A geometric study of circle packings and ideal class groups

TL;DR

This paper introduces fractal circle arrangements linked to imaginary quadratic fields, providing geometric criteria related to the local-global principle and class groups, revealing new connections between geometry and algebraic number theory.

Contribution

It presents a novel geometric framework for understanding circle packings and their relation to ideal class groups in imaginary quadratic fields.

Findings

Arrangements encompass all integral curvature circle sets with dense symmetry.

Provides a geometric criterion for the almost local-global principle.

Connects circle arrangements to ideal class group properties.

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.