A Hasse principle for higher Chow groups of curves over a global field

Let $X$ be a smooth projective curve over a global field $F$, and let $V(X)$ denote the kernel of the push-forward map $CH^2(X,1)\to CH^1(F,1) = F^\times$ of the higher Chow groups. For an odd prime $l$ different from the characteristic of $F$, we describe the structure of $V(X)/lV(X)$ by combining