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

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.