Beilinson–Bloch–Kato conjecture for polarized Chow motives

Let $K$ be a number field and let $\mathsf{M}$ be a polarized Chow motive in $\mathsf{Mot}_K^\mathrm{rat}$. Let $\operatorname{CH}(\mathsf{M})^0$ be the homologically trivial degree-zero Chow group, let $\operatorname{AJ}_p$ be its $p$-adic étale Abel–Jacobi map, let $\mathrm{H}^1_f(K,\mathsf{M}_p)$ denote the Bloch–Kato Selmer group, and let $L(s,\mathsf{M})$ be the associated $L$-function. Beilinson–Bloch–Kato conjecture. The following assertions hold: (1) for every prime $p$, the map

$$\operatorname{AJ}_p\colon\operatorname{CH}(\mathsf{M})^0\otimes_{\mathbb{Q}}\mathbb{Q}_p\xrightarrow{\sim}\mathrm{H}^1_f(K,\mathsf{M}_p)$$

is an isomorphism; (2) $L(s,\mathsf{M})$ has a meromorphic continuation to the entire complex plane and satisfies

$$L(s,\mathsf{M})=\epsilon(\mathsf{M})c(\mathsf{M})^{-s}L(-s,\mathsf{M});$$

(3) for all $p$,

$$\operatorname{ord}_{s=0}L(s,\mathsf{M})=\dim_{\mathbb{Q}_p}\mathrm{H}^1_f(K,\mathsf{M}_p)-\dim_{\mathbb{Q}_p}\mathrm{H}^0(K,\mathsf{M}_p).$$

Here $\epsilon(\mathsf{M})\in\{\pm1\}$ and $c(\mathsf{M})$ is a positive integer. This combines the Beilinson–Bloch and Bloch–Kato conjectures; the assertions are open in general.