On a Filtration of CH_{0} for an Abelian Variety A

TL;DR

This paper introduces a new filtration on the zero-cycle group of an abelian variety and establishes an isomorphism with a K-group, also computing specific kernels in the case of split multiplicative reduction over p-adic fields.

Contribution

It defines a filtration of CH_0 for abelian varieties and proves an isomorphism with a Somekawa K-group, extending understanding of zero-cycles and K-theory.

Findings

Established a filtration F^r of CH_0(A)

Proved an isomorphism with K(k;A,...,A)/Sym

Computed kernels for A with split multiplicative reduction over p-adic fields

Abstract

Let $A$ be an abelian variety defined over a field $k$. In this paper we define a filtration $F^{r}$ of the group $CH_{0}(A)$ and prove an isomorphism $\frac{K(k;A,...,A)}{\Sym}\otimes\mathbb{Z}[\frac{1}{r!}]\simeq F^{r}/F^{r+1}\otimes\mathbb{Z}[\frac{1}{r!}]$, where $K(k;A,...,A)$ is the Somekawa K-group attached to $r$-copies of the abelian variety $A$.\\ In the special case when $k$ is a finite extension of $\mathbb{Q}_{p}$ and $A$ has split multiplicative reduction, we compute the kernel of the map $CH_{0}(A)\otimes\Z[\frac{1}{2}]\rightarrow \rm{Hom}(Br(A),\Q/\Z)\otimes\Z[\frac{1}{2}]$, induced by the pairing $CH_{0}(A)\times Br(A)\rightarrow\mathbb{Q}/\Z$.