论文标题
广义类多项式
Generalized class polynomials
论文作者
论文摘要
希尔伯特类多项式是根源的椭圆曲线的J-不变，其内态环是一个给定的想象二次秩序。它可用于在有限数量的有限场上计算椭圆曲线。由于其系数通常相当大，因此一直有兴趣寻找替代模块化功能，其相应的类多项式较小。最著名的是韦伯的功能，它的大小为假想二次判别物的正密度子集降低了72倍。另一方面，布鲁克（Bröker）和史蒂文哈根（Stevenhagen）表明，没有模块化功能的能力比100.83的倍数更好。我们引入了类多项式的概括，其还原因子不受布鲁克 - 斯坦哈根结合的限制。我们提供与韦伯还原因子相匹配的示例。对于无限的判别因子家族，其还原因子超过了所有以前已知的模块化功能的因子，至少一个因素。
The Hilbert class polynomial has as roots the j-invariants of elliptic curves whose endomorphism ring is a given imaginary quadratic order. It can be used to compute elliptic curves over finite fields with a prescribed number of points. Since its coefficients are typically rather large, there has been continued interest in finding alternative modular functions whose corresponding class polynomials are smaller. Best known are Weber's functions, that reduce the size by a factor of 72 for a positive density subset of imaginary quadratic discriminants. On the other hand, Bröker and Stevenhagen showed that no modular function will ever do better than a factor of 100.83. We introduce a generalization of class polynomials, with reduction factors that are not limited by the Bröker-Stevenhagen bound. We provide examples matching Weber's reduction factor. For an infinite family of discriminants, their reduction factors surpass those of all previously known modular functions by a factor at least 2.
