The cubic analogue of the Ankeny–Artin–Chowla–Mordell conjecture

Let $p\neq 3$ be a prime number, let $P=p$ or $2p$, and let

$$\epsilon=\frac{\alpha+\beta p^{1/3}+\gamma p^{2/3}}{3}\in\mathbb{R}_{>1}$$

be the fundamental unit of $\mathbb{Q}(P^{1/3})$, where $\alpha,\beta,\gamma\in\mathbb{Z}$. The cubic analogue of the Ankeny–Artin–Chowla–Mordell conjecture. Then $\beta\not\equiv 0\pmod p$. The authors verified this conjecture for all $p<10^5$ by Magma. It is proposed as a cubic-field analogue of the classical Ankeny–Artin–Chowla–Mordell conjecture for real quadratic fields, and is used to produce infinite families of counterexamples to the local-global principle for non-singular plane curves of every relevant odd degree.