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arXiv · 2202.10530

A geometric study of circle packings and ideal class groups

Daniel Martin
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Abstract

A family of fractal arrangements of circles is introduced for each imaginary quadratic field $K$. Collectively, these arrangements contain (up to an affine transformation) every set of circles in the extended complex plane with integral curvatures and Zariski dense symmetry group. When that set is a circle packing, we show how the ambient structure of our arrangement gives a geometric criterion for satisfying the almost local-global principle. Connections to the class group of $K$ are also explored. Among them is a geometric property that guarantees certain ideal classes are group generators.

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Daniel Martin. 2022-02-21. A geometric study of circle packings and ideal class groups. https://arxiv.org/abs/2202.10530

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