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 Izv. RAN. Ser. Mat., 2009, Volume 73, Issue 5, Pages 105–170 (Mi izv2749)

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

L. V. Kuz'min

Russian Research Centre "Kurchatov Institute"

Abstract: For an algebraic number field $k$ that is either a field of CM-type (real or imaginary) or a field having Abelian completions at all places over $\ell$ and satisfying the feeble conjecture on the $\ell$-adic regulator [1] and its cyclotomic $\mathbb{Z}_\ell$-extension $k_\infty$, we obtain formulae that represent for all sufficiently large $n$ the $\ell$-adic exponent of the number $R_\ell(k_{n+1})/R_\ell(k_n)$, where $R_\ell(k_n)$ is the $\ell$-adic regulator in the sense of [1]. We discuss the meaning of the Iwasawa invariants occurring in these formulae and the resemblance between our results and the Brauer–Siegel theorem.

Keywords: Iwasawa theory, cyclotomic $Z_\ell$-extensions, $\ell$-adic regulator, Iwasawa invariants.

DOI: https://doi.org/10.4213/im2749

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English version:
Izvestiya: Mathematics, 2009, 73:5, 959–1021

Bibliographic databases:

UDC: 519.4
MSC: 11S85, 11S25

Citation: 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. RAN. Ser. Mat., 73:5 (2009), 105–170; Izv. Math., 73:5 (2009), 959–1021

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv2749
• https://doi.org/10.4213/im2749
• http://mi.mathnet.ru/eng/izv/v73/i5/p105

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This publication is cited in the following articles:
1. L. V. Kuz'min, “The feeble conjecture on the 2-adic regulator for some 2-extensions”, Izv. Math., 76:2 (2012), 346–355
2. L. V. Kuz'min, “On a new type of $\ell$-adic regulator for algebraic number fields (the $\ell$-adic regulator without logarithms)”, Izv. Math., 79:1 (2015), 109–144
3. L. V. Kuz'min, “On a new type of $\ell$-adic regulator for algebraic number fields. II”, St. Petersburg Math. J., 27:6 (2016), 977–984
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