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

Asakura, Masanori; Saito, Shuji

doi:10.2140/ant.2007.1.163

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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

DOI: 10.2140/ant.2007.1.163

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 ${CH}_{0} (X)$ , 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.

Keywords

Chow group, torsion $0$-cycles on surface

Mathematical Subject Classification 2000

Primary: 14C25

Secondary: 14G20, 14C30

Milestones

Received: 30 January 2007

Revised: 15 August 2007

Accepted: 15 September 2007

Published: 1 May 2007

Authors

Masanori Asakura
Faculty of Mathematics Kyushu University Fukuoka 812-8581 Japan

Shuji Saito
Graduate School of Mathematical Sciences The University of Tokyo Tokyo 153-8914 Japan

Vol. 1, No. 2, 2007