A Positive Proportion of Hasse Principle Failures in a Family of Châtelet Surfaces

Abstract

We investigate the family of surfaces defined by the affine equation $$Y^2+Z^2=(a{T}^2+b)(c{T}^2+d)$$ where $|ad-bc|=1$ 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 Bretèche and Browning.

13 pages, comments welcome. To appear in International Journal of Number Theory