Modular forms and ellipsoidal T-designs

Abstract

In recent work, Miezaki introduced the notion of a $spherical$ $T$-d$esign$ in ${\mathbb{R}}^2$, where $T$ is a potentially infinite set. As an example, he offered the ${\mathbb{Z}}^2$-lattice points with fixed integer norm (a.k.a. shells). These shells are $maximal$ spherical $T$-designs, where $T={\mathbb{Z}}^{+}\setminus 4{\mathbb{Z}}^{+}$. We generalize the notion of a spherical $T$-design to special ellipses, and extend Miezaki's work to the norm form shells for rings of integers of imaginary quadratic fields with class number 1.