@article{a33d45f5abd945379eb0ea7dd3df77f5,

title = "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 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.",

keywords = "Diophantine equations, cubic fields, local-global principle, primes represented by polynomials",

author = "Yoshinosuke Hirakawa and Yosuke Shimizu",

note = "Funding Information: This research was supported by JSPS KAKENHI Grant Number JP15J05818, the Research Grant of Keio Leading-edge Laboratory of Science & Technology (Grant Numbers 2018-2019 000036 and 2019-2020 000074). This research was supported in part by KAKENHI 18H05233. This research was conducted as part of the KiPAS program FY2014–2018 of the Faculty of Science and Technology at Keio University as well as the JSPS Core-to-Core program “Foundation of a Global Research Cooperative Center in Mathematics focused on Number Theory and Geometry”. The first author is the corresponding author. Publisher Copyright: {\textcopyright} 2022 American Mathematical Society. All rights reserved.",

year = "2022",

doi = "10.1090/proc/15306",

language = "English",

volume = "150",

pages = "1821--1835",

journal = "Proceedings of the American Mathematical Society",

issn = "0002-9939",

publisher = "American Mathematical Society",

number = "5",

}