Some remarks on the $l$-adic regulator. II

Given an algebraic number field $k$ with unit group $U(k)$ and a prime number $l$, consider the bilinear form $S\colon(U(k)\otimes\mathbf Z_l)(U(k)\otimes\mathbf Z_l)\to\mathbf Q_l$, $S(x,y)=\operatorname{Sp}_{k/\mathbf Q}(\log x\cdot\log y)$ where $\log$ is the $l$-adic logarithm. For certain types of fields it is shown that the form $S$ is nondegenerate. We investigate the behavior of the rank of the kernel of $S$ on the family of intermediate fields in a $\mathbf Z_l$-extension $k_\infty/k$.
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