论文标题
在等级没有跳跃的椭圆表面的纤维上
On the fibres of an elliptic surface where the rank does not jump
论文作者
论文摘要
对于$ \ mathbb {p}^1 $定义的非恒定椭圆表面，定义在$ \ mathbb {q} $上的$，这是Silverman的结果，即莫尔德尔（Mordell） - 纤维等级至少是该组的排名，这是一组部分的等级，以限制许多纤维。如果椭圆表面是非异位表面的，那么它的结合对于无限的许多纤维来说是一个相等的相等性，尽管没有任何例子是无条件的。在Bunyakovsky的猜想下，诺伊曼（Neumann）和塞策（Setzer）构建了这样的例子。在本说明中，我们表明，Legendre椭圆表面具有所需的特性，其条件是存在于无限的许多Mersenne Prime。
For a non-constant elliptic surface over $\mathbb{P}^1$ defined over $\mathbb{Q}$, it is a result of Silverman that the Mordell--Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the elliptic surface is non-isotrivial one expects that this bound is an equality for infinitely many fibres, although no example is known unconditionally. Under the Bunyakovsky conjecture, such an example has been constructed by Neumann and Setzer. In this note we show that the Legendre elliptic surface has the desired property, conditional on the existence of infinitely many Mersenne primes.
