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

We continue to examine the bilinear form $U(K_n)\times U(K_n)\to\mathbb{Q}_\ell$, $(x,y)\to\operatorname{Sp}_{K_n/\mathbb{Q}_\ell}(\log x\cdot\log y)$ where $K_n$ runs through all intermediate subfields of the cyclotomic $\mathbb{Z}_\ell$-extension $K_\infty/K$, $K$ is an arbitrary finite extension of $\mathbb{Q}_\ell$, and $\log$ is the $\ell$-adic logarithm. We give applications to the weak conjecture on the $\ell$-adic regulator. In particular, we prove this conjecture for $\ell$-extensions of Abelian number fields.