Stark--Tate conjecture for rank-one abelian extensions

Let $H/F$ be an abelian extension of number fields, let $W$ be the number of roots of unity in $H$, and let $S$ be a finite set of places of $F$ containing the ramified and infinite places, with $|S|\geq 2$. Suppose a place $\mathfrak p\in S$ splits completely in $F$, put $T=S\setminus\{\mathfrak p\}$, and let $U_{S,H}^T$ be the group of elements satisfying the valuation conditions in the source. For $\sigma\in\operatorname{Gal}(H/F)$ and a place $\mathfrak P\mid\mathfrak p$, let $\zeta_S^{\operatorname{Gal}}(\sigma,s)$ be the Galois-theoretic partial zeta function. Stark--Tate conjecture. There exists $\e\in U_{S,H}^T$ such that

$$\log|\sigma(\e)|_{\mathfrak P}=-W\zeta_S^{\operatorname{Gal}\prime}(\sigma,0)$$

for every $\sigma$ and $\mathfrak P\mid\mathfrak p$, and such that $H(\e^{1/W})$ is abelian over $F$. Tate's formulation refines Stark's published assertion and is used to obtain abelian extensions containing the relevant RM values; the conjecture remains unresolved in the source.