论文标题
数字字段上椭圆表面的平均Mordell-Weil等级
The average Mordell-Weil rank of elliptic surfaces over number fields
论文作者
论文摘要
令$ k $为$ \ mathbb {q} $的有限生成的字段。令$ \ mathcal {x} \ to \ mathcal {b} $是$ k $上的椭圆表面的家庭，使每个椭圆纤维都具有相同的奇异纤维配置。让$ r $成为这个家庭中Mordell-Weil等级的最低水平。然后，我们证明了$ | \ Mathcal {b} | $中的基因座，其中mordell-weil等级至少为$ r+1 $是一个稀疏的子集。 通过这种方式，我们证明了Cowan在$ \ Mathbb {Q} $上的平均Mordell-Weil等级上的猜想，并证明了在任意数字字段上椭圆表面的相似结果。
Let $K$ be a finitely generated field over $\mathbb{Q}$. Let $\mathcal{X}\to \mathcal{B}$ be a family of elliptic surfaces over $K$ such that each elliptic fibration has the same configuration of singular fibers. Let $r$ be the minimum of the Mordell-Weil rank in this family. Then we show that the locus inside $|\mathcal{B}|$ where the Mordell-Weil rank is at least $r+1$ is a sparse subset. In this way we prove Cowan's conjecture on the average Mordell-Weil rank of elliptic surfaces over $\mathbb{Q}$ and prove a similar result for elliptic surfaces over arbitrary number fields.
