrossRefMATHGoogle Scholar
  5. Fukuda,T., Komatsu, K.: Weber’s class number problem in the cyclotomic Z 2-extension of Q. Exp. Math. 18(2), 213–222 (2009)Google Scholar
  6. Fukuda, T., Komatsu, K.: Weber’s class number problem in the cyclotomic Z 2-extension of Q, II. J. Théor. Nombres Bordeaux 22(2), 359–368 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. Fukuda, T., Komatsu, K.: Weber’s class number problem in the cyclotomic Z 2-extension of Q, III. Int. J. Number Theory 7(6), 1627–1635 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. Horie, K.: Ideal class groups of Iwasawa-theoretical abelian extensions over the rational field. J. Lond. Math. Soc. 66, 257–275 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. Horie, K.: Primary components of the ideal class group of the Z p-extension over Q for Typical Inert Primes. Proc. Jpn. Acad. Ser. A Math. Sci. 81(3), 40–43 (2005a)MathSciNetCrossRefMATHGoogle Scholar
  10. Horie, K: The ideal class group of the basic Z p-extension over an imaginary quadratic field. Tohoku Math. J. (2) 57, 375–394 (2005b)Google Scholar
  11. Horie, K.: Certain primary components of the ideal class group of the Z p-extension over the rationals. Tohoku Math. J. 59, 259–291 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. Horie, K., Horie, M.: The narrow class groups of some Z p-extensions over the rationals. Acta Arith. 135(2), 159–180 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. Horie, K., Horie, M.: The ideal class group of the Z p-extension over the rationals. Tohoku Math. J. 61, 551–570 (2009a)MathSciNetCrossRefMATHGoogle Scholar
  14. Horie, K., Horie, M.: The ideal class group of the Z 23-extension over the rational field. Proc. Jpn. Acad. Ser. A 85, 155–159 (2009b)MathSciNetCrossRefMATHGoogle Scholar
  15. Horie, K., Horie, M.: The narrow class groups of the Z 17- and Z 19-extensions over the rational field. Abh. Math. Semin. Univ. Hambg. 80(1), 47–57 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. Inatomi, A.: On Z p-extensions of real abelian fields. Kodai Math. J. 12, 420–422 (1986)MathSciNetCrossRefMATHGoogle Scholar
  17. Iwasawa, K.: A note on class numbers of algebraic number fields. Abh. Math. Sem. Univ. Hambg. 20, 257–258 (1956)MathSciNetMATHGoogle Scholar
  18. Masley, J.M.: Class numbers of real cyclic number fields with small conductor. Compos. Math. 37, 297–319 (1978)MathSciNetMATHGoogle Scholar
  19. Mazur, B., Wiles, A.: Class fields of abelian extensions of Q. Invent. Math. 76, 179–330 (1984)MathSciNetCrossRefMATHGoogle Scholar
  20. Morisawa, T.: A class number problem in the cyclotomic Z 3-extension of Q. Tokyo J. Math. 32, 549–558 (2009)MathSciNetCrossRefMATHGoogle Scholar
  21. Morisawa, T.: Mahler measure of the Horie unit and Weber’s class number problem in the cyclotomic Z 3-extension of Q. Acta Arith. 153(1), 35–49 (2012)MathSciNetCrossRefMATHGoogle Scholar
  22. Morisawa, T.: On the -part of the \(\mathbf{Z}_{p_{1}} \times \cdots \times \mathbf{Z}_{p_{s}}\)-extension of Q. J. Number Theory 133(6), 1814–1826 (2013)MathSciNetCrossRefMATHGoogle Scholar
  23. Morisawa, T., Okazaki, R.: Mahler measure and Weber’s class number problem in the cyclotomic Z p-extension of Q for odd prime number p. Tohoku Math. J. 65(2), 253–272 (2013)Google Scholar
  24. Okazaki, R.: On a lower bound for relative units, Schinzel’s lower bound and Weber’s class number problem (preprint)Google Scholar
  25. van der Linden, F.J.: Class number computations of real abelian number fields. Math. Comput. 39, 693–707 (1982)MathSciNetCrossRefMATHGoogle Scholar
  26. Washington, L.C.: The non-p-part of the class number in a cyclotomic Z p-extension. Invent. Math. 49, 87–97 (1978)MathSciNetCrossRefMATHGoogle Scholar
  27. Washington, L.C.: Introduction to Cyclotomic Fields. Graduate Texts in Mathematics, vol. 83, 2nd edn. Springer, New York/Heidelberg/Berlin (1997)Google Scholar
  28. Weber, H.: Theorie der Abel’schen Zahlkörper. Acta Math. 8, 193–263 (1886)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Takashi Fukuda
    • 1
  • Keiichi Komatsu
    • 2
  • Takayuki Morisawa
    • 3
  1. 1.Department of Mathematics, College of Industrial TechnologyNihon UniversityNarashinoJapan
  2. 2.Department of Mathematics, School of Fundamental Science and EngineeringWaseda UniversityShinjukuJapan
  3. 3.Department of Mathematics, Faculty of Science and TechnologyKeio UniversityKohoku, YokohamaJapan

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