COUNTEREXAMPLES TO THE LOCAL-GLOBAL PRINCIPLE FOR NON-SINGULAR PLANE CURVES AND A CUBIC ANALOGUE OF ANKENY-ARTIN-CHOWLA-MORDELL CONJECTURE

Yoshinosuke Hirakawa, Yosuke Shimizu

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)1821-1835
Number of pages15
JournalProceedings of the American Mathematical Society
Volume150
Issue number5
DOIs
Publication statusPublished - 2022

Keywords

  • Diophantine equations
  • cubic fields
  • local-global principle
  • primes represented by polynomials

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