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

Abstract

A generalized Mordell curve of degree $n\ge 3$ over $\bQ$ is the smooth projective model of the affine curve of the form $A{z}^2=B{x}^n+C$, where $A,B,C$ are nonzero integers. A generalized Fermat curve of signature $(n,n,n)$ with $n\ge 3$ over $\bQ$ is the smooth projective curve of the form $A{x}^n+B{y}^n+C{z}^n=0$ for some nonzero integers $A,B,C$. In this paper, we show that for each prime $p$ with $p\equiv 1(\mathrm{mod}8)$ and $p\equiv 2(\mathrm{mod}3)$, there exists a threefold $\cX_p \subseteq \bP^6$ such that certain rational points on $\cX_p$ 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 $\cX_p$ producing generalized Mordell curves and generalized Fermat curves that are counterexamples to the Hasse principle is infinite, and can be constructed explicitly.

arXiv comment45 pages