Some remarks on Leopoldt's conjecture

Let $k$ be a number field, and let $p$ be a fixed prime number. Then the vanishing of the Leopoldt kernel $\mathscr{L}_p(k)$ is shown to be equivalent to the validity of a "Strong Local-Global Principle on units of $k$". This adds a problem of effectivity to Leopoldt's conjecture (an example to which is provided by the classical Kummer lemma on the $p$th powers of units in the field of the $p$th roots of unity). Some further remarks pertain to $\mathscr{L}_p(k)$ as a Galois module. For example, if $k/{\mathbb Q}$ is an Abelian $p$-extension, then the triviality of $\mathscr{L}_p(k)$ can be shown quite easily (in particular, without using Brumer's transcendency theorem).