A Hasse principle for $GL_2(\mathbb{F}_p)$ and Bloch's exact sequence for elliptic curves over number fields
Toshiro Hiranouchi
TL;DR
The paper investigates higher Chow groups of elliptic curves via $SK_1$ and the kernel $V(X)$ of the norm, focusing on the elliptic case $V(E)$ over number fields. It combines Bloch’s exact sequence with a Hasse principle for mod $p$ Galois representations to relate global torsion in $V(E)$ to the mod $p$ action on $E[p]$ and local data at good and bad primes. A central result shows that if $E[p]_{G_F} eq 0$, then the mod $p$ boundary map $ar ullbar{ ho}_{E,p}$ has finite kernel and cokernel, and in fact the $p$-primary torsion can be infinite in dimension for some $p$, governed by $ ho_{E,p}$. The paper also develops explicit local computations, including Hilbert symbol and Tate parameter analysis, and provides detailed examples over $Q$ (notably curves in conductor $651$) illustrating the interplay between global Galois representations and local data in Bloch’s sequence. Overall, it advances understanding of Bloch’s exact sequence for elliptic curves and reveals how a Hasse principle constrains the mod $p$-structure of $V(E)$ in terms of $E[p]_{G_F}$.
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.