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Proceedings of the American Mathematical Society

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



Class numbers and Iwasawa invariants
of quadratic fields

Author: James S. Kraft
Journal: Proc. Amer. Math. Soc. 124 (1996), 31-34
MSC (1991): Primary 11R11, 11R23, 11R29
MathSciNet review: 1301510
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Abstract: Let $\mathbf{Q}(\sqrt {-d}\hskip 2pt)$ and $\mathbf{Q}(\sqrt {3d}\hskip 2pt)$ be quadratic fields with $d \equiv $ 2 (mod 3) a positive integer. Let $\lambda ^-, \lambda ^+$ be the respective Iwasawa $\lambda $-invariants of the cyclotomic $\mathbf{Z}_3$-extension of these fields. We show that if $\lambda ^- =1$, then 3 does not divide the class number of $\mathbf{Q}(\sqrt {3d}\hskip 2pt)$ and $\lambda ^+ = 0$.

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

  • 1 Kuniaki Horie, A note on basic Iwasawa 𝜆-invariants of imaginary quadratic fields, Invent. Math. 88 (1987), no. 1, 31–38. MR 877004,
  • 2 K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257--258, MR 18:644d.
  • 3 N. Jochnowitz, A $p$-adic conjecture about derivatives of $L$-series attatched to modular forms, Proceedings of the Boston University Conference on $p$-Adic Monodromy and the $p$-Adic Birch and Swinnerton-Dyer Conjecture (to appear), CMP 94:13.
  • 4 ------, An alternative approach to non-vanishing theorems for coefficients of half integral weight forms mod $p$ and implications for Iwasawa's $\lambda $-invariant for quadratic fields (to appear).
  • 5 Lawrence C. Washington, Zeros of 𝑝-adic 𝐿-functions, Seminar on Number Theory, Paris 1980-81 (Paris, 1980/1981) Progr. Math., vol. 22, Birkhäuser, Boston, Mass., 1982, pp. 337–357. MR 693329
  • 6 Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674

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Additional Information

James S. Kraft

Received by editor(s): September 1, 1993
Received by editor(s) in revised form: August 1, 1994
Communicated by: William Adams
Article copyright: © Copyright 1996 American Mathematical Society