Primes of the form $X^{3}+NY^{3}$ and a family of non-singular plane curves which violate the local-global principle

Abstract

Let $n$ be an integer such that $n = 5$ or $n \geq 7$. 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 $X^{3}+NY^{3}$ due to Heath-Brown and Moroz and the Fermat type equation of the form $x^{3}+Ny^{3} = Lz^{n}$, 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 which satisfies 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 obtain infinitely many non-singular plane curves of every degree $n = 5$ or $n \geq 7$ 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 originated by Selmer (1951) and extended by Fujiwara (1972) and others.