Arithmetic of Rational Points and Zero-cycles on Products of Kummer Varieties and K3 Surfaces

Francesca Balestrieri,Rachel Newton

doi:10.1093/imrn/rny303

Abstract

AbstractLet $k$ 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.

Highlights

Let X be a smooth, proper, geometrically integral variety over a number field k with a fixed choice of algebraic closure k, let X = X ×Spec k Spec k, and let Br X := H2ét(X, Gm) be the Brauer group of X
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, 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
In [25], Skorobogatov conjectured that the Brauer–Manin obstruction is the only obstruction to the Hasse principle for rational points on K3 surfaces over number fields

Summary

Introduction

In [25], Skorobogatov conjectured that the Brauer–Manin obstruction is the only obstruction to the Hasse principle for rational points on K3 surfaces over number fields This leads to the following conditional result for Kummer surfaces. This paper was inspired by Liang’s work in [14, Thm. 3.2.1], in which he takes X to be a geometrically rationally connected variety over a number field k He shows that if the Brauer–Manin obstruction is the only obstruction to the Hasse principle for rational points on X over every finite extension of k, the Brauer–Manin obstruction is the only obstruction to the Hasse principle for 0-cycles of degree 1 on X over k. Since the geometric Brauer group of a geometrically rationally connected variety is finite, the growth of the whole of Br X/ Br0 X under extensions of the base field is controlled in Liang’s paper. If the Brauer–Manin obstruction to the Hasse principle is the only one for k-rational points, it is the only one for l-rational points, for all field extensions l/k as above

Structure of the paper

Some Sets of Varieties

Preliminaries on Brauer Groups of Products of Varieties

Preliminaries on Zero-cycles

From Rational Points to Zero-cycles

Transferring Emptiness of the Brauer–Manin Set Over Field Extensions