## Algebra & Number Theory

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

#### Abstract

We give an example of a projective smooth surface $X$ over a $p$-adic field $K$ such that for any prime $ℓ$ different from $p$, the $ℓ$-primary torsion subgroup of $CH0(X)$, 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 $ℓ$-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.

#### Article information

Source
Algebra Number Theory, Volume 1, Number 2 (2007), 163-181.

Dates
Revised: 15 August 2007
Accepted: 15 September 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513797126

Digital Object Identifier
doi:10.2140/ant.2007.1.163

Mathematical Reviews number (MathSciNet)
MR2361939

Zentralblatt MATH identifier
1161.14300

#### Citation

Asakura, Masanori; Saito, Shuji. Surfaces over a p-adic field with infinite torsion in the Chow group of 0-cycles. Algebra Number Theory 1 (2007), no. 2, 163--181. doi:10.2140/ant.2007.1.163. https://projecteuclid.org/euclid.ant/1513797126

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