Arithmetic of Chatelet surfaces under extensions of base fields

Abstract

For Ch\^atelet 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 Ch\^atelet surface over $K$ which does not satisfy weak approximation over any intermediate field of $L/K,$ and a Ch\^atelet surface over $K$ which satisfies the Hasse principle over an intermediate field $L'$ if and only if $[L' : K]$ is even.