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

Hirakawa, Yoshinosuke; Shimizu, Yosuke

doi:10.1090/proc/15306

Counterexamples to the local-global principle for non-singular plane curves and a cubic analogue of Ankeny-Artin-Chowla-Mordell conjecture
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by Yoshinosuke Hirakawa and Yosuke Shimizu PDF

Proc. Amer. Math. Soc. 150 (2022), 1821-1835 Request permission

Abstract:

In this article, we introduce a systematic and uniform construction of non-singular plane curves of odd degrees $n \geq 5$ 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 $p \geq 5$ which satisfies a conjecture on a $p$-adic property of the fundamental unit of $\mathbb {Q}(p^{1/3})$ and $\mathbb {Q}((2p)^{1/3})$. This conjecture is a natural cubic analogue of the classical Ankeny-Artin-Chowla-Mordell conjecture for $\mathbb {Q}(p^{1/2})$ and easily verified numerically.

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