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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Iwasawa invariants and class numbers
of quadratic fields for the prime $3$


Author: Hisao Taya
Journal: Proc. Amer. Math. Soc. 128 (2000), 1285-1292
MSC (1991): Primary 11R23, 11R11, 11R29
Published electronically: August 3, 1999
MathSciNet review: 1641133
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Abstract: Let $d$ be a square-free integer with $d \equiv 1 \pmod{3}$ and $d > 0$. Put $k^{+}=\Bbb Q(\sqrt{d})$ and $k^{-}=\Bbb Q(\sqrt{-3d})$. For the cyclotomic $\Bbb Z_3$-extension $k^{+}_\infty$ of $k^{+}$, we denote by $k^{+}_n$ the $n$-th layer of $k^{+}_\infty$ over $k^{+}$. We prove that the $3$-Sylow subgroup of the ideal class group of $k^{+}_n$ is trivial for all integers $n \geq 0$ if and only if the class number of $k^{-}$ is not divisible by the prime $3$. This enables us to show that there exist infinitely many real quadratic fields in which $3$ splits and whose Iwasawa $\lambda _3$-invariant vanishes.


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Additional Information

Hisao Taya
Email: taya@math.is.tohoku.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05177-1
PII: S 0002-9939(99)05177-1
Keywords: Iwasawa invariants, real quadratic fields, class numbers
Received by editor(s): August 27, 1997
Received by editor(s) in revised form: June 22, 1998
Published electronically: August 3, 1999
Additional Notes: This research was partially supported by the Grant-in-Aid for Encouragement of Young Scientists, The Ministry of Education, Science, Sports and Culture, Japan.
Dedicated: Dedicated to Professor Koji Uchida on his 60th birthday
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2000 American Mathematical Society