论文标题
形式的$ x^{3}+ny^{3} $的素数和一个违反本地全球原理的非斑点平面曲线的家族
Primes of the form $X^{3}+NY^{3}$ and a family of non-singular plane curves which violate the local-global principle
论文作者
论文摘要
令$ n $为一个整数，以便$ n = 5 $或$ n \ geq 7 $。在本文中，我们介绍了某种违反本地全球原则的非符号平面曲线的无限家族的配方。此外，每个家庭都包含无限的许多成员，这些成员彼此之间并非同构。我们的构造基于两个算术对象。也就是说，由于Heath-Brown和Moroz以及表格$ x^{3}+ny^{3} = lz^{n} $，在$ n $和$ n $和$ l $的情况下，$ x^{3}+ny^{3} = lz^{ny^{3}+ny^{3}+ny^{3} = l $是$ n $和$ l $是适当选择的整数。从这个意义上讲，我们的构建是奇数$ n $家族的延伸，这是Shimizu和作者以前发现的。以前的构造仅在给定的$ n $具有主要分离器的情况下才能有效，这满足了Ankeny-Artin-Artin-Chowla-Mordell类型的某些不可分割的猜想。在这个时候，我们专注于互补案例，即均匀度和特殊奇数的情况。因此，我们的食谱整体运作良好。这意味着我们可以无条件地获得每个度的无限非单明平面曲线$ n = 5 $或$ n \ geq 7 $，违反了本地全球原则。这得出了搜索显式三元形式的经典故事，违反了局部全球原则，该原理由Selmer（1951）（1951年），由Fujiwara（1972）等人扩展。
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.
