Spherical Designs for Finite Quaternionic Unit Groups and Their Applications to Modular Forms

For a finite subset X of the d-dimensional unit sphere, the harmonic strength T(X) of X is the set of ℓ∈ℕ such that ∑_x∈ X P(x)=0 for all harmonic polynomials P of homogeneous degree ℓ. 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 }∪𝕆^+, where 𝕆^+ denotes the set of all positive odd integers. This result provides the first characterization of 2O from the spherical design viewpoint. For G∈{2T,2O,2I}, we consider the lattice 𝒪_G generated by G over R_G on which the group G acts on by multiplication, where R_2T=ℤ, R_2O=ℤ[√(2)], R_2I=ℤ[(1+√(5))/2] are the ring of integers. We introduce the spherical theta function θ_G,P(z) attached to the lattice 𝒪_G and a harmonic polynomial P of degree ℓ 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 ℂ-vector space spanned by all θ_G,P(z) of harmonic polynomials P of homogeneous degree ℓ has dimension zero–without relying on the theory of modular forms.