Leopoldt's conjecture

Let $K$ be a number field with $r_1$ real embeddings and $r_2$ pairs of complex embeddings, let $\mathcal{O}_K$ be its ring of integers, and let $E = \mathcal{O}_K^{\times}$ be its group of global units. Fix a rational prime $p$. For each prime $P$ of $K$ with $P \mid p$, let $K_P$ denote the completion of $K$ at $P$, let $U_P$ denote the group of units of the valuation ring of $K_P$, and let $U_{1,P} \subseteq U_P$ denote the subgroup of principal units (those congruent to $1$ modulo the maximal ideal). Set

$$U_{1}=\prod _{P\,\mid \,p}U_{1,P},$$

regarded as a subgroup of $\prod_{P \mid p} U_P$, and let

$$E_{1}=\Bigl\{\varepsilon \in E \;:\; (\varepsilon)_{P \mid p} \in U_{1}\Bigr\}$$

be the group of global units whose image under the diagonal embedding $E \to \prod_{P \mid p} U_P$ lies in $U_1$. Then $E_1$ has finite index in $E$, so $E_1$ is a finitely generated abelian group of rank $r_1 + r_2 - 1$.

Give $U_1$ its natural $p$-adic topology, so that $U_1$ is a $\mathbb{Z}_p$-module, and let $\overline{E_1}$ denote the closure in $U_1$ of the image of $E_1$ under the diagonal embedding. Then

$$\operatorname{rank}_{\mathbb{Z}_p} \overline{E_1} = r_1 + r_2 - 1.$$

Equivalently, if $\varepsilon_1,\dots,\varepsilon_{r}$ with $r = r_1 + r_2 - 1$ is a system of generators modulo torsion of $E_1$, then their images in $U_1$ are independent over $\mathbb{Z}_p$, i.e. the $p$-adic regulator of $K$, formed from the values $\log_p$ of the conjugates of $\varepsilon_1,\dots,\varepsilon_r$, is nonzero.