Weak approximation on Ch\^{a}telet surfaces

TL;DR

This paper investigates weak approximation properties of Châtelet surfaces over number fields, demonstrating conditions under which the Brauer-Manin obstruction vanishes, thereby ensuring weak approximation holds.

Contribution

It establishes new criteria for weak approximation on Châtelet surfaces by analyzing the Brauer-Manin obstruction and its behavior over extensions.

Findings

Weak approximation holds when all singular fibers are rational points.

The Brauer-Manin obstruction vanishes under certain conditions.

Results extend to finite extensions of the base field.

Abstract

We study weak approximation for Ch\^{a}telet surfaces over number fields when all singular fibers are defined over rational points. We consider Ch\^{a}telet surfaces which satisfy weak approximation over every finite extension of the ground field. We prove many of these results by showing that the Brauer-Manin obstruction vanishes, then apply results of Colliot-Th\'el\`ene, Sansuc, and Swinnerton-Dyer.