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Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 1, Pages 115–152 (Mi izv8177)  

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

On a new type of $\ell$-adic regulator for algebraic number fields (the $\ell$-adic regulator without logarithms)

L. V. Kuz'min

National Research Centre "Kurchatov Institute"

Abstract: For an algebraic number field $K$ such that a prime $\ell$ splits completely in $K$, we define a regulator $\mathfrak R_\ell(K)\in\mathbb Z_\ell$ that characterizes the subgroup of universal norms from the cyclotomic $\mathbb Z_\ell$-extension of $K$ in the completed group of $S$-units of $K$, where $S$ consists of all prime divisors of $\ell$. We prove that the inequality $\mathfrak R_\ell(K)\ne0$ follows from the $\ell$-adic Schanuel conjecture and holds for some Abelian extensions of imaginary quadratic fields. We study the connection between the regulator $\mathfrak R_\ell(K)$ and the feeble conjecture on the $\ell$-adic regulator, and define analogues of the Gross regulator.

Keywords: $\ell$-adic regulator, $S$-units, global universal norm, Schanuel conjecture, Iwasawa theory.

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-00588-a
This paper was written with the financial support of the Russian Foundation for Basic Research (grant no. 11-01-00588-a).


DOI: https://doi.org/10.4213/im8177

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English version:
Izvestiya: Mathematics, 2015, 79:1, 109–144

Bibliographic databases:

UDC: 511.236.3
MSC: 11R23, 11R18
Received: 16.10.2013

Citation: L. V. Kuz'min, “On a new type of $\ell$-adic regulator for algebraic number fields (the $\ell$-adic regulator without logarithms)”, Izv. RAN. Ser. Mat., 79:1 (2015), 115–152; Izv. Math., 79:1 (2015), 109–144

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. 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  mathnet  crossref  mathscinet  isi  elib
    2. L. V. Kuz'min, “Local and global universal norms in the cyclotomic $\mathbb Z_\ell$-extension of an algebraic number field”, Izv. Math., 82:3 (2018), 532–548  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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