Triviality of Iwasawa module associated to some abelian fields of prime conductors

  • Humio Ichimura


Let p be an odd prime number and \(\ell \) an odd prime number dividing \(p-1\). We denote by \(F=F_{p,\ell }\) the real abelian field of conductor p and degree \(\ell \), and by \(h_F\) the class number of F. For a prime number \(r \ne p,\,\ell \), let \(F_{\infty }\) be the cyclotomic \(\mathbb {Z}_r\)-extension over F, and \(M_{\infty }/F_{\infty }\) the maximal pro-r abelian extension unramified outside r. We prove that \(M_{\infty }\) coincides with \(F_{\infty }\) and consequently \(h_F\) is not divisible by r when r is a primitive root modulo \(\ell \) and r is smaller than an explicit constant depending on p.


Iwasawa module Class number Cyclotomic field 

Mathematics Subject Classification

Primary 11R23 Secondary 11R18 



The author thanks the referee for several valuable comments which improved the presentation of the whole paper and for suggesting him the simple proofs of Lemma 1(II) and Lemma 5 and informing him of the paper of Hardy and Littlewood [11]


  1. 1.
    Cornacchia, P.: The parity of class number of the cyclotomic fields of prime conductor. Proc. Am. Math. Soc. 125(11), 3163–3168 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cornacchia, P., Greither, C.: Fitting ideals of class groups of real fields of prime power conductor. J. Number Theory 73(2), 459–471 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Davis, D.: Computing the number of totally positive circular units which are square. J. Number Theory 10(1), 1–9 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Estes, D.R.: On the parity of the class number of the field of \(q\)th roots of unity. Rocky Mt. J. Math. 19(3), 675–682 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Friedman, E.: Ideal class groups in basic \({\mathbb{Z}}_{p_1}\times \cdots \times {\mathbb{Z}}_{p_s}\)-extensions of abelian number fields. Invent. Math. 65(3), 425–440 (1982)CrossRefGoogle Scholar
  6. 6.
    Fujima, S., Ichimura, H.: Note on the class number of the \(p\)th cyclotomic field II. Exp. Math. (2016). doi: 10.1080/10586458.2016.1230528 zbMATHGoogle Scholar
  7. 7.
    Gillard, R.: Remarques sur les unités cyclotomiques et les unités elliptiques. J. Number Theory 11(1), 21–48 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gillard, R.: Unités cyclotomiques, unités semi-locales et \({\mathbb{Z}}_{\ell }\)-extensions II. Ann. Inst. Fourier (Grenoble) 29(4), 1–15 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Greenberg, R.: On \(2\)-adic \(L\)-functions and cyclotomic invariants. Math. Z. 159(1), 37–45 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Greither, C.: Class groups of abelian extensions, and the main conjecture. Ann. Inst. Fourier (Grenoble) 42(3), 449–499 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hardy, G.H., Littlewood, J.E.: Some problems of Partitio numerorum; III: On the expression of a number as a sum of primes. Acta Math. 44(1), 1–70 (1923)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hasse, H.: Über die Klassenzahl abelscher Zahlkörper. Akademia Verlag, Berlin (1952)zbMATHGoogle Scholar
  13. 13.
    Horie, K.: Ideal class groups of the Iwasawa-theoretical extensions over the rationals. J. Lond. Math. Soc. 66(2), 257–275 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Horie, K.: Triviality in ideal class groups of Iwasawa-theoretical abelian number fields. J. Math. Soc. Jpn. 57(3), 827–857 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ichimura, H.: On the class group of a cyclotomic \({\mathbb{Z}}_p\times {\mathbb{Z}}_{\ell }\)-extension. Acta Arith. 150(3), 263–283 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ichimura, H.: Class number parity of a quadratic twist of a cyclotomic field of prime power conductor. Osaka J. Math. 50(2), 563–572 (2013)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ichimura, H.: Refined version of Hasse’s Satz 45 on class number parity. Tsukuba J. Math. 38(2), 189–199 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ichimura, H.: On a duality of Gras between totally positive and primary cyclotomic units. Math. J. Okayama Univ. 58, 125–132 (2016)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ichimura, H.: Note on Bernoulli numbers associated to some Dirichlet character of conductor \(p\). Arch. Math. (Basel) 107(2), 595–601 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Iwasawa, K.: Lectures on \(p\)-Adic \(L\)-Functions. Annals of Mathematics Studies, vol. 74. Princeton University Press, Princeton (1972)zbMATHGoogle Scholar
  21. 21.
    Iwasawa, K.: On \({\mathbb{Z}}_{\ell }\)-extensions of algebraic number fields. Ann. Math. 98, 246–326 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jakubec, S.: On divisibility of class number of real abelian fields of prime conductors. Abh. Math. Univ. Hambg. 63, 67–86 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jakubec, S.: On divisibility of class number \(h^+\) of the real cyclotomic fields of prime degree \(\ell \). Math. Comput. 67, 369–398 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Jakubec, S.: On the class number of some real abeian number fields of prime conductors. Acta Arith. 145(4), 315–318 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Jakubec, S., Pasteka, M., Schinzel, A.: Class number of real abelian fields. J. Number Theory 148, 365–371 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Koyama, Y., Yoshino, K.: Prime divisors of the class numbers of the real \(p^r\)th cyclotomic field and characteristic polynomial attached to them. RIMS Kôkyûroku Bessatsu B12, 149–172 (2009)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Metsänkylä, T.: An application of \(p\)-adic class number formula. Manuscr. Math. 93(4), 481–498 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Metsänkylä, T.: On the parity of the class number of real abelian fields. Acta Math. Inform. Univ. Ostrav. 6(1), 159–166 (1998)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers, 3rd edn. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  30. 30.
    Sinnott, W.: On the Stickelberger ideal and the circular units of an abelian field. Invent. Math. 62(2), 181–234 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Stevenhagen, P.: Class number parity of the \(p\)th cyclotomic field. Math. Comput. 63, 773–784 (1994)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Trojovský, P.: On divisibility of the class number \(h^+\) of the real cyclotomic fields \({\mathbb{Q}}({\zeta }_p+{\zeta }_p^{-1})\) by primes \(q < 10000\). Math. Slovaca 50(5), 541–555 (2000)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Washington, L.C.: Introduction to Cyclotomic Fields, 2nd edn. Springer, New York (1997)CrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Faculty of ScienceIbaraki UniversityMitoJapan

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