论文标题
立方场的正比例并不是单一的，但没有局部障碍
A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so
论文作者
论文摘要
我们表明，尽管没有局部阻塞是单基因的，但立方场的正比例不是单基因。我们的证明涉及比较$ 2 $ deScent和$ 3 $ deScent在某个家族的Mordell Curves $ e_k \ colon y^2 = x^3 + k $中的比较。作为我们方法的副产品，我们表明，对于每一个$ r \ geq 0 $，曲线的正比例为$ e_k $具有tate-shafarevich Group，其中$ 3 $ rank至少$ r $。
We show that a positive proportion of cubic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof involves the comparison of $2$-descent and $3$-descent in a certain family of Mordell curves $E_k \colon y^2 = x^3 + k$. As a by-product of our methods, we show that, for every $r \geq 0$, a positive proportion of curves $E_k$ have Tate--Shafarevich group with $3$-rank at least $r$.
