论文标题
重新访问模块组对假想二次数字字段的子组的作用
Revisiting the action of a subgroup of the modular group on imaginary quadratic number fields
论文作者
论文摘要
考虑模块化组$ \ mbox {psl}（2，\ mathbb {z}）= \ langle x，\，\，y \，| \，x^2 = y^3 = y^3 = 1 \ rangle $由转换$ x：z \ mapsto -1/z $ y＆y z $：z $ y：z \ mapsto（z -1 $）产生。令$ h $为适当的子组$ \ langle y，\，\，| \，y^3 = v^3 = 1 \ rangle $ of $ \ mbox {psl}（2，\ mathbb {z}）$，其中$ v = xyx $。引用（M. Ashiq和Q. Mushtaq，{\ em {\ em的动作模块组在虚构的二次场上}，quasigropus及相关系统{\ bf 14}（2006），133---146）提出了有关$ h $在子集上的结果$ \ {\ frac {a+\ sqrt { - n}} {c} \，| \，a，b = \ frac {a^2+n} {c}，c \ in \ in \ mathbb {z} $ \ mathbb {q}（\ sqrt { - n}）$用于一个不含正方形的整数$ n $。在当前文章中，作者指出并纠正了上述参考中出现的错误。最重要的是，给出了此动作产生的轨道数量的校正估计。
Consider the modular group $\mbox{PSL}(2,\mathbb{Z})=\langle x, \, y \,|\, x^2=y^3=1\rangle$ generated by the transformations $x: z\mapsto -1/z$ and $y:z\mapsto (z-1)/z$. Let $H$ be the proper subgroup $\langle y,\,v\,|\, y^3=v^3=1\rangle$ of $\mbox{PSL}(2,\mathbb{Z})$, where $v=xyx$. The reference (M. Ashiq and Q. Mushtaq, {\em Actions of a subgroup of the modular group on an imaginary quadratic field}, Quasigropus and Related Systems {\bf 14} (2006), 133--146) proposed results concerning the action of $H$ on the subset $\{\frac{a+\sqrt{-n}}{c}\,|\, a,b=\frac{a^2+n}{c}, c \in \mathbb{Z}, c\neq 0\}$ of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-n})$ for a positive square-free integer $n$. In the current article, the author points out and corrects errors appearing in the aforementioned reference. Most importantly, the corrected estimate for the number of orbits arising from this action is given.
