Algebra & Number Theory

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

Masanori Asakura and Shuji Saito

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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.

Article information

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

Received: 30 January 2007
Revised: 15 August 2007
Accepted: 15 September 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14C25: Algebraic cycles
Secondary: 14G20: Local ground fields 14C30: Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture

Chow group torsion $0$-cycles on surface


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.

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