论文标题
$ \ ell^\ infty $ -selmer groups in Leg
The distribution of $\ell^\infty$-Selmer groups in degree $\ell$ twist families I
论文作者
论文摘要
在本文及其续集中，我们开发了一种技术，用于查找$ \ ell^{\ infty} $ - selmer ofter的分布，$ \ ell $ $ \ ell $ twist twist twist twist five galois模块在数字字段上。给定在满足某些技术条件的许多字段的椭圆曲线e，该技术可用于表明E的二次曲折的100％曲折最多排名1。鉴于prime $ \ ell $和一个数字f含有$μ__{2 \ ell} $与Cohen-Lenstra-Gerth启发式方法一致的分布。 对于这项工作，我们开发了固定点Selmer组的理论，该理论是$ \ ell^{\ infty} $ - Selmer组的基础层。第一篇论文提供了一种技术，可以找到$ \ ell^{\ infty} $ - selmer群体的分布，其中某些曲折的家族固定点Selmer组稳定。在续集纸中，我们将提供一种控制固定点Selmer组的技术。
In this paper and its sequel, we develop a technique for finding the distribution of $\ell^{\infty}$-Selmer groups in degree $\ell$ twist families of Galois modules over number fields. Given an elliptic curve E over a number field satisfying certain technical conditions, this technique can be used to show that 100% of the quadratic twists of E have rank at most 1. Given a prime $\ell$ and a number field F not containing $μ_{2\ell}$, this method also shows that the $\ell^{\infty}$-class groups in the family of degree $\ell$ cyclic extensions of F have a distribution consistent with the Cohen-Lenstra-Gerth heuristics. For this work, we develop the theory of the fixed point Selmer group, which serves as the base layer of the $\ell^{\infty}$-Selmer group. This first paper gives a technique for finding the distribution of $\ell^{\infty}$-Selmer groups in certain families of twists where the fixed point Selmer group is stable. In the sequel paper, we will give a technique for controlling fixed point Selmer groups.
