Document Type

Article

Publication Date

2-21-2024

Published In

Transactions Of The American Mathematical Society Series B

Abstract

We study Apollonian circle packings using the properties of a certain rank 4 indefinite Kac-Moody root system ϕ. We introduce the generating function Z(s) of a packing, an exponential series in four variables with an Apollonian symmetry group, which is a symmetric function for ϕ. By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of ϕ, with automorphic Weyl denominators, we express Z(s) in terms of Jacobi theta functions and the Siegel modular form Δ₅. We also show that the domain of convergence of Z(s) is the Tits cone of ϕ, and discover that this domain inherits the intricate geometric structure of Apollonian packings.

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Creative Commons Attribution 3.0 License
This work is licensed under a Creative Commons Attribution 3.0 License.

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