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Izv. RAN. Ser. Mat., 1999, Volume 63, Issue 6, Pages 29–82 (Mi izv268)  

This article is cited in 5 scientific papers (total in 5 papers)

Some remarks on the $\ell$-adic regulator. III

L. V. Kuz'min

Russian Research Centre "Kurchatov Institute"

Abstract: Let $K$ be a finite extension of the field of rational $\ell$-adic numbers $\mathbb Q_\ell$, and let $K_\infty$ be the cyclotomic $\mathbb Z_\ell$-extension of $K$. For an intermediate field $K_n$ in $K_\infty/K$, let $U(K_n)$ be the group of units of $K_n$ and put $U(K_n)^\perp=\{x\in K_n\mid\operatorname{Sp}_{K_n/\mathbb Q_\ell}(x\log u)\in {\mathbb Z}_\ell$ for all $u\in U(K_n)\}$, where $\log\colon U(K_n)\to K_n$ is the $\ell$-adic logarithm. We give an approximate characterization of $U(K_n)^\perp$. The proofs are based on the use of Laurent series with integer coefficients and infinite principal part.

DOI: https://doi.org/10.4213/im268

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English version:
Izvestiya: Mathematics, 1999, 63:6, 1089–1138

Bibliographic databases:

MSC: 11S85, 11R23, 11R37
Received: 13.01.1998

Citation: L. V. Kuz'min, “Some remarks on the $\ell$-adic regulator. III”, Izv. RAN. Ser. Mat., 63:6 (1999), 29–82; Izv. Math., 63:6 (1999), 1089–1138

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. L. V. Kuz'min, “Some remarks on the $\ell$-adic regulator. IV”, Izv. Math., 64:2 (2000), 265–310  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. L. V. Kuz'min, “A property of the $\ell$-adic logarithms of units of non-abelian local fields”, Izv. Math., 70:5 (2006), 949–974  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. 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
    4. L. V. Kuz'min, “Some remarks on the $\ell$-adic regulator. V. Growth of the $\ell$-adic regulator in the cyclotomic $Z_\ell$-extension of an algebraic number field”, Izv. Math., 73:5 (2009), 959–1021  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. L. V. Kuz'min, “The feeble conjecture on the 2-adic regulator for some 2-extensions”, Izv. Math., 76:2 (2012), 346–355  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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