论文标题
椭圆曲线的Ostrowski商
The Ostrowski quotient of an elliptic curve
论文作者
论文摘要
对于$ k/f $，数字字段的有限galois扩展，相对pólya$ \ po（k/f）$是所有强烈模棱两可的理想类在$ k/f $中产生的理想类$ k $的子组。最近在\ cite {srm}中引入了ostrowski商$ \ ost（k/f）$的概念。 在本文中，使用González-Avilés\ cite {aviles}的一些结果，我们找到了一种定义$ \ po（k/f）$和$ \ ost（k/f）$的新方法，这是我们在椭圆曲线设置中研究类似概念的主要动机。对于$ e $ $ f $定义的椭圆曲线，我们定义了Ostrowski商$ \ ost（e，k/f）$和粗糙的Ostrowski商$ \ ost_c（e，k/f）的$ $ e $相对于$ k/f $，在后一个组中，我们对这是一个不好的prime crime cartime carter的折扣。我们的主要结果是针对组$ \ ost_c（e，k/f）$的非平凡结构定理，我们对curves $ e $的类别分析了该定理$ k/f $。
For $K/F$ a finite Galois extension of number fields, the relative Pólya group $\Po(K/F)$ is the subgroup of the ideal class group of $K$ generated by all the strongly ambiguous ideal classes in $K/F$. The notion of Ostrowski quotient $\Ost(K/F)$, as the cokernel of the capitulation map into $\Po(K/F)$, has been recently introduced in \cite{SRM}. In this paper, using some results of González-Avilés \cite{Aviles}, we find a new approach to define $\Po(K/F)$ and $\Ost(K/F)$ which is the main motivation for us to investigate analogous notions in the elliptic curve setting. For $E$ an elliptic curve defined over $F$, we define the Ostrowski quotient $\Ost(E,K/F)$ and the coarse Ostrowski quotient $\Ost_c(E,K/F)$ of $E$ relative to $K/F$, for which in the latter group we do not take into account primes of bad reduction. Our main result is a non-trivial structure theorem for the group $\Ost_c(E,K/F)$ and we analyze this theorem, in some detail, for the class of curves $E$ over quadratic extensions $K/F$.
