On the anticyclotomic $\mathbb Z_{p}$-extension of an imaginary quadratic field

Jangheon Oh

doi:10.11568/kjm.2015.23.3.323

Korean J. Math. Vol. 23 No. 3 (2015) pp.323-326
DOI: https://doi.org/10.11568/kjm.2015.23.3.323

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Published: Sep 30, 2015

Keywords:

Iwasawa theory, anticylotomic extension, Hilbert class field.

Mathematics Subject Classification:

11R23.

Jangheon Oh

Faculty of Mathematics and Statistics Sejong University Seoul 143-747, Korea

Abstract

We prove that if a subfield of the Hilbert class field of an imaginary quadratic field

k

meets the anticyclotomic

Z_{p}

-extension

k_{\infty}^{a}

of

k

, then it is contained in

k_{\infty}^{a}

. And we give an example of an imaginay quadratic field

k

with

λ_{3} (k_{\infty}^{a}) \geq 8.

References

[1] S.Fujii, On a bound of λ and the vanishing of μ of Zp-extensions of an imaginary quadratic field, J.Math.Soc.Japan. 65 (1) (2013), 277–298. Google Scholar

[2] J.Minardi, Iwasawa modules for Zdp-extensions of algebraic number fields, Ph.D dissertation, University of Washington, 1986. Google Scholar

[3] J.Oh, On the first layer of anti-cyclotomic Zp-extension over imaginary quadratic fields, Proc. Japan Acad. Ser.A Math.Sci. 83 (3) (2007), 19–20. Google Scholar

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[5] J.Oh, Construction of 3-Hilbert class field of certain imaginary quadratic fields, Proc. Japan Aca. Ser.A Math. Sci. 86 (1) (2010), 18–19. Google Scholar

[6] J.Oh, Anti-cyclotomic extension and Hilbert class field, Journal of the Chungcheong Math. Society 25 (1) (2012), 91–95 . Google Scholar

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