On the denominators of rational points on elliptic curves
On the denominators of rational points on elliptic curves
复制标题
关于椭圆曲线上有理点的分母
DOI：
10.1112/blms/bdm061
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发表时间：
2007
影响因子：
0.9
通讯作者：
S. Stevens
中科院分区：
文献类型：
--
作者：
G. Everest;J. Reynolds;S. Stevens
设x（P）= AP/B2 P表示Weierstrass形式的椭圆曲线上有理点P的x坐标。我们考虑BP何时可以是完全幂或素数。利用Faltings定理，我们证明了对于一个固定的f > 1，只有100个有理点P，BP等于1的1次方。在下降通过一个issues是可能的，我们表明，只有200多个合理的点P与BP等于一个素数，这些点是有界的数量在一个明确的方式，他们是有效的可计算的。最后，我们证明了一个更强的版本，这一结果的曲线齐次形式。
Let x(P) = AP/B2P denote the x‐coordinate of the rational point P on an elliptic curve in Weierstrass form. We consider when BP can be a perfect power or a prime. Using Faltings' theorem, we show that for a fixed f > 1, there are only finitely many rational points P with BP equal to an fth power. Where descent via an isogeny is possible, we show that there are only finitely many rational points P with BP equal to a prime, that these points are bounded in number in an explicit fashion, and that they are effectively computable. Finally, we prove a stronger version of this result for curves in homogeneous form.