A Positive Proportion of Hasse Principle Failures in a Family of Ch\^atelet Surfaces

TL;DR

This paper studies a family of Châtelet surfaces and finds that approximately 23.7% of them fail the Hasse principle, providing an asymptotic formula for this frequency.

Contribution

It establishes a positive proportion of Hasse principle failures in a specific family of Châtelet surfaces, extending previous research.

Findings

Approximately 23.7% of surfaces fail the Hasse principle

Develops an asymptotic formula for the frequency of failures

Builds on prior work of la Bretèche and Browning

Abstract

We investigate the family of surfaces defined by the affine equation $$Y^2 + Z^2 = (aT^2 + b)(cT^2 +d)$$ where $\vert ad-bc \vert=1$ and develop an asymptotic formula for the frequency of Hasse principle failures. We show that a positive proportion (roughly 23.7%) of such surfaces fail the Hasse principle, by building on previous work of la Bret\`{e}che and Browning.