Background: Katherine Stange describes Schmidt arrangements in “Visualising the arithmetic of imaginary quadratic fields”, arXiv:1410.0417. Given an imaginary quadratic field $K$, we study the Bianchi group $\mathrm{PSL}_2(\mathcal{O}_K)$, which is the group of Möbius transformations with coefficients in the ring of integers of $K$. The image of $\mathbb R$ under a group element is called a […]

Quadratic integer ring $\mathcal{O}$ is defined by \begin{equation} \mathcal{O}=\begin{cases} \mathbb{Z}[\sqrt{D}] & \text{if}\ D=2,3\ \pmod 4\\ \mathbb{Z}\left[\frac{1+\sqrt{D}}{2}\right]\ & \text{if}\ D=1\pmod 4 \end{cases} \end{equation} where $D$ is square-free. I understand $\mathbb{Z}\left[\frac{1+\sqrt{D}}{2}\right]$ is not closed under multiplication if $D=2,3\pmod 4$. But still, isn’t it more natural to define $\mathcal{O}=\mathbb{Z}[\sqrt{D}]$ for all square-free $D$? (in that case, it really […]

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