论文标题
椭圆形曲线的循环循环等异基因固定二次场
Cyclic isogenies of elliptic curves over fixed quadratic fields
论文作者
论文摘要
Kenku建立在Mazur 1978年的Prime Isegenies上的基础上，于1981年确定了椭圆曲线的所有可能的循环均超过$ \ Mathbb {Q} $。尽管已经过去了40多年，但迄今尚未实现椭圆曲线的环状曲线的循环性异基因。 在本文中，我们制定了一个程序，以帮助确定给定二次领域的确定。用$ | D |在所有二次字段上执行此过程<10^4 $我们获得的，以广义的Riemann假设为条件，确定椭圆曲线超过$ 19 $二次字段的循环循环，包括$ \ MATHBB {Q}（\ sqrt {213}）$和$ \ \ \ \ \ \ {Q}（Q}（Q}（Q}（Q}））$为了使此过程起作用，我们确定模块化曲线上的所有有限的二次点$ x_0（125）$和$ x_0（169）$，这可能具有独立的利益。
Building on Mazur's 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over $\mathbb{Q}$. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields $\mathbb{Q}(\sqrt{d})$ with $|d| < 10^4$ we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over $19$ quadratic fields, including $\mathbb{Q}(\sqrt{213})$ and $\mathbb{Q}(\sqrt{-2289})$. To make this procedure work, we determine all of the finitely many quadratic points on the modular curves $X_0(125)$ and $X_0(169)$, which may be of independent interest.
