椭圆曲线的算术及分布研究

The Birch-Swinnerton-Dyer conjecture (BSD conjecture for short) is the central problem in the arithmetic of elliptic curves. Like many phonomena in number theory, the general methods for p-part of BSD conjecture become ineffective to attack 2-part of BSD conjecture. The goal of this project is to use all known techniques in the attack of 2-part of BSD conjecture to study more general situation; another goal of this project is to use Cassels pairing to characterize 2-part of Shafarevich-Tate groups of general elliptic curves; this project also uses the equidistribution of residue symbols to study the distribution of arithmetic groups of elliptic curves. In particular, the last two techniques play an important role in the breakthrough of Smith on 2^k-Selmer groups of quadratic twists of elliptic curves. The last goal of this project is to combine our results and Smith's governing field idea to study the refined version of distribution of 2^k-Selmer groups of quadratic twists of elliptic curves.

Birch-Swinnerton-Dyer猜想（简称BSD猜想）是椭圆曲线算术的核心问题。同数论中的诸多现象一样，BSD猜想的2-部分需要不同于BSD猜想的p部分的工具进行研究。本项目希望通过对当前解决BSD猜想2-部分的各种技术的研究来解决更一般的椭圆曲线的BSD猜想的2-部分；并且借助Cassels配对理论来刻画一般的椭圆曲线的Shafarevich-Tate群的2-部分；同时本项目拟对剩余符号一致分布性研究，从而讨论椭圆曲线的分布结果。最后这两项技术在Smith的关于椭圆曲线二次扭族的2^k-Selmer群的分布结果中起着重要的作用。最后，本项目拟结合自己的科研成果和Smith的控制域技术来研究给定素因子个数或者素因子类型的椭圆曲线二次扭族的2^k-Selmer群的分布。

本项目源于当今椭圆曲线的算术的中心问题：Birch-Swinnerton-Dyer猜想。本项目的主要研究内容是椭圆曲线y^2=x(x-a^2)(x+b^2)的二次扭族{ y^2=x(x-a^2n)(x+b^2n): n 是任何自然数} 的Shafarevich-Tate群的2- 部分和它们的Birch-Swinnerton-Dyer猜想的2-部分及其分布行为的研究。 本项目的重要结果如下：设a, b, c 是方程a^2+b^2=2c^2的一个本原解，椭圆曲线y^2=x(x-a^2)(x+b^2)的2-Selmer群的阶为4：当平方自由的正整数n满足一定的条件时： 椭圆曲线y^2=x(x-a^2n)(x+b^2n)的Shafarevich-Tate群的2- 部分和Mordell-Weil 群均同构于(Z/2Z)^2的充要条件是虚二次数域Q(\sqrt{-n})的理想类群的4- 秩为1且8-秩满足一定的条件；对于任意的正整数k， X以内的满足上述条件的恰有k个素因子的n的个数在差一个误差项的意义下是某个正的常数乘以 X(loglog X)^{k-1}/log X。