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 $\ell$ different from $p$, the $\ell$-primary torsion subgroup of ${CH}_0\left(X\right)$, 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 $\ell$-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.