Counterexamples to the local-global principle for non-singular plane curves and a cubic analogue of Ankeny-Artin-Chowla-Mordell conjecture

In this article, we introduce a systematic and uniform construction of non-singular plane curves odd degrees n ≥ 5 n \geq 5 which violate the local-global principle. Our works unconditionally for alttext="n"> encoding="application/x-tex">n divisible by alttext="p squared"> p 2 encoding="application/x-tex">p^{2} some prime number alttext="p"> encoding="application/x-tex">p . Moreover, our also encoding="application/x-tex">p satisfies conjecture on -adic property fundamental unit alttext="double-struck upper Q left-parenthesis p Superscript 1 slash 3 Baseline right-parenthesis"> Q ( 1 / 3 stretchy="false">) encoding="application/x-tex">\mathbb {Q}(p^{1/3}) 2 right-parenthesis {Q}((2p)^{1/3}) This is natural cubic analogue classical Ankeny-Artin-Chowla-Mordell {Q}(p^{1/2}) easily verified numerically.