The generalized Coleman–Ihara formula

Let $K$ be a totally real field in which $p$ is unramified, put $r=[K:\mathbb{Q}]$, and let $S$ be the archimedean places together with the places above $p$. Fix an odd integer $j>1$. Let $\eta_{K,S}(1-j)$ and $\eta_{K,S}(j)$ be the generalized Stark elements, let $\operatorname{loc}_p$ be localization, let $D_K$ be the discriminant of $K$, let $\Phi_j$ be the higher-rank Coates–Wiles element, and let $\chi_{\rm cyc}$ and $\mathcal{L}$ have the meanings given in the paper.

Generalized Coleman–Ihara formula. For every odd integer $j>1$,

$$\operatorname{loc}_p\left(\eta_{K,S}(1-j) \right)=\pm \chi_{\rm cyc}^{1-j}(\mathcal{L})\cdot D_K^j \cdot \Phi_j \quad\text{in }\mathbb{C}_p\bigwedge_{\mathbb{Z}_p}^r H^1(K_p,\mathbb{Z}_p(j)).$$

The immediately preceding theorem proves the analogous formula with $\eta_{K,S}(j)$ in place of $\chi_{\rm cyc}^{1-j}(\mathcal{L})$; this prediction follows on combining that theorem with the $p$-adic Beilinson conjecture, and is therefore not established in the supplied passage.