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Izv. RAN. Ser. Mat., 2006, Volume 70, Issue 5, Pages 97–122 (Mi izv707)  

This article is cited in 2 scientific papers (total in 2 papers)

A property of the $\ell$-adic logarithms of units of non-abelian local fields

L. V. Kuz'min

Russian Research Centre "Kurchatov Institute"

Abstract: We continue to examine the finite abelian $\ell$-groups ${\mathcal A}_n^{(p)}$ and ${\mathcal B}_n^{(p)}$, which were introduced in [7] to characterize the bilinear form $U(K_n)\times U(K_n)\to {\mathbb Q}_\ell$, $(x,y)\to {\operatorname{Sp}}_{K_n/{\mathbb Q}_\ell} (\log x\cdot\log y)$, where $K_n$ is an intermediate subfield of the cyclotomic ${\mathbb Z}_\ell$-extension $K_\infty/K$, $K$ is a finite extension of ${\mathbb Q}_\ell$, $U(K_n)$ is the group of units of $K_n$ and $\log$ is the $\ell$-adic logarithm. If $\ell\ge 3$ and $K$ is a non-abelian field, we prove that ${\mathcal A}_n^{(p)}\neq 0$ and ${\mathcal B}_n^{(p)}\neq0$ except in the case when $\ell=3$ and the $K$ is a quadratic extension of a cyclotomic field. We also investigate this exceptional case.

DOI: https://doi.org/10.4213/im707

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English version:
Izvestiya: Mathematics, 2006, 70:5, 949–974

Bibliographic databases:

UDC: 519.4
MSC: 11S85, 11S25
Received: 27.04.2005

Citation: L. V. Kuz'min, “A property of the $\ell$-adic logarithms of units of non-abelian local fields”, Izv. RAN. Ser. Mat., 70:5 (2006), 97–122; Izv. Math., 70:5 (2006), 949–974

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. L. V. Kuz'min, “On coherent families of uniformizing elements in some towers of Abelian extensions of local number fields”, J. Math. Sci., 166:5 (2010), 670–674  mathnet  crossref  mathscinet  elib  elib
    2. Kuz'min A. G., “Self-sustained oscillations and bifurcations of transonic flow past simple airfoils”, J. Appl. Mech. Tech. Phys., 49:6 (2008), 919–925  crossref  adsnasa  isi  elib  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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