On the distribution of shapes of octic Kummer extensions

The shape of a number field $K$ of degree $n$ is defined as the equivalence class of the lattice of integers under linear operations generated by rotations, reflections, and positive scalar dilations. It may be viewed as a point in the space of shapes $\mathscr{S}_{n-1} = \mathrm{GL}_{n-1}(\mathbb{Z})\backslash \mathrm{GL}_{n-1}(\mathbb{R})/\mathrm{GO}_{n-1}(\mathbb{R})$. In this paper, we study the distribution of shapes of octic Kummer extensions $L=\mathbb{Q}(i,\sqrt[4]{m})$, where $m\in\mathbb{Z}[i]$ is fourth-power-free. We parametrize these shapes by explicit invariants known as shape parameters and establish an asymptotic formula for their joint distribution ordered by absolute discriminant. The limiting distribution is given by an explicit measure that factors as the product of a continuous measure and a discrete measure arising from local arithmetic conditions.