1 results
On a filtration of
$\mathit{CH}_{0}$ for an abelian variety
- Part of
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- Journal:
- Compositio Mathematica / Volume 151 / Issue 3 / March 2015
- Published online by Cambridge University Press:
- 22 December 2014, pp. 435-460
- Print publication:
- March 2015
-
- Article
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Let
$A$ be an abelian variety defined over a field 
$k$. In this paper we define a descending filtration 
$\{F^{r}\}_{r\geqslant 0}$ of the group 
$\mathit{CH}_{0}(A)$ and prove that the successive quotients 
$F^{r}/F^{r+1}\otimes \mathbb{Z}[1/r!]$ are isomorphic to the group 
$(K(k;A,\dots ,A)/Sym)\otimes \mathbb{Z}[1/r!]$, where 
$K(k;A,\dots ,A)$ 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 
$\mathit{CH}_{0}(A)\otimes \mathbb{Z}[\frac{1}{2}]\rightarrow \text{Hom}(Br(A),\mathbb{Q}/\mathbb{Z})\otimes \mathbb{Z}[\frac{1}{2}]$, induced by the pairing 
$\mathit{CH}_{0}(A)\times Br(A)\rightarrow \mathbb{Q}/\mathbb{Z}$.
