# A filtration on the higher Chow group of zero cycles on an abelian variety

## TL;DR

This paper introduces a new filtration on the higher Chow group of zero-cycles on abelian varieties, linking it to K-groups and étale cohomology, with applications to local fields.

## Contribution

It extends Gazaki's results to higher Chow groups and establishes a new filtration connected to K-groups and étale cohomology.

## Key findings

- Connected the filtration to Somekawa type K-groups
- Compared the filtration with étale cohomology via Hochschild-Serre spectral sequence
- Provided an estimate of the kernel of the reciprocity map over local fields

## Abstract

In this paper we extend Gazaki's results on the Chow groups of abelian varieties to the higher Chow groups. We introduce a Gazaki type filtration on the higher Chow group of zero-cycles on an abelian variety, whose graded quotients are connected to the Somekawa type $K$-group. Via the \'{e}tale cycle map, we will compare this filtration with a filtration on the \'{e}tale cohomology induced by the Hochschild-Serre spectral sequence. As an application over local fields, we obtain an estimate of the kernel of the reciprocity map.