Hasse principle for the Chow groups on quadric fibrations
Kazuki Sato
TL;DR
The paper addresses the injectivity of the global-to-local map for the relative Chow group of zero-cycles on quadric fibrations over curves over number fields, focusing on dimensions 2 and 3 with the generic fiber defined over the base field. It introduces the delta map $\delta$ linking $\mathrm{CH}_0(X/C)$ to norm-related groups $k(C)^*/(k^*N_q(k(C)))$, and shows that $\delta$ becomes an isomorphism under admissibility, enabling a norm-based reformulation of injectivity. The main result proves that the global-to-local map $\Phi$ is injective under these hypotheses by reducing to a local-global problem for norm groups, employing a decomposition $q=\langle 1,a,b,abd\rangle$ and the field $L=k(\sqrt d)$, together with $I^3$-injectivity over function fields. The work also discusses the limitation of restricted real-place injectivity for low-rank cases and clarifies the necessity of the generic-fiber defined-over-$k$ condition, contributing to a Hasse-principle understanding for zero-cycles on quadric fibrations.
Abstract
We give a sufficient condition for the injectivity of the global-to-local map of the relative Chow group of zero-cycles on a quadric fibration of dimension 2 or 3 defined over a number field.