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

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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.

- John R. Bloom and Frank Gerth III,
*The Iwasawa invariant $\mu $ in the composite of two ${\bf Z}_{l}\,$-extensions*, J. Number Theory**13**(1981), no. 2, 262–267. MR**612687**, DOI https://doi.org/10.1016/0022-314X%2881%2990009-3
N. Bourbaki, - Albert A. Cuoco and Paul Monsky,
*Class numbers in ${\bf Z}^{d}_{p}$-extensions*, Math. Ann.**255**(1981), no. 2, 235–258. MR**614400**, DOI https://doi.org/10.1007/BF01450674 - D. S. Dummit, D. Ford, H. Kisilevsky, and J. W. Sands,
*Computation of Iwasawa lambda invariants for imaginary quadratic fields*, J. Number Theory**37**(1991), no. 1, 100–121. MR**1089792**, DOI https://doi.org/10.1016/S0022-314X%2805%2980027-7 - Bruce Ferrero and Lawrence C. Washington,
*The Iwasawa invariant $\mu _{p}$ vanishes for abelian number fields*, Ann. of Math. (2)**109**(1979), no. 2, 377–395. MR**528968**, DOI https://doi.org/10.2307/1971116 - Eduardo Friedman,
*Iwasawa invariants*, Math. Ann.**271**(1985), no. 1, 13–30. MR**779602**, DOI https://doi.org/10.1007/BF01455793 - Frank Gerth III,
*Upper bounds for an Iwasawa invariant*, Compositio Math.**39**(1979), no. 1, 3–10. MR**538999** - Robert Gold,
*The nontriviality of certain $Z_{1}$-extensions*, J. Number Theory**6**(1974), 369–373. MR**369316**, DOI https://doi.org/10.1016/0022-314X%2874%2990034-1 - Daniel Gorenstein,
*Finite groups*, Harper & Row, Publishers, New York-London, 1968. MR**0231903** - Ralph Greenberg,
*On the Iwasawa invariants of totally real number fields*, Amer. J. Math.**98**(1976), no. 1, 263–284. MR**401702**, DOI https://doi.org/10.2307/2373625 - Kenkichi Iwasawa,
*A note on class numbers of algebraic number fields*, Abh. Math. Sem. Univ. Hamburg**20**(1956), 257–258. MR**83013**, DOI https://doi.org/10.1007/BF03374563
---, - Lawrence C. Washington,
*Introduction to cyclotomic fields*, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR**718674** - Hideo Yokoi,
*On the class number of a relatively cyclic number field*, Nagoya Math. J.**29**(1967), 31–44. MR**207681**

*Commutative algebra*, Addison-Wesley, Reading, MA, 1972.

*On*$\Gamma$

*-extensions of algebraic number fields*, Bull. Amer. Math. Soc.

**65**(1959), 183-226. ---,

*On*${\mathbb {Z}_l}$

*-extensions of algebraic number fields*, Ann. of Math.

**98**(1973), 246-326.

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Keywords:
Iwasawa invariant,
distinguished polynomial,
class field

Article copyright:
© Copyright 1991
American Mathematical Society