论文标题
在$ 2 $ -Selmer组和椭圆曲线的二次曲折上
On $2$-Selmer groups and quadratic twists of elliptic curves
论文作者
论文摘要
令$ k $为一个数字字段，$ e/k $是椭圆曲线，没有$ 2 $ - torsion积分。在本文中，我们以$ 2 $ -SELMER等级为$ e $的下限和上限，就$ 2 $ e $的$ 2 $ torsion而言，其特定立方扩展名为$ e $的$ k $。作为应用程序，我们证明（根据轻度假设），$ e $的主要导体二次曲折的正比例为$ 2 $ -SELMER GROUP。
Let $K$ be a number field and $E/K$ be an elliptic curve with no $2$-torsion points. In the present article we give lower and upper bounds for the $2$-Selmer rank of $E$ in terms of the $2$-torsion of a narrow class group of a certain cubic extension of $K$ attached to $E$. As an application, we prove (under mild hypotheses) that a positive proportion of prime conductor quadratic twists of $E$ have the same $2$-Selmer group.
