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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Generalized Fesenko reciprocity map


Authors: K. I. Ikeda and E. Serbest
Original publication: Algebra i Analiz, tom 20 (2008), nomer 4.
Journal: St. Petersburg Math. J. 20 (2009), 593-624
MSC (2000): Primary 11S37
Published electronically: June 1, 2009
MathSciNet review: 2473746
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Abstract: The paper is a natural continuation and generalization of the works of Fesenko and of the authors. Fesenko's theory is carried over to infinite APF Galois extensions $ L$ over a local field $ K$ with a finite residue-class field $ \kappa_K$ of $ q=p^f$ elements, satisfying $ \pmb{\mu}_p(K^{\mathrm{sep}})\subset K$ and $ K\subset L\subset K_{\varphi^d}$, where the residue-class degree $ [\kappa_L:\kappa_K]$ is equal to $ d$. More precisely, for such extensions $ L/K$ and a fixed Lubin-Tate splitting $ \varphi$ over $ K$, a $ 1$-cocycle

$\displaystyle \pmb{\Phi}_{L/K}^{(\varphi)}:\mathrm{Gal}(L/K)\rightarrow K^\times/N_{L_0/K}L_0^\times\times U_{\widetilde{\mathbb{X}}(L/K)}^\diamond/Y_{L/L_0}, $

where $ L_0=L\cap K^{nr}$, is constructed, and its functorial and ramification-theoretic properties are studied. The case of $ d=1$ recovers the theory of Fesenko.


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Additional Information

K. I. Ikeda
Affiliation: Department of Mathematics, Yeditepe University, 26 Aǧustos Yerleşimi, İnönü Mah., Kayışdaǧı Cad., 34755 Kadıköy, Istanbul, Turkey
Email: ilhan.ikeda@yeditepe.edu.tr

E. Serbest
Affiliation: Gümüş Pala Mahallesi, Gümüş Sok., Öz Aksu Sitesi, C-2/39, 34160 Avcılar, Istanbul, Turkey
Email: erols73@yahoo.com

DOI: http://dx.doi.org/10.1090/S1061-0022-09-01063-2
PII: S 1061-0022(09)01063-2
Keywords: Local fields, higher-ramification theory, APF extensions, Fontaine--Wintenberger field of norms, Fesenko reciprocity map, generalized Fesenko reciprocity map, non-Abelian local class field theory
Received by editor(s): October 20, 2007
Published electronically: June 1, 2009
Article copyright: © Copyright 2009 American Mathematical Society