Geodesic
Some remarks on the $\ell$-adic regulator. V.
Izvestiya. Mathematics , Tome 73 (2009) no. 5, pp. 959-1021

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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, $\ell$-adic regulator, Iwasawa invariants.
Mots-clés : cyclotomic $Z_\ell$-extensions 
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     author = {L. V. Kuz'min},
     title = {Some remarks on the $\ell$-adic regulator. {V.}},
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L. V. Kuz'min. Some remarks on the $\ell$-adic regulator. V.. Izvestiya. Mathematics , Tome 73 (2009) no. 5, pp. 959-1021. http://geodesic.mathdoc.fr/item/IM2_2009_73_5_a4/