论文标题
一曲线的投降判别因素
Capitulation discriminants of genus one curves
论文作者
论文摘要
在本文中，我们研究了属于$ \ mathbb {p}^{n-1} $的属性和不变理论。我们概括了属属的概念$ n $的概念，由克雷莫纳，费舍尔和斯托尔以$ n \ leq 5 $的形式介绍给任意的奇数$ n $，并描述了以$ n $嵌入的属属属属的不变理论，以$ \ \ \ \ \ \ \ \ \ mthbb {p}^p}^{n-1} $进行了自由划分。我们证明，在本地可溶的属一曲线上，$ \ mathbb {q} $允许最小的积分模型，其不变性与雅各布椭圆曲线的最小模型相同。然后，我们将这些结果应用于研究椭圆曲线$ e/\ mathbb {q} $的Tate-Shafarevich组的投降问题。我们证明，$ \ text {sha}（e/\ mathbb {q}）的每个元素奇数索引$ n $的$ splies $ n $ n $ n $ n $ nubm nubm field $ k $，最多是$ c（n）h_e h_e^h_e^e^{2n）$ h_e $ h_e $ h_e $ n是$ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ c（n）$ n $ n $ n $ n $ c（
In this paper we study the arithmetic and invariant theory of genus one normal curves embedded in $\mathbb{P}^{n-1}$. We generalize the notion of genus one model of degree $n$, introduced by Cremona, Fisher and Stoll for $n \leq 5$, to arbitrary odd $n$, and describe the invariant theory of a genus one curve of degree $n$ embedded in $ \mathbb{P}^{n-1}$ in terms of the minimal graded free resolution of its homogeneous ideal. We prove that everywhere locally soluble genus one curves over $ \mathbb{Q}$ admit minimal integral models, with the same invariants as those of the minimal model of their Jacobian elliptic curve. We then apply these results to study the capitulation problem for the Tate-Shafarevich group of an elliptic curve $E/\mathbb{Q}$. We prove that every element of $\text{Sha}(E/\mathbb{Q})[n]$ of odd index $n$ splits over a degree $n$ number field $K$, of absolute discriminant at most $c(n) H_E^{2n-2}$, where $H_E$ is the naive height of $E$ and $c(n)$ is a constant only depending on $n$.
