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

We give an example of a projective smooth surface X over a p-adic field K such that for any prime different from p, the -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 -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.

Chow group, torsion $0$-cycles on surface
Mathematical Subject Classification 2000
Primary: 14C25
Secondary: 14G20, 14C30
Received: 30 January 2007
Revised: 15 August 2007
Accepted: 15 September 2007
Published: 1 May 2007
Masanori Asakura
Faculty of Mathematics
Kyushu University
Fukuoka 812-8581
Shuji Saito
Graduate School of Mathematical Sciences
The University of Tokyo
Tokyo 153-8914