Generalized Mordell curves, generalized Fermat curves, and the Hasse principle

A generalized Mordell curve of degree $n\ge 3$ over $\text{bQ}$ is the smooth projective model of the affine curve of the form $A{z}^2=B{x}^n+C$, where $A, B, C$arenonzerointegers\mathrm{.}AgeneralizedFermatcurveofsignature$(n, n, n)$with$n \ge 3$over$\bQ$isthesmoothprojectivecurveoftheform$Ax^n + By^n + Cz^n = 0$forsomenonzerointegers$A, B, C$. In this paper, we show that for each prime $p$ with $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}8)$ and $p\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}3)$, there exists a threefold ${\text{cX}}_p\subseteq {\text{bP}}^6$ such that certain rational points on ${\text{cX}}_p$ produce infinite families of non-isomorphic generalized Mordell curves of degree $12n$ and infinite families of generalized Fermat curves of signature $(12n,12n,12n)$ for each $n\ge 2$ that are counterexamples to the Hasse principle explained by the Brauer-Manin obstruction. We also show that the set of special rational points on ${\text{cX}}_p$ producing generalized Mordell curves and generalized Fermat curves that are counterexamples to the Hasse principle is infinite, and can be constructed explicitly.