论文标题
椭圆曲线的刚性固体局部全球原理
Rigidity in Elliptic Curve Local-Global Principles
论文作者
论文摘要
我们研究了椭圆形曲线在数字字段上的两个众所周知的局部全球原理中当地条件的刚性。特别是，我们考虑了由于Serre和Katz引起的局部扭转原理，而sutherland造成的同基因。对于这些本地全球原则中的每一个，我们证明，如果椭圆曲线$ e $上的$ k $ $ k $，以至于它无法满足至少一个$ k $良好减少的$ k $的当地条件，那么$ e $就可以满足本地条件的不超过75％的主要理想。我们还（猜想）在没有复杂的乘法的情况下（猜想）所有椭圆曲线，即满足上述局部条件的素数的密度。
We study the rigidity of the local conditions in two well-known local-global principles for elliptic curves over number fields. In particular, we consider a local-global principle for torsion due to Serre and Katz, and one for isogenies due to Sutherland. For each of these local-global principles, we prove that if an elliptic curve $E$ over a number field $K$ is such that it fails to satisfy the local condition for at least one prime ideal of $K$ of good reduction, then $E$ can satisfy the local condition at no more than 75% of prime ideals. We also give for (conjecturally) all elliptic curves over the rationals without complex multiplication, the densities of primes that satisfy the local conditions mentioned above.
