A Hasse principle for the 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 $C{H}^2(X,1)\to F^{\times }$. We study the mod-$l$ structure of $V(X)$ by combining Bloch's exact sequence with a Hasse principle in Galois cohomology associated with the mod-$l$ representation of the Jacobian $J$ of $X$. We obtain an exact sequence that describes the kernel and cokernel of the boundary map in terms of local reduction data and the coinvariant quotient $J[l{]}_{G_F}$. As a consequence, if ${\mathrm{End}}_{\overline{F}}(J)=\mathbb{Z}$ and $J$ has semistable reduction of toric dimension one at some place of $F$, then the mod-$l$ boundary map is an isomorphism for all but finitely many primes $l\ne \mathrm{char}(F)$. We also give explicit computations for elliptic curves.