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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Class numbers and Iwasawa invariants of quadratic fields
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by James S. Kraft
Proc. Amer. Math. Soc. 124 (1996), 31-34
DOI: https://doi.org/10.1090/S0002-9939-96-03085-7
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Abstract:

Let $\mathbf {Q}(\sqrt {-d})$ and $\mathbf {Q}(\sqrt {3d})$ 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 $\mathbf {Q}(\sqrt {3d})$ and $\lambda ^+ = 0$.
References
    K. Horie, A note on basic Iwasawa $\lambda$-invariants of imaginary quadratic fields, Invent. Math. 88 (1987), 31–38, MR 88i:11073. K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257–258, MR 18:644d. N. Jochnowitz, A $p$-adic conjecture about derivatives of $L-series attatched to modular forms\closeXMLelement {italic}, Proceedings of the Boston University Conference on$p$-Adic Monodromy and the$p$-Adic Birch and Swinnerton-Dyer Conjecture (to appear), CMP \textbf {94:13}.$ N. Jochnowitz, 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). L. Washington, Zeroes of $p$-adic $L$-functions, Sém Delange-Pisot- Poitou, Théorie des Nombres, 1980/1981, Birkhäuser, Boston, Basel, and Stuttgart, 1982, MR 84f:12008. L. Washington, Introduction to cyclotomic fields, Graduate Texts in Math., Springer-Verlag, New York, 1982, MR 85g:11001.
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Bibliographic Information
  • James S. Kraft
  • Email: kraft@ithaca.edu
  • Received by editor(s): September 1, 1993
  • Received by editor(s) in revised form: August 1, 1994
  • Communicated by: William Adams
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 31-34
  • MSC (1991): Primary 11R11, 11R23, 11R29
  • DOI: https://doi.org/10.1090/S0002-9939-96-03085-7
  • MathSciNet review: 1301510