Hasse Principles in algebraic equations

Fujiwara, Masahiko

doi:10.4064/aa-22-3-267-276

Cited by 8 publications

(16 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For forms of degree 3, there have been found several counter examples ( [2], [4], [5], [6]). Such a form of degree 5 was discovered by the first author [3]. In this paper, we prove the following theorem.…”

mentioning

confidence: 67%

“…Throughout this section, by a solution, we always mean a nontrivial one. Solvability in Q q for q Φ 2, 3, 5 is found in the remark at the end of § 1 of [3]. Therefore we will deal with the cases q = 2 (Prop.…”

Section: Is Any Nonnegative Integer and P Is In Pmentioning

confidence: 99%

“…This assures a 3-adic solution of (E) by HenseΓs lemma. If y Q is a solution of the congruence (3) or (4), then take (0, y Q , ±1) according to (3) or (4) and we can easily show JFXO, y 0 , ±1) = 0 (mod 27) and (dF/dz)(Q, y 0 , ±1) = 0 (mod 3) ^ 0 (mod 3 2 ). Thus we obtain a 3-adic solution of (E).…”

Section: Proof Of Proposition 2 (I)mentioning

confidence: 99%

“…Though the method of the proof is basically analogous to that of [3], local solvability needs more careful and involved treatment.…”

Section: Is Any Nonnegative Integer and P Is In Pmentioning

confidence: 99%

“…Our aim in this paper is to push forward the method in [3] and thus producing a family of forms of odd degree for which H. P. fails. As is well known, the Hasse-Minkowski theorem assures the validity of H. P. for any quadratic forms.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Some forms of odd degree for which the Hasse principle fails

Fujiwara¹,

Sudo²

1976

Pacific J. Math.

Self Cite

View full text Add to dashboard Cite

mentioning

confidence: 67%

Section: Is Any Nonnegative Integer and P Is In Pmentioning

confidence: 99%

Section: Proof Of Proposition 2 (I)mentioning

confidence: 99%

“…Though the method of the proof is basically analogous to that of [3], local solvability needs more careful and involved treatment.…”

Section: Is Any Nonnegative Integer and P Is In Pmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Some forms of odd degree for which the Hasse principle fails

Fujiwara¹,

Sudo²

1976

Pacific J. Math.

Self Cite

View full text Add to dashboard Cite

The Twenties

Narkiewicz

2012

Springer Monographs in Mathematics

View full text Add to dashboard Cite

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

Hirakawa

Shimizu

2022

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

Hasse Principles in algebraic equations

Cited by 8 publications

References 0 publications

Some forms of odd degree for which the Hasse principle fails

Some forms of odd degree for which the Hasse principle fails

The Twenties

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

Contact Info

Product

Resources

About