Arithmetic of Ch\^atelet surface bundles revisited

TL;DR

This paper investigates the arithmetic properties of Châtelet surface bundles over curves, revealing that the Brauer-Manin obstruction's role in the Hasse principle violation varies under field extensions, assuming Tate-Shafarevich group finiteness.

Contribution

It demonstrates that the Brauer-Manin obstruction's effectiveness in explaining Hasse principle violations is not invariant under field extensions for certain algebraic varieties.

Findings

The Brauer-Manin obstruction's role varies with field extensions.

Assuming Tate-Shafarevich group finiteness, the violation of the Hasse principle is not stable.

Study applies Lagrange interpolation to fibrations of algebraic varieties.

Abstract

We study arithmetic of the algebraic varieties defined over number fields by applying Lagrange interpolation to fibrations. Assuming the finiteness of the Tate-Shafarevich group of a certain elliptic curve, we show, for Ch\^atelet surface bundles over curves, that the violation of Hasse principle being accounted for by the Brauer-Manin obstruction is not invariant under an arbitrary finite extension of the ground field.