论文标题
椭圆形曲线的Selmer组在功能场上的几何分布
The geometric distribution of Selmer groups of elliptic curves over function fields
论文作者
论文摘要
修复一个正整数$ n $和有限字段$ \ mathbb f_q $。我们研究了$ e $的排名，$ e $ $ e $的$ n $ -selmer集团以及$ e $ $ e $的$ n $ torsion as $ e $的$ n $ torsion a $ e $ a $ a $ e $远比固定高度$ d \ geq 2 $ by $ \ geq 2 $ + $ \ mathbb f_q（t）$变化。我们在$ Q $限制中计算此联合分配。我们还表明，此分布的“大$ Q $，然后大的高度”限制与Bhargava-Kane-Kane-Lenstra-Poonen-Rains预测的分布相符。
Fix a positive integer $n$ and a finite field $\mathbb F_q$. We study the joint distribution of the rank of $E$, the $n$-Selmer group of $E$, and the $n$-torsion in the Tate-Shafarevich group of $E$ as $E$ varies over elliptic curves of fixed height $d \geq 2$ over $\mathbb F_q(t)$. We compute this joint distribution in the large $q$ limit. We also show that the "large $q$, then large height" limit of this distribution agrees with the one predicted by Bhargava-Kane-Lenstra-Poonen-Rains.
