Geodesic
Some remarks on the $l$-adic regulator.~II
Izvestiya. Mathematics , Tome 35 (1990) no. 1, pp. 113-144

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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 $k_\infty/k$. Bibliography: 11 titles. 
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     author = {L. V. Kuz'min},
     title = {Some remarks on the $l$-adic {regulator.~II}},
     journal = {Izvestiya. Mathematics },
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L. V. Kuz'min. Some remarks on the $l$-adic regulator.~II. Izvestiya. Mathematics , Tome 35 (1990) no. 1, pp. 113-144. http://geodesic.mathdoc.fr/item/IM2_1990_35_1_a5/