On the Chow group of elliptic surfaces over number fields

TL;DR

This paper investigates the structure of the Chow group of codimension 2 cycles on elliptic surfaces over number fields, establishing finite generation of a certain kernel related to their models.

Contribution

It proves that the kernel of the Chow group map from a smooth projective model to the generic fiber is finitely generated, a new result in the context of elliptic surfaces over number fields.

Findings

Kernel of the Chow group map is finitely generated.

Results apply to elliptic surfaces with singular fibers and sections.

Advances understanding of algebraic cycles on elliptic surfaces.

Abstract

Let $X$ be a smooth projective surface over a number field $K$. Assume that $X$ has an elliptic fibration over $\mathbb{P}^1_K$ 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 $\ker\bigl(\mathrm{CH}^2(\mathcal{X}) \rightarrow \mathrm{CH}^2(X)\bigr)$ is a finitely generated group.