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

Abstract

Let $X$ be a smooth projective curve over a global field $F$, and let $V(X)$ denote the kernel of the push-forward map $C{H}^2(X,1)\to C{H}^1(F,1)=F^{\times }$ 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 Bloch's exact sequence with a Hasse principle in Galois cohomology for the mod-$l$ Galois representation of the Jacobian variety $J$ of $X$. We also give explicit computations for elliptic curves.