Arithmetic of Chatelet surfaces under extensions of base fields

TL;DR

This paper investigates how the arithmetic properties of Châtelet surfaces, specifically the Hasse principle and weak approximation, change when extending the base field, revealing new phenomena related to intermediate fields.

Contribution

It generalizes previous constructions to show that Châtelet surfaces can fail weak approximation over all intermediate fields and satisfy the Hasse principle only over certain even-degree extensions.

Findings

Existence of Châtelet surfaces failing weak approximation over all intermediate fields.

Construction of surfaces satisfying the Hasse principle only over even-degree extensions.

Extension-dependent behavior of arithmetic properties of Châtelet surfaces.

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.