Arithmetic of Chatelet surfaces under extensions of base fields

Abstract

For Châtelet surfaces defined over number fields, we study two arithmetic properties, the Hasse principle and weak approximation, when passing to an extension of the base field. Generalizing a construction of Y. Liang, we show that for an arbitrary extension of number fields $L/K,$ there is a Châtelet surface over $K$ which does not satisfy weak approximation over any intermediate field of $L/K,$ and a Châtelet surface over $K$ which satisfies the Hasse principle over an intermediate field $L^{\prime }$ if and only if $[L^{\prime }:K]$ is even.

This is a part of our paper "Châtelet surfaces and non-invariance of the Brauer-Manin obstruction for $3$-folds" arXiv:2010.04919. It is interesting for us, but the editor of journal think there are too technical. So we reorganize it