Equality of the three Brumer–Stark unit formulas

Let $u_{\mathfrak p}$ be the Brumer–Stark element attached to the data $(S,T,H,\mathfrak P)$, and let $u_1,u_2,u_3\in F_{\mathfrak p}^*\otimes\mathbb Z[G]$ be the three formulas defined in the paper: respectively the $p$-adic analytic formula and the two cohomological Eisenstein-cocycle formulas. Equality of the three formulas. For $i=1,2,3$,

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

These formulas were conjectured by the first author and by the first author with Spieß as formulas for the image of the Brumer–Stark element. The paper proves their equality and verifies their validity using the Brumer–Stark result, so this conjectural package is solved.