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 nonsingular plane curves of odd degrees n ≥ 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 ≥ 5 which satisfies a conjecture on a p-adic property of the fundamental unit of Q(p 1/3 ). This conjecture is a natural cubic analogue of the classical Ankeny-Artin-Chowla-Mordell conjecture for Q(p 1/2 ) and easily verified numerically.

Primes of the form $X^{3}+NY^{3}$ and a family of non-singular plane curves which violate the local-global principle

“…(2) Since gcd(l, b 0 c 0 ) = 1 and all of l, b 0 , c 0 are odd, two polynomials X 3 + P 2 Y 3 and b 0 X 3 + lc 0 Y 3 are both irreducible in Q[X, Y ] and cannot have a common root in C. The infinitude of the geometric isomorphy classes of plane curves follows from Schwarz's theorem on the finiteness of the automorphism group of a non-singular algebraic curve of genus ≥ 2. For the detail, see [9,Lemma 4.1]. This completes the proof in the case u ≡ 0 mod 3.…”

“…The infinitude of the geometric isomorphy classes of plane curves follows from Schwarz's theorem on the finiteness of the automorphism group of a non-singular algebraic curve of genus ≥ 2. For the detail, see [9,Lemma 4.1]. This completes the proof.…”

“…Theorem 1.2) and a property of the primitive solutions of the Fermat type equation x 3 + P 2 y 3 = l m z n . Moreover, the proof of the infinitude of geometric isomorphy is essentially given in the previous work [9,Lemma 4.1].…”

“…[2,17]). In fact, the authors verified in [9] that Conjecture 1.4 for all p < 10 5 by numerical examination using Magma [3]. Anyway, Theorem 1.3 gives, in a certain uniform manner, an explicit construction of infinitely many non-singular plane curves which violate the local-global principle, but it works only conditionally for higher odd degrees.…”

“…Remark 4.3). On the other hand, Shimizu and the author [9] formulated the following conjecture, which implies again ι = 1 for all p = 3.…”