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

We study the higher Chow group $C{H}^2(X,1)$ of a smooth projective curve $X$ over a global field $F$, focusing on the kernel $V(X)$ of the push-forward map $C{H}^2(X,1)\to C{H}^1(F,1)=F^{\times }$. Our main purpose is to investigate the structure of the torsion subgroup of $V(X)$ and its relation to the arithmetic of the curve. Using Bloch’s exact sequence together with a Hasse principle for Galois cohomology arising from mod-$l$ Galois representations, we show that the mod-$l$ quotient $V(X)\mathrm{/}lV(X)$ is governed by the mod-$l$ Galois representation on the $l$-torsion subgroup $J[l]$ of the Jacobian variety $J$ of $X$.