The $\ell$-adic regulator conjecture

Let $K$ be an algebraic number field, let $\ell$ be a prime, and let $\varepsilon_1,\ldots,\varepsilon_r$ be a system of fundamental units of $K$. Set $\varepsilon_0=1+\ell$, with $\varepsilon_0=5$ if $\ell=2$, and let $\log_\ell$ be the vector of $\ell$-adic logarithms. Let $\operatorname{Sp}_{K/\mathbb Q}:K\otimes_{\mathbb Q}\mathbb Q_\ell\to\mathbb Q_\ell$ be induced by the field trace.

The $\ell$-adic regulator conjecture. Define

$$R_\ell(K)=\det\bigl(\operatorname{Sp}_{K/\mathbb Q}(\log_\ell\varepsilon_i\cdot\log_\ell\varepsilon_j)\bigr)_{0\leq i,j\leq r}.$$

Then $R_\ell(K)\neq 0$ for every $K$ and $\ell$.

The paper presents this as a strengthening of Leopoldt's conjecture and attributes it to the cited source. It implies non-degeneracy of the associated logarithmic pairing and is not known in general.