Bloch–Beilinson-type conjecture for zero-cycles over p-adic fields

Let $V$ be a smooth projective geometrically integral variety over a $p$-adic field $k$. Let $A_0(V)$ be the degree-zero subgroup of the Chow group of zero-cycles, and let

$$\phi_V:A_0(V)\longrightarrow \operatorname{Alb}_V(k)$$

be the Albanese map. Bloch–Beilinson-type conjecture over p-adic fields. The group $\operatorname{Ker}(\phi_V)$ is the direct sum of a finite group and a divisible group.

This is the paper’s proposed $p$-adic analogue of the Bloch–Beilinson conjecture for number fields. The source says that very little is known about the analogous number-field conjecture and studies this $p$-adic formulation.