#### Vol. 1, No. 2, 2007

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Surfaces over a p-adic field with infinite torsion in the Chow group of 0-cycles

### Masanori Asakura and Shuji Saito

Vol. 1 (2007), No. 2, 163–181
##### Abstract

We give an example of a projective smooth surface $X$ over a $p$-adic field $K$ such that for any prime $\ell$ different from $p$, the $\ell$-primary torsion subgroup of ${CH}_{0}\left(X\right)$, the Chow group of $0$-cycles on $X$, is infinite. A key step in the proof is disproving a variant of the Bloch–Kato conjecture which characterizes the image of an $\ell$-adic regulator map from a higher Chow group to a continuous étale cohomology of $X$ by using $p$-adic Hodge theory. With the aid of the theory of mixed Hodge modules, we reduce the problem to showing the exactness of the de Rham complex associated to a variation of Hodge structure, which is proved by the infinitesimal method in Hodge theory. Another key ingredient is the injectivity result on the cycle class map for Chow group of $1$-cycles on a proper smooth model of $X$ over the ring of integers in $K$, due to K. Sato and the second author.

##### Keywords
Chow group, torsion $0$-cycles on surface
##### Mathematical Subject Classification 2000
Primary: 14C25
Secondary: 14G20, 14C30