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

References in corpus (1)

• Power-free values of polynomials on symmetric varieties

Cited by in corpus (2)

• Schinzel Hypothesis on average and rational points
• Local-global principles for semi-integral points on Markoff orbifold pairs