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 ` 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 etale 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.