论文标题
枚举属4
Algorithms to enumerate superspecial Howe curves of genus 4
论文作者
论文摘要
Howe曲线是属属$ 4 $的曲线，作为两个属的光纤产品 - $ 1 $ $ \ MATHBF {p}^1 $的双重封面。在本文中，我们提出了一种用于测试Howe曲线同构的简单算法，我们提出了两种用于查找和枚举这些曲线的主要算法：一种涉及解决来自Cartier-Manin矩阵的多变量系统，而另一种用途，而另一种用途Richelot Isegenies richelot isenies of Certus centus cenus $ 2 $ $ 2 $。通过实施和复杂性分析比较两种算法，我们得出的结论是，后者列举了曲线的效率。使用这些算法，我们表明，每个prime $ p $的特征性$ p $ $ 4 $ p $，$ 7 <p <p <20000 $。
A Howe curve is a curve of genus $4$ obtained as the fiber product of two genus-$1$ double covers of $\mathbf{P}^1$. In this paper, we present a simple algorithm for testing isomorphism of Howe curves, and we propose two main algorithms for finding and enumerating these curves: One involves solving multivariate systems coming from Cartier--Manin matrices, while the other uses Richelot isogenies of curves of genus $2$. Comparing the two algorithms by implementation and by complexity analyses, we conclude that the latter enumerates curves more efficiently. Using these algorithms, we show that there exist superspecial curves of genus $4$ in characteristic $p$ for every prime $p$ with $7 < p < 20000$.
