Archiv der Mathematik

, Volume 113, Issue 6, pp 593–601 | Cite as

The structure of higher-dimensional Iwasawa modules under a far-fetched assumption

  • Cornelius Greither
  • Sören KleineEmail author


From the assumption that Leopoldt’s conjecture fails and some mild extra assumptions, we deduce the existence of multiple \(\mathbb {Z}_p\)-extensions whose Iwasawa modules are “large” in a precise sense. We are not aware of any constructions of such extensions that avoid our preposterously strong hypothesis.


Leopoldt’s conjecture Pseudo-null module Unbounded lambda invariants 

Mathematics Subject Classification




  1. 1.
    Babaĭcev, V.A.: On some questions in the theory of \(\Gamma \)-extensions of algebraic number fields. II. Math. USSR Izvestiya 10(4), 675–685 (1976)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bhargava, M., Varma, I.: The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders. Proc. Lond. Math. Soc. 112, 235–266 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cuoco, A., Monsky, P.: Class numbers in \(\mathbb{Z}_p^d\)-extensions. Math. Ann. 255, 235–258 (1981)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kida, Y.: \(l\)-extensions of CM-fields and cyclotomic invariants. J. Number Theory 12, 519–528 (1980)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Monsky, P.: Some invariants of \(\mathbb{Z}_p^d\)-extensions. Math. Ann. 255, 229–233 (1981)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of Number Fields, 2nd edn. Springer, Berlin (2008)CrossRefGoogle Scholar
  7. 7.
    Schoof, R.: Minus class groups of the fields of the \(l\)-th roots of unity. Math. Comput. 67, 1225–1245 (1998)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Washington, L.C.: Introduction to Cyclotomic Fields, 2nd edn. Springer, New York (1997)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für theor. Informatik, Mathematik und ORUniversität der Bundeswehr MünchenNeubibergGermany

Personalised recommendations