Density of Ch\^atelet surfaces failing the Hasse principle

TL;DR

This paper studies how often certain Châtelet surfaces over the rationals fail the Hasse principle, contributing to understanding rational points on algebraic surfaces.

Contribution

It analyzes the frequency of Hasse principle failures within a special class of Châtelet surfaces over the rationals.

Findings

Quantifies the density of counter-examples to the Hasse principle.

Provides new insights into the distribution of rational points on Châtelet surfaces.

Enhances understanding of rational solutions on algebraic surfaces.

Abstract

Ch\^atelet surfaces provide a rich source of geometrically rational surfaces which do not always satisfy the Hasse principle. Restricting attention to a special class of Ch\^atelet surfaces, we investigate the frequency that such counter-examples arise over the rationals.