Certain generalized Mordell curves over the rational numbers are pointless (Q2788669)

A generalized Mordell curve is a curve of the form \(Ay^2=Bx^n+C\), where \(n\), \(A\), \(B\), and \(C\) are nonzero integers with \(n\geq 3\). One expects in general that such curves should have few rational solutions -- although for small \(n\) the solution set might be infinite -- and even more so, that most curves of high degree should have no rational solutions at all. A conjecture of Scharaschkin and Skorobogatov suggests that any smooth, geometrically irreducible curve over \(\mathbb{Q}\) has a rational point if and only if there is no Brauer-Manin obstruction. The purpose of the paper under review is to verify this conjecture for a large collection of generalized Mordell curves. The main technique is to construct an infinite collection of such curves for which there \textit{is} a Brauer-Manin obstruction to the existence of rational points, and so for all curves so constructed, the conjecture of Scharaschkin and Skorobogatov is true.