Iwasawa's main conjecture via the Rankin–Selberg p-adic height

Assume that $Pi$ is incoherent and Hypothesis $ref{hy:galois}$ holds. Let $\mathcal{E}/E$ be a $mathtt{P}$-extension. Height-form main conjecture. The following are equivalent: (i) $\mathscr{L}^1_\mathcal{E}(Pi)$ is nonzero; (ii) $\mathscr{X}(\mathcal{E},\mathbb{W}_Pi)$ has rank one; (iii) $\mathscr{S}(\mathcal{E},\mathbb{W}_Pi)$ has rank one. When these conditions hold, for every $\mathbb{Z}_p$-linear map $\phi\colon\Gamma_mathtt{P}^F\mathtt{o}\mathbb{Z}_p$, the characteristic ideal of $\mathscr{X}(\mathcal{E},\mathbb{W}_Pi)^\mathrm{tor}$ is generated by the image of $\phi\mathscr{L}^1_\mathcal{E}(Pi)$. This gives a formulation of the rank-one main conjecture without using $\mathscr{K}$.