Local-Global Principles for Zero-Cycles on Homogeneous Spaces over Arithmetic Function Fields

TL;DR

This paper investigates conditions under which zero-cycles of degree one exist on varieties over function fields of curves, establishing local-global principles that relate zero-cycle existence to rational points over extensions.

Contribution

It extends local-global principles for zero-cycles from number fields to function fields over complete discretely valued fields, including the henselian case.

Findings

Local-global principles hold for zero-cycles under certain conditions.

Results generalize known number field results to function fields.

Principles also apply in the henselian setting.

Abstract

We study the existence of zero-cycles of degree one on varieties that are defined over a function field of a curve over a complete discretely valued field. In particular, we show that local-global principles hold for such zero-cycles provided that local-global principles hold for the existence of rational points over extensions of the function field. This assertion is analogous to a known result concerning varieties over number fields. We also show that our results hold more generally in the henselian case.