Bloch–Kato and Chow-rank conjecture for the motives $\mathsf{M}_{E,A}$

Let $(E,A)$ be an arbitrary pair, with associated Chow motive $\mathsf{M}_{E,A}$, Chow group $\operatorname{CH}(\mathsf{M}_{E,A})$, Bloch–Kato Selmer groups $\mathrm{H}^1_f(\mathbb{Q},(\mathsf{M}_{E,A})_p)$, $L$-function $L(s,\mathsf{M}_{E,A})$, and Chow class $\Delta_{E,A}$. Chow-rank conjecture for $\mathsf{M}_{E,A}$. (1) If at least one of $\dim_{\mathbb{Q}_p}\mathrm{H}^1_f(\mathbb{Q},(\mathsf{M}_{E,A})_p)=0$ for some prime $p$, $\dim_{\mathbb{Q}}\operatorname{CH}(\mathsf{M}_{E,A})=0$, or $\operatorname{ord}_{s=0}L(s,\mathsf{M}_{E,A})=0$ holds, then for every prime $p$,

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

(2) If at least one of $\dim_{\mathbb{Q}_p}\mathrm{H}^1_f(\mathbb{Q},(\mathsf{M}_{E,A})_p)=1$ for some prime $p$, $\dim_{\mathbb{Q}}\operatorname{CH}(\mathsf{M}_{E,A})=1$, or $\operatorname{ord}_{s=0}L(s,\mathsf{M}_{E,A})=1$ holds, then $\Delta_{E,A}\neq0$ and, for every prime $p$,

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

The class $\Delta_{E,A}$ may be constructed in a similar way. This proposal packages the expected equality between Chow rank, Selmer rank, and the order of vanishing at zero; it is not proved in the supplied text.