On the Chow group of elliptic surfaces over number fields

Abstract

Let $X$ be a smooth projective surface over a number field $K$. Assume that $X$ has an elliptic fibration over ${\mathbb{P}}_K^1$ with at least one singular fibre and a section. Let $\mathcal{X}/U$ be a smooth projective model of $X$ over some open subset $U\subset \mathrm{Spec}({\mathcal{O}}_K)$. We show that $\ker ({\mathrm{CH}}^2(\mathcal{X})\to {\mathrm{CH}}^2(X))$ is a finitely generated group.

11 pages, comments are welcome