The third Brumer–Stark formula

Let $u_3\in F_{\mathfrak p}^*\otimes\mathbb Z[G]$ be the element defined by the cap product

$$u_3=c_{\operatorname{id},\mathfrak p}\cap(\kappa_\lambda\cap\eta_{S_p}).$$

Let $u_{\mathfrak p}$ denote the Brumer–Stark element. Third Brumer–Stark formula.

$$u_3=u_{\mathfrak p}.$$

This is adapted from a conjecture of the first author and Spieß. The paper's main result establishes equality of the three formulas, but the supplied candidate carries no explicit resolution evidence for this individually.