Arithmetic of rational points and zero-cycles on products of Kummer varieties and K3 surfaces

TL;DR

This paper explores the relationship between rational points and zero-cycles on Kummer varieties, showing how obstructions to the Hasse principle for points influence those for zero-cycles, with extensions to products and other varieties.

Contribution

It establishes a link between the Brauer-Manin obstruction for rational points over extensions and for zero-cycles on Kummer varieties and related classes, extending previous results.

Findings

If the Brauer-Manin obstruction is the only obstacle for rational points over all extensions, it also governs zero-cycles of odd degree.

Results apply to products of Kummer varieties, K3 surfaces, and rationally connected varieties.

The work generalizes Liang's results to broader classes of varieties.

Abstract

Let $k$ be a number field. In the spirit of a result by Yongqi Liang, we relate the arithmetic of rational points over finite extensions of $k$ to that of zero-cycles over $k$ for Kummer varieties over $k$. For example, for any Kummer variety $X$ over $k$, we show that if the Brauer-Manin obstruction is the only obstruction to the Hasse principle for rational points on $X$ over all finite extensions of $k$, then the ($2$-primary) Brauer-Manin obstruction is the only obstruction to the Hasse principle for zero-cycles of any given odd degree on $X$ over $k$. We also obtain similar results for products of Kummer varieties, K3 surfaces and rationally connected varieties.