论文标题
$ \ MATHBB Q $ - 曲线上的奇数数字字段
$\mathbb Q$-curves over odd degree number fields
论文作者
论文摘要
通过重新定义和扩展麋鹿的结果，我们在$ \ Mathbb Q $ curves上证明了一些奇数字段的结果。我们表明，在此类领域，唯一的主要等级学位〜$ \ ell $，没有CM的椭圆曲线可能拥有的是那些已经可能超过〜$ \ $ \ mathbb q $本身的学位（尤其是$ \ ell \ ell \ el \ le37 $），我们仅在$ \ \ q $ \ q. $ \ q $ \ q $ \ math的范围内表明了限制的限制。我们还证明，$ \ mathbb Q $ culves的唯一可能的扭转组超过了数字字段，而不是由prime $ \ ell \ el \ leq 7 $排除的数字字段，是$ 15 $的$ 15 $组，这些$ 15 $组以$ \ mathbb Q $ $ \ \ mathbb q $的扭转组而出现。与这些理论结果相辅相成，我们给出了一种算法，以确定任何给定的椭圆曲线$ e $是否是$ \ mathbb q $ -curve，它涉及仅在$ \ mathbb q（j（e））$上工作。
By reformulating and extending results of Elkies, we prove some results on $\mathbb Q$-curves over number fields of odd degree. We show that, over such fields, the only prime isogeny degrees~$\ell$ which an elliptic curve without CM may have are those degrees which are already possible over~$\mathbb Q$ itself (in particular, $\ell\le37$), and we show the existence of a bound on the degrees of cyclic isogenies between $\mathbb Q$-curves depending only on the degree of the field. We also prove that the only possible torsion groups of $\mathbb Q$-curves over number fields of degree not divisible by a prime $\ell\leq 7$ are the $15$ groups that appear as torsion groups of elliptic curves over $\mathbb Q$. Complementing these theoretical results we give an algorithm for establishing whether any given elliptic curve $E$ is a $\mathbb Q$-curve, which involves working only over $\mathbb Q(j(E))$.
