Leopoldt's conjecture on the rank of global units

Let $M$ be a number field. For each prime $P$ of $M$ dividing $p$, let $\mathcal{O}_P^{\times}$ and $\mathcal{O}_{P,1}^{\times}$ denote the local units and principal local units, respectively. Let

$$D:\mathcal{O}_M^{\times}\longrightarrow\prod_{P\mid p}\mathcal{O}_P^{\times}$$

be the diagonal embedding, and set

$$U=D^{-1}\left(\prod_{P\mid p}\mathcal{O}_{P,1}^{\times}\right).$$

Leopoldt's conjecture. The $\mathbb{Z}$-rank of $\mathcal{O}_M^{\times}$ equals the $\mathbb{Z}_p$-rank of the topological closure of $D(U)$ in $\prod_{P\mid p}\mathcal{O}_{P,1}^{\times}$. This is the classical $p$-adic independence conjecture for units; the paper notes that the $p$-adic Schanuel conjecture would strengthen it, while the general case remains open.