论文标题
$ p^{n} $的不规则素数的椭圆形类似物 - 曲线的difric tifer字段$ y^{2} = x^{3} {3} - （s^{4}+t^{2}）x $
Elliptic analogue of irregular prime numbers for the $p^{n}$-division fields of the curves $y^{2} = x^{3}-(s^{4}+t^{2})x$
论文作者
论文摘要
如果将$ p $ - p $ - th cyclotomic field $ \ mathbb {q}（ζ__{p}）= \ mathbb {q}（\ mathbb {g} _m [p] _m [p]）$ prime Number $ p $划分。在本文中，我们研究了其用于椭圆曲线的分裂场的椭圆类似物。更确切地说，对于质量数量$ p \ geq 5 $和一个正整数$ n $，我们研究了$ p $ - $ p^{n} $ - disement field $ \ mathbb {q}（e [e [p^{n}]）$ elliptic curve curve $ e $的$ y^$ y^y^2} x^{3} - （s^{4}+t^{2}）x $。特别是，我们构建了一定的无限亚科，由曲线组成，具有新颖的属性，它们是mordell-weil等级1，其$ p^{n} $ - division-division-division-division-division-division字段的类数则由$ p^{2n} $排除。此外，我们可以证明这些分区字段彼此不是同构。在我们的施工中，我们使用第一作者获得的最新结果。
A prime number $p$ is said to be irregular if it divides the class number of the $p$-th cyclotomic field $\mathbb{Q}(ζ_{p}) = \mathbb{Q}(\mathbb{G}_m[p])$. In this paper, we study its elliptic analogue for the division fields of an elliptic curve. More precisely, for a prime number $p \geq 5$ and a positive integer $n$, we study the $p$-divisibility of the class number of the $p^{n}$-division field $\mathbb{Q}(E[p^{n}])$ of an elliptic curve $E$ of the form $y^{2} = x^{3}-(s^{4}+t^{2})x$. In particular, we construct a certain infinite subfamily consisting of curves with novel properties that they are of Mordell-Weil rank 1 and the class numbers of their $p^{n}$-division fields are divisible by $p^{2n}$. Moreover, we can prove that these division fields are not isomorphic to each other. In our construction, we use recent results obtained by the first author.
