In this article, we introduce a systematic and uniform construction of non-singular plane curves of odd degrees n ≥ 5 which violate the localglobal principle. Our construction works unconditionally for n divisible by p2 for some odd prime number p. Moreover, our construction also works for n divisible by some p ≥ 5 which satisfies a conjecture on a p-adic property of the fundamental unit of Q(p1/3) and Q((2p)1/3). This conjecture is a natural cubic analogue of the classical Ankeny-Artin-Chowla-Mordell conjecture for Q(p1/2) and easily verified numerically.
- Diophantine equations
- cubic fields
- local-global principle
- primes represented by polynomials