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

In this talk I would like to demonstrate how Hodge theory can play a crucial role in an arithmetic question. The issue is to construct 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 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. By the aid of the theory of mixed Hodge modules, we reduce the problem to showing the exactness of de Rham complex associated to a certain variation of Hodge structure, which follows from Nori's connectivity theorem.