论文标题
CM椭圆曲线的局部全球原则
The local-global principle for divisibility in CM elliptic curves
论文作者
论文摘要
我们考虑在数字字段定义的CM椭圆曲线的Mordell-Weil组中的局部全球原理。对于每个Prime $ p $，我们给出了一个数字字段的$ d $的急剧下限，那里存在CM椭圆形曲线，该曲线与$ p $的电源相反。作为推论，我们推断出最多有限的椭圆曲线（有或没有CM），它们是$ p> 2d+1 $的反例。我们还推断出，$ 7 $ $ 7 $的划分的本地全球原则在二次领域中持有。
We consider the local-global principle for divisibility in the Mordell-Weil group of a CM elliptic curve defined over a number field. For each prime $p$ we give sharp lower bounds on the degree $d$ of a number field over which there exists a CM elliptic curve which gives a counterexample to the local-global principle for divisibility by a power of $p$. As a corollary we deduce that there are at most finitely many elliptic curves (with or without CM) which are counterexamples with $p > 2d+1$. We also deduce that the local-global principle for divisibility by powers of $7$ holds over quadratic fields.
