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

Abstract

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