Abstract

In this work, we present an algorithm running over SAGE, which allows users to deal with group actions on Riemann surfaces up to topological equivalence. Our algorithm allows us to study the equisymmetric stratification of the branch locus ${\mathcal{B}}_g$ of the moduli space ${\mathcal{M}}_g$ of compact Riemann surfaces of genus $g\ge 2,$ corresponding to group actions with orbit genus 0. That is, it works for actions on surfaces of any genus in the case the genus of the quotient surface is zero, except for obvious hardware constraints. Our approach is toward studying inclusions and intersections of (closed) strata of ${\mathcal{B}}_g.$ We apply our algorithm to describe part of the geometry of the branch locus ${\mathcal{B}}_9,$ in terms of equisymmetric stratification. We also use it to compute all group actions up to topological equivalence for genus 5–10, this completes the lists. Finally, we add an optimized version of an algorithm, which allows us to identify Jacobian varieties of CM-type. As a byproduct, we obtain a Jacobian variety of dimension 11 which is isogenous to $E_i^9\times E_{i\sqrt{3}}^2,$ where E_i and $E_{i\sqrt{3}}$ are elliptic curves with complex multiplication.

摘要

在这项工作中，我们提出了一种在 SAGE 上运行的算法，它允许用户处理黎曼曲面上的群作用直至拓扑等价。我们的算法允许我们研究分支轨迹的等对称分层${\mathcal{乙}}_G$模空间的${\mathcal{米}}_G$紧黎曼曲面的属$G\ge \mathrm{2个},$对应于轨道亏格为 0 的群动作。也就是说，它适用于任何亏格曲面上的动作，前提是商曲面的亏格为零，明显的硬件限制除外。我们的方法是研究（封闭）地层的夹杂物和交叉点${\mathcal{乙}}_G.$我们应用我们的算法来描述分支轨迹的部分几何形状${\mathcal{乙}}_9,$在等对称分层方面。我们还使用它来计算所有的群行为，直到亏格 5-10 的拓扑等价，这样就完成了列表。最后，我们添加了一个算法的优化版本，它允许我们识别 CM 类型的雅可比变体。作为副产品，我们获得了维数为 11 的雅可比变体，它与${乙}_{我}^9\times {乙}_{我\sqrt{\mathrm{3个}}}^{\mathrm{2个}},$其中E _i和${乙}_{我\sqrt{\mathrm{3个}}}$是具有复数乘法的椭圆曲线。