双陪集的组合结构与模群子群的基本域
Combinatorics of double cosets and fundamental domains for the subgroups of the modular group
AI总结：
该研究通过双陪集组合结构建立模群子群与二分立方图的双射对应，提出构造模群子群基本域的高效算法，并证明了模曲线自同构群的相关结果。
摘要：
正如R.库尔卡尼（R. Kulkarni）所指出的，模群的子群的共轭类与二分立方图存在双射对应关系。我们将说明如何通过$\text{PSL}_2(\boldsymbol{Z})$在右陪集$G\backslash\text{PSL}_2(\boldsymbol{Z})$上的右作用组合结构，恢复对应于$\text{PSL}_2(\boldsymbol{Z})$子群$G$的图，这为构造$G$在上半平面作用的良好基本域（库尔卡尼称之为“特殊多边形”）提供了方法。对于经典同余子群$\boldsymbol{\tau}_0(N)$、$\boldsymbol{\tau}_1(N)$、$\boldsymbol{\tau}(N)$等，该方法所需的操作数为指数乘以一个不超过$\boldsymbol{\tau}$多项式增长的量，约为朴素方法所需操作数的平方根。我们给出了定位上半平面中某点在基本域上的算法，以及将$G$的给定元素表示为独立生成元乘积的算法，还（重新）证明了关于模曲线自同构群的若干相关结果，例如给出了$X(N)$的自同构群为$\text{SL}_2(\boldsymbol{Z}/N)/{\boldsymbol{\tau}I}$的简洁证明。
英文摘要：
As noticed by R.~Kulkarni, the conjugacy classes of subgroups of the modular group correspond bijectively to bipartite cuboid graphs. We'll explain how to recover the graph corresponding to a subgroup $G$ of $\mathrm{PSL}_2(\mathbb{Z})$ from the combinatorics of the right action of $\mathrm{PSL}_2(\mathbb{Z})$ on the right cosets $G\backslash\mathrm{PSL}_2(\mathbb{Z})$. This gives a method of constructing nice fundamental domains (which Kulkarni calls "special polygons") for the action of $G$ on the upper half plane. For the classical congruence subgroups $Γ_0(N)$, $Γ_1(N)$, $Γ(N)$ etc. the number of operations the method requires is the index times something that grows not faster than a polynomial in $\log N$. This is roughly the square root of the number of operations required by the naive procedure. We give algorithms to locate an element of the upper half-plane on the fundamental domain and to write a given element of $G$ as a product of independent generators. We also (re)prove a few related results about the automorphism groups of modular curves. For example, we give a simple proof that the automorphism group of $X(N)$ is $\mathrm{SL}_2(\mathbb{Z}/N)/\{\pm I\}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
