Some forms of odd degree for which the Hasse principle fails

Fujiwara, Masahiko; Sudo, Masaki

doi:10.2140/pjm.1976.67.161

Cited by 7 publications

(4 citation statements)

References 5 publications

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“…violate the local-global principle for every k ∈ Z ≥6 . However, their argument cannot give any specific (conjectural) counterexamples because the abc conjecture is ineffective to estimate the candidates of A, B, C. It may be surprising that there seems to be no other concrete counterexamples to the local-global principle for non-singular plane curves of odd degrees ≥ 5 than [11,12].…”

Section: Introductionmentioning

confidence: 99%

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

Hirakawa¹,

Shimizu²

2019

Preprint

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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.

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Section: Introductionmentioning

confidence: 99%

“…For example, if we make effort for 3-adic and p-adic solubility in the proof of Lemma 2.2, then we can relax the conditions on the coefficients and generalize our construction so that it includes the constructions of[11,12]. Moreover, the use of Theorem 1.4 in the proof of Theorem 3.1 is not essential.…”

mentioning

confidence: 99%

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

Hirakawa¹,

Shimizu²

2019

Preprint

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“…For other counterexamples, see e.g. [1,[4][5][6][8][9][10]. Among them, Mordell [9] and Jahnel [10] generalized the above construction by Swinnerton-Dyer to infinite families of cubic forms of 4 variables.…”

Section: Introductionmentioning

confidence: 99%

Counterexamples to the local-global principle associated with Swinnerton-Dyer's cubic form

Hirakawa¹

2019

Preprint

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In this paper, we imitate a classical construction of a counterexample to the local-global principle of cubic forms of 4 variables which was discovered first by Swinnerton-Dyer (Mathematica (1962)). Our construction gives new explicit families of counterexamples in homogeneous forms of 4, 5, 6, ..., 2n + 2 variables of degree 2n + 1 for infinitely many integers n. It is contrastive to Swinnerton-Dyer's original construction that we do not need any concrete calculation in the proof of local solubility.

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“…In [5], the author constructed an arithmetic family of quartic forms over Q that are counterexamples to the Hasse principle, which affirmatively answers the question when n = 4. Fujiwara and Sudo [7] produced many forms of degree n with n ≡ 5 (mod 10) that violate the Hasse principle. In [6], the author proved that there are algebraic families of degree n with n ≡ 2 (mod 4) that violate the Hasse principle.…”

Section: Introductionmentioning

confidence: 99%

Certain Forms Violate the Hasse Principle

Nguyen¹

2017

Tokyo J. Math.

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A family of smooth geometrically irreducible curves violates the Hasse principle if they have local points everywhere, but they possesses no global points. In this paper, we show how to construct non-constant algebraic families of forms of degree 4k that violate the Hasse principle. Some examples of non-constant algebraic families of forms of degrees 12 and 24 that violate the Hasse principle are given to illustrate the method.

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Some forms of odd degree for which the Hasse principle fails

Cited by 7 publications

References 5 publications

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

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

Counterexamples to the local-global principle associated with Swinnerton-Dyer's cubic form

Certain Forms Violate the Hasse Principle

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