Bloch–Kato conjecture for unitary Galois representations

Let $\pi$ be an automorphic representation with associated Galois representation $\rho_\pi$, let $\chi$ be a character, and let $\mathcal{K}$ be the relevant number field. Write $\tilde{\rho}_\pi$ for the representation occurring in the $L$-function and $\mathrm{Sel}(\mathcal{K},\rho_\pi\otimes\chi^{-1})$ for the corresponding Selmer group. Bloch–Kato conjecture. The vanishing order of $L(\tilde{\rho}_\pi\otimes\chi, s)$ at $s=1$ is equal to the rank of the Selmer group $\mathrm{Sel}(\mathcal{K}, \rho_\pi\otimes\chi^{-1})$. This is the representation-theoretic central-value formulation used in the paper; its status is not specified in the source.