Let $E/\mathbb{Q}$ and $A/\mathbb{Q}$ be elliptic curves. We can construct modular points derived from A via the modular parametrisation of E. With certain assumptions we can show that these points are of infinite order and are not divisible by a prime p. In particular, using Kolyvagin's construction of derivative classes, we can find elements in certain Shafarevich-Tate groups of order $p^n$.

中文翻译：

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椭圆曲线上模点的 Kolyvagin 导数

让 $乙/\mathbb{问}$ 和 $\mathrm{一种}/\mathbb{问}$是椭圆曲线。我们可以通过E的模参数化构造从A导出的模点。通过某些假设，我们可以证明这些点是无限阶的，并且不能被素数p整除。特别是，使用 Kolyvagin 的派生类构造，我们可以找到某些 Shafarevich-Tate 阶群中的元素${磷}^n$.