Spherical designs for finite quaternionic unit groups and their applications to modular forms

TL;DR

This paper investigates special finite quaternionic groups acting on spheres, characterizes their harmonic strength and minimality as spherical designs, and connects these structures to modular forms via spherical theta functions.

Contribution

It provides the first spherical design characterization of the octahedral group $2O$, determines harmonic strengths of three quaternionic groups, and links these to modular forms through spherical theta functions.

Findings

$2O$ is uniquely minimal with specific harmonic strength.

Harmonic strengths of $2T$, $2O$, and $2I$ are explicitly determined.

Spherical theta functions associated with these groups are shown to be modular forms.

Abstract

For a finite subset $X$ of the $d$-dimensional unit sphere, the harmonic strength $T(X)$ of $X$ is the set of $\ell\in \mathbb{N}$ such that $\sum_{x\in X} P(x)=0$ for all harmonic polynomials $P$ of homogeneous degree $\ell$. We will study three exceptional finite groups of unit quaternions, called the binary tetrahedral group $2T$ of order 24, the octahedral group $2O$ of order 48, and the icosahedral group $2I$ of order 120, which can be viewed as a subset of the 3-dimensional unit sphere. For these three groups, we determine the harmonic strength and show the minimality and the uniqueness as spherical designs. In particular, the group $2O$ is unique as a minimal subset $X$ of the 3-dimensional unit sphere with $T(X)=\{22,14,10,6,4,2 \}\cup \mathbb{O}^+$, where $\mathbb{O}^+$ denotes the set of all positive odd integers. This result provides the first characterization of $2O$ from the spherical design viewpoint. For $G\in \{2T,2O,2I\}$, we consider the lattice $\mathcal{O}_{G}$ generated by $G$ over $R_G$ on which the group $G$ acts on by multiplication, where $R_{2T}=\mathbb{Z},\ R_{2O}=\mathbb{Z}[\sqrt{2}],\ R_{2I}=\mathbb{Z}[(1+\sqrt{5})/2]$ are the ring of integers. We introduce the spherical theta function $\theta_{G,P}(z)$ attached to the lattice $\mathcal{O}_G$ and a harmonic polynomial $P$ of degree $\ell$ and prove that they are modular forms. By applying our results on the characterization of $G$ as a spherical design, we determine the cases in which the $\mathbb{C}$-vector space spanned by all $\theta_{G,P}(z)$ of harmonic polynomials $P$ of homogeneous degree $\ell$ has dimension zero--without relying on the theory of modular forms.