A Hasse principle of the higher chow groups for an elliptic curve over a global function field

TL;DR

This paper explores the structure of higher Chow groups for elliptic curves over global function fields, focusing on the torsion part related to Galois representations of torsion points.

Contribution

It provides new insights into the kernel of the push-forward map in higher Chow groups for elliptic curves over global function fields, linking it to Galois representations.

Findings

Characterization of the torsion subgroup V(E)

Relation between V(E) and Galois representations of E[l]

Structural results on higher Chow groups over function fields

Abstract

We investigate the structure of the higher Chow groups $CH^2(E,1)$ for an elliptic curve $E$ over a global function field $F$. Focusing on the kernel $V(E)$ of the push-forward map $CH^2(E,1)\to F^{\times}$ associated to the structure map $E\to \operatorname{Spec}(F)$, we analyze the torsion part $V(E)$ based on the mod $l$ Galois representations associated to the $l$-torsion points $E[l]$.