论文标题
使用Elkies的方法，在有限场上的Abelian表面上计数点
Counting points on abelian surfaces over finite fields with Elkies's method
论文作者
论文摘要
我们将Elkies的方法推广，这是海洋算法中的一种基本要素，可将椭圆曲线上的椭圆曲线上的点数计算到有限特征的有限磁场上，并将其设置为P.P.阿贝尔表面。在与Elkies Primes分布相关的合理假设下，我们在两种情况下获得了Schoof方法的改进。如果Abelian表面A超过FQ的RM由固定的二次场F f，我们达到了与SEA算法相同的渐近复杂性Otilde（log4 Q），直至恒定因素，具体取决于F。如果A在数量字段上定义了A，则我们在otilde（log6 Q）中足够多的Modulo值（log6 Q）的prime（log6 Q）Birinary Biarry Operations Biarrions by naperiations Biarriations Brienlations Biential biver。数值实验证明了我们方法的实际可用性。
We generalize Elkies's method, an essential ingredient in the SEA algorithm to count points on elliptic curves over finite fields of large characteristic, to the setting of p.p. abelian surfaces. Under reasonable assumptions related to the distribution of Elkies primes, we obtain improvements over Schoof's method in two cases. If the abelian surface A over Fq has RM by a fixed quadratic field F, we reach the same asymptotic complexity Otilde(log4 q) as the SEA algorithm up to constant factors depending on F. If A is defined over a number field, we count points on A modulo sufficiently many primes in Otilde(log6 q) binary operations on average. Numerical experiments demonstrate the practical usability of our methods.
