The Iwasawa main conjecture for Kato's Euler system

Let $\mathbb{Q}_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, let $\Lambda$ be its Iwasawa algebra, and assume that $\rho_f$ has large image. For a height-one prime $\mathfrak{P}$ of $\Lambda$, consider the Kato Euler-system class $\kappa^{\mathrm{Kato},k-r,\infty}_1$, the Iwasawa cohomology module $\mathrm{H}^1_{\mathrm{Iw}}(\mathbb{Q},T_f(k-r))$, and the dual Selmer module $\mathrm{Sel}_0(\mathbb{Q}_\infty,W_{\overline f}(r))^\vee$. The Iwasawa main conjecture. The localized equality

$$\operatorname{ord}_{\mathfrak{P}}\!\left(\operatorname{char}_{\Lambda}\!\left(\frac{\mathrm{H}^1_{\mathrm{Iw}}(\mathbb{Q},T_f(k-r))}{\Lambda\kappa^{\mathrm{Kato},k-r,\infty}_1}\right)\right)=\operatorname{ord}_{\mathfrak{P}}\!\left(\operatorname{char}_{\Lambda}\!\left(\mathrm{Sel}_0(\mathbb{Q}_\infty,W_{\overline f}(r))^\vee\right)\right)$$

should hold for every height-one prime ideal $\mathfrak{P}$ of $\Lambda$. This is the central Iwasawa-theoretic input for the paper’s non-vanishing and refined Tamagawa-number results; the source does not state that it is proved in full generality.