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

Authors: Nick Rome
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 asymptotic formulae 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.
 Comments: 13 pages, comments welcome Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG) MSC classes: 14G05, 11D25, 11G35 Cite as: arXiv:1803.07017 [math.NT] (or arXiv:1803.07017v1 [math.NT] for this version)

## Submission history

From: Nick Rome [view email]
[v1] Mon, 19 Mar 2018 16:21:34 GMT (12kb)