Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On small Iwasawa invariants and imaginary quadratic fields


Author: Jonathan W. Sands
Journal: Proc. Amer. Math. Soc. 112 (1991), 671-684
MSC: Primary 11R23
DOI: https://doi.org/10.1090/S0002-9939-1991-1057961-4
MathSciNet review: 1057961
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If $ p$ is an odd prime that does not divide the class number of the imaginary quadratic field $ k$, and the cyclotomic $ {\mathbb{Z}_p}$-extension of $ k$ has $ \lambda $-invariant less than or equal to two, we prove that every totally ramified $ {\mathbb{Z}_p}$-extension of $ k$ has $ \mu $-invariant equal to zero and $ \lambda $-invariant less than or equal to two. Combined with a result of Bloom and Gerth, this has the consequence that $ \mu = 0$ for every $ {\mathbb{Z}_p}$-extension of $ k$, under the same assumptions. In the principal case under consideration, Iwasawa's formula for the power of $ p$ in the class number of the $ n$th layer of a $ {\mathbb{Z}_p}$-extension becomes valid for all $ n$ , and is completely explicit.


References [Enhancements On Off] (What's this?)

  • [1] J. R. Bloom and F. Gerth III, The Iwasawa invariant $ \mu $ in the composite of two $ {\mathbb{Z}_l}$extensions, J. Number Theory 13 (1981), 262-267. MR 612687 (82j:12006)
  • [2] N. Bourbaki, Commutative algebra, Addison-Wesley, Reading, MA, 1972.
  • [3] A. Cuoco and P. Monsky, Class numbers in $ \mathbb{Z}_p^d$-extensions, Math. Ann. 255 (1981), 235-258. MR 614400 (82h:12010)
  • [4] D. S. Dummit, D. Ford, H. Kisilevsky, and J. W. Sands, Computation of Iwasawa lambda invariants for imaginary quadratic fields, J. Number Theory (to appear). MR 1089792 (92a:11124)
  • [5] B. Ferrero and L. Washington, The Iwasawa invariant $ {\mu _p}$ vanishes for abelian number fields, Ann. Math. 109 (1979), 377-395. MR 528968 (81a:12005)
  • [6] E. Friedman, Iwasawa invariants, Math. Ann. 271 (1985), 13-30. MR 779602 (86h:11093)
  • [7] F. Gerth III, Upper bounds for an Iwasawa invariant, Compositio Math. 39 (1979), 3-10. MR 538999 (80f:12006)
  • [8] R. Gold, The non-triviality of certain $ {\mathbb{Z}_l}$-extensions, J. Number Theory 6 (1974), 369-373. MR 0369316 (51:5551)
  • [9] D. Gorenstein, Finite groups, Harper and Row, New York, 1968. MR 0231903 (38:229)
  • [10] R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), 263-284. MR 0401702 (53:5529)
  • [11] K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. U. Hamburg 20 (1956), 257-258. MR 0083013 (18:644d)
  • [12] -, On $ \Gamma $-extensions of algebraic number fields, Bull. Amer. Math. Soc. 65 (1959), 183-226.
  • [13] -, On $ {\mathbb{Z}_l}$-extensions of algebraic number fields, Ann. of Math. 98 (1973), 246-326.
  • [14] L. C. Washington, Introduction to cyclotomic fields, Springer-Verlag, New York, 1982. MR 718674 (85g:11001)
  • [15] H. Yokoi, On the class number of a relatively cyclic number field, Nagoya Math. J. 29 (1967), 31-44. MR 0207681 (34:7496)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11R23

Retrieve articles in all journals with MSC: 11R23


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1057961-4
Keywords: Iwasawa invariant, distinguished polynomial, class field
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society