Chow group of 0-cycles on surface over a p-adic field with infinite torsion subgroup

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 Block-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. By aid of theory of mixed Hodge modules, we reduce the problem to showing the exactness of de Rham complex associated to a variation of Hodge structure, which follows from Nori’s connectivity theorem. Another key ingredient is the injectivity result on étale cycle class map for Chow group of 1cycles on a proper smooth model of X over the ring of integers in K of due to K. Sato and the second author.