Quickly constructing curves of genus 4 with many points
Quickly constructing curves of genus 4 with many points
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快速构造多点的 4 格曲线
DOI：
10.1090/conm/663/13353
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Everett W. Howe
中科院分区：
文献类型：
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作者：
Everett W. Howe
有限域上曲线的“缺陷”是曲线上有理点的数目与曲线的Weil-Serre界之间的差。给出了有限域上亏格为2的曲线的亏格为4的双覆盖的构造，使得双覆盖的亏格不比亏格为2的曲线的亏格多多少。我们给出了一个算法，使用这种结构来产生亏格为4的曲线与小缺陷。启发式地，对于所有足够大的素数和几乎所有的素数幂q，该算法预计将产生一个亏格为4的曲线在F_q上的时间q^{3/4}，最多为对数因子。
作为算法分析的一部分，我们提出了一个重新解释的结果Hayashida的数的属-2曲线的雅可比同构的平方给定的椭圆曲线与复数乘法的最大阶。我们证明了在这样的椭圆曲线的平方上的一类主极化等价于在某个四元数阶上的一类右理想。
The "defect" of a curve over a finite field is the difference between the number of rational points on the curve and the Weil-Serre bound for the curve. We present a construction for producing genus-4 double covers of genus-2 curves over finite fields such that the defect of the double cover is not much more than the defect of the genus-2 curve. We give an algorithm that uses this construction to produce genus-4 curves with small defect. Heuristically, for all sufficiently large primes and for almost all prime powers q, the algorithm is expected to produce a genus-4 curve over F_q with defect at most 4 in time q^{3/4}, up to logarithmic factors.
As part of the analysis of the algorithm, we present a reinterpretation of results of Hayashida on the number of genus-2 curves whose Jacobians are isomorphic to the square of a given elliptic curve with complex multiplication by a maximal order. We show that a category of principal polarizations on the square of such an elliptic curve is equivalent to a category of right ideals in a certain quaternion order.