A property of the $\ell$-adic logarithms of units of

L. V. Kuz'min

Izvestiya. Mathematics , Tome 70 (2006) no. 5, pp. 949-974

Résumé

We continue to examine the finite abelian $\ell$-groups ${\mathcal A}_n^{(p)}$ and ${\mathcal B}_n^{(p)}$, which were introduced in [7] to characterize 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 $K_n$ is an intermediate subfield of the cyclotomic ${\mathbb Z}_\ell$-extension $K_\infty/K$, $K$ is a finite extension of ${\mathbb Q}_\ell$, $U(K_n)$ is the group of units of $K_n$ and $\log$ is the $\ell$-adic logarithm. If $\ell\geqslant 3$ and $K$ is a non-abelian field, we prove that ${\mathcal A}_n^{(p)}\neq 0$ and ${\mathcal B}_n^{(p)}\neq0$ except in the case when $\ell=3$ and the $K$ is a quadratic extension of a cyclotomic field. We also investigate this exceptional case.