A Hasse principle for $GL_2(\mathbb{F}_p)$ and Bloch's exact sequence for elliptic curves over number fields

Abstract

We investigate the higher Chow groups, specifically $SK_1(E)$ for elliptic curves $E$ over number fields $F$. Focusing on the kernel $V(E)$ of the norm map $SK_1(E)\to F^{\times}$, we analyze its mod $p$ structure. We provide conditions, based on the mod $p$ Galois representations associated to $E$, under which the torsion subgroup of $V(E)$ is infinite.