  In this article, we introduce a systematic and uniform construction of
non-singular plane curves of odd degrees $n \geq 5$ which violate the
local-global principle. Our construction works unconditionally for $n$
divisible by $p^2$ for some odd prime number $p$. Moreover, our construction
also works for $n$ divisible by some $p \geq 5$ which satisfies a conjecture on
$p$-adic properties of the fundamental units of $\mathbb{Q}(p^{1/3})$ and
$\mathbb{Q}((2p)^{1/3})$. This conjecture is a natural cubic analogue of the
classical Ankeny-Artin-Chowla-Mordell conjecture for $\mathbb{Q}(p^{1/2})$ and
easily verified numerically.
