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Kate Stange : Visualising the arithmetic of imaginary quadratic fields

Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$. The Schmidt arrangement of $K$ is the orbit of the extended real line in the extended complex plane under the Mobius transformation action of the Bianchi group $\operatorname{PSL}(2,\mathcal{O}_K)$. The arrangement takes the form of a dense collection of intricately nested circles. Aspects of the number theory of $\mathcal{O}_K$ can be characterised by properties of this picture: for example, the arrangement is connected if and only if $\mathcal{O}_K$ is Euclidean. I'll explore this structure and its connection to Apollonian circle packings. Specifically, the Schmidt arrangement for the Gaussian integers is a disjoint union of all primitive integral Apollonian circle packings. Generalizing this relationship to all imaginary quadratic $K$, the geometry naturally defines some new circle packings and thin groups of arithmetic interest.

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