Bloch–Kato conjecture for geometric irreducible representations

Let $p$ be a prime, let $cK$ be a $p$-adic field, and let $V$ be a geometric irreducible representation of $\mathrm{Gal}(\overline{\mathbf Q}/\mathbf Q)$ over $\mathcal K$. Write $V^\vee$ for the dual Galois representation and $\mathrm H^1_{\mathrm f}(\mathbf Q,V^\vee(1))$ for the Bloch–Kato Selmer group with coefficients in $V^\vee(1)$. Bloch–Kato conjecture. The Hasse–Weil $L$-function $L(V,s)$ has a meromorphic continuation to $\mathbf C$ and satisfies

$$\operatorname{rank}_{s=0}L(V,s)=\dim_{\mathcal K}\mathrm H^1_{\mathrm f}(\mathbf Q,V^\vee(1))-\dim_{\mathcal K}\mathrm H^0(\mathbf Q,V^\vee(1)).$$

This conjecture predicts a relation between the analytic rank of the Hasse–Weil $L$-function and the Bloch–Kato Selmer group. The paper proves it for the $p$-adic Galois representations associated with full-level Hecke eigen cuspforms under the depth–weight compatibility on the motivic fundamental Lie algebra, while the stated general form remains open.