Torsion points on elliptic curves with complex multiplication
Torsion points on elliptic curves with complex multiplication
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复数乘法椭圆曲线上的扭转点
DOI：
10.1142/s1793042112501436
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
James Stankewicz
中科院分区：
文献类型：
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作者：
P. L. Clark;Brian Cook;James Stankewicz
本文给出了数域上CM椭圆曲线上素数阶扭点结构的七个定理。前三个结果通过考虑CM阶的类数和CM域中素数的分裂来改进Silverberg和Prasad-Yogananda的界。在许多情况下，我们可以证明我们的改进的界限是最优的或渐近最优的。我们还得到了X_1（N）上CM-点的最小次数的渐近上下界。通过与X_1（N）上存在无穷多个有理点的最小次数的界的比较，我们推出：对于足够大的N，X_1（N）将有一个有理CM点，其次数小于除N之外的至少所有非CM点的次数。
We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results refine bounds of Silverberg and Prasad-Yogananda by taking into account the class number of the CM order and the splitting of the prime in the CM field. In many cases we can show that our refined bounds are optimal or asymptotically optimal. We also derive asymptotic upper and lower bounds on the least degree of a CM-point on X_1(N). Upon comparison to bounds for the least degree for which there exist infinitely many rational points on X_1(N), we deduce that, for sufficiently large N, X_1(N) will have a rational CM point of degree smaller than the degrees of at least all but finitely many non-CM points.