English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

A generalized Mordell curve of degree $n \ge 3$ over $\bQ$ is the smoothprojective 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 showthat 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 rationalpoints on $\cX_p$ produce infinite families of non-isomorphic generalizedMordell curves of degree $12n$ and infinite families of generalized Fermatcurves of signature $(12n, 12n, 12n)$ for each $n \ge 2$ that arecounterexamples to the Hasse principle explained by the Brauer-Maninobstruction. We also show that the set of special rational points on $\cX_p$producing generalized Mordell curves and generalized Fermat curves that arecounterexamples to the Hasse principle is infinite, and can be constructedexplicitly.

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