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

TL;DR

This paper constructs infinite families of generalized Mordell and Fermat curves over $Q$ that violate the Hasse principle due to the Brauer-Manin obstruction, using rational points on a specific threefold.

Contribution

It demonstrates the existence of explicit infinite families of counterexamples to the Hasse principle for generalized Mordell and Fermat curves, linked to rational points on a constructed threefold.

Findings

Infinite families of counterexamples to the Hasse principle are constructed.

Counterexamples are explained by the Brauer-Manin obstruction.

Explicit rational points generating these families are identified.

Abstract

A generalized Mordell curve of degree $n \ge 3$ over $\bQ$ is the smooth projective model of the affine curve of the form $Az^2 = Bx^n + C$, 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 $Ax^n + By^n + Cz^n = 0$ for some nonzero integers $A, B, C$. In this paper, we show that for each prime $p$ with $p \equiv 1 \pmod{8}$ and $p \equiv 2 \pmod{3}$, 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.