Approximation faible pour les 0-cycles sur un produit de vari\'et\'es rationnellement connexes

TL;DR

This paper investigates weak approximation for 0-cycles on products of certain rationally connected varieties over number fields, showing it is governed by the Brauer group under specific conditions, even without 0-cycles of degree one.

Contribution

It proves that weak approximation for 0-cycles on products of a Châtelet surface or similar and a rationally connected variety is controlled by the Brauer group, assuming this holds for the second variety over all finite extensions.

Findings

Weak approximation for 0-cycles on the product is controlled by the Brauer group.

The result holds without the existence of degree 1 0-cycles on the factors.

The property is preserved under finite field extensions.

Abstract

Consider weak approximation for 0-cycles on a smooth proper variety defined over a number field, it is conjectured to be controlled by its Brauer group. Let $X$ be a Ch\^atelet surface or a smooth compactification of a homogeneous space of a connected linear algebraic group with connected stabilizer. Let $Y$ be a rationally connected variety. We prove that weak approximation for 0-cycles on the product $X\times Y$ is controlled by its Brauer group if it is the case for $Y$ after every finite extension of the base field. We do not suppose the existence of 0-cycles of degree $1$ neither on $X$ nor on $Y$.