Journal of the Mathematical Society of Japan
Online ISSN : 1881-1167
Print ISSN : 0025-5645
ISSN-L : 0025-5645
Primes of the form 𝑋3 + 𝑁𝑌3 and a family of non-singular plane curves which violate the local-global principle
Yoshinosuke Hirakawa
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2024 年 76 巻 2 号 p. 451-471

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Let 𝑛 be an integer such that 𝑛 = 5 or 𝑛 ≥ 7. In this article, we introduce a recipe for a certain infinite family of non-singular plane curves of degree 𝑛 which violate the local-global principle. Moreover, each family contains infinitely many members which are not geometrically isomorphic to each other. Our construction is based on two arithmetic objects; that is, prime numbers of the form 𝑋3 + 𝑁𝑌3 due to Heath-Brown and Moroz and the Fermat type equation of the form 𝑥3 + 𝑁𝑦3 = 𝐿𝑧𝑛, where 𝑁 and 𝐿 are suitably chosen integers. In this sense, our construction is an extension of the family of odd degree 𝑛 which was previously found by Shimizu and the author. The previous construction works only if the given degree 𝑛 has a prime divisor 𝑝 for which the pure cubic fields ℚ(𝑝1/3) or ℚ((2𝑝)1/3) satisfy a certain indivisibility conjecture of Ankeny–Artin–Chowla–Mordell type. In this time, we focus on the complementary cases, namely the cases of even degrees and exceptional odd degrees. Consequently, our recipe works well as a whole. This means that we can unconditionally produce infinitely many explicit non-singular plane curves of every degree 𝑛 = 5 or 𝑛 ≥ 7 which violate the local-global principle. This gives a conclusion of the classical story of searching explicit ternary forms violating the local-global principle, which was initiated by Selmer (1951) and extended by Fujiwara (1972) and others.

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