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A note on Greenberg's conjecture and the abc conjecture

Author(s): Humio Ichimura
Journal: Proc. Amer. Math. Soc. 126 (1998), 1315-1320.
MSC (1991): Primary 11R23
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Abstract: For any totally real number field $k$ and any prime number $p$, Greenberg's conjecture for $(k,p)$ asserts that the Iwasawa invariants $\lambda _p(k)$ and $\mu _p(k)$ are both zero. For a fixed real abelian field $k$, we prove that the conjecture is ``affirmative'' for infinitely many $p$ (which split in $k)$ if we assume the abc conjecture for $k$.

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Additional Information:

Humio Ichimura
Affiliation: Department of Mathematics, Yokohama City University, 22-2, Seto, Kanazawa-ku, Yokohama, 236 Japan
Email: ichimura@yokohama-cu.ac.jp

DOI: 10.1090/S0002-9939-98-04196-3
PII: S 0002-9939(98)04196-3
Received by editor(s): June 23, 1996
Received by editor(s) in revised form: October 30, 1996
Additional Notes: The author was partially supported by the Grants-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan.
Communicated by: William W. Adams
Copyright of article: Copyright 1998, American Mathematical Society

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