Arithmetic of rational points and zero-cycles on Kummer varieties

TL;DR

This paper explores the relationship between rational points and zero-cycles on Kummer varieties over number fields, showing how obstructions to rational points influence zero-cycle existence, extending Liang's results.

Contribution

It establishes a link between the Brauer-Manin obstruction for rational points over extensions and zero-cycles on Kummer varieties, providing new insights into their arithmetic properties.

Findings

If the Brauer-Manin obstruction is the only obstacle to rational points over all extensions, then the 2-primary Brauer-Manin obstruction is the only obstacle to zero-cycles of degree δ.

The results generalize Liang's work to Kummer varieties, connecting rational points and zero-cycles.

The paper demonstrates a conditional equivalence between obstructions for rational points and zero-cycles on Kummer varieties.

Abstract

Let $k$ be a number field, let $X$ be a Kummer variety over $k$, and let $\delta$ be an odd integer. 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 $X$. For example, we show that if the Brauer-Manin obstruction is the only obstruction to the existence of rational points on $X$ over all finite extensions of $k$, then the $2$-primary Brauer-Manin obstruction is the only obstruction to the existence of a zero-cycle of degree $\delta$ on $X$ over $k$.