On the distribution of shapes of octic Kummer extensions
Anuj Jakhar, Anwesh Ray
TL;DR
The paper addresses the distribution of shapes for a non-generic, sparse family of number fields, focusing on octic Kummer extensions $L=\mathbb{Q}(i,\sqrt[4]{m})$. It develops a relative shape via the projection to $J(\mathcal{O}_L)^{\perp}$ and parameterizes shapes by $(\lambda_1,\lambda_2)$ derived from the factorization $m=f g^2 h^3$, proving that, as the discriminant grows, the shapes equidistribute according to a product measure $\hat{\mu}=\mu_{\infty}\times\mu_{\mathrm{sf}}$ on the $(\lambda_1,\lambda_2)$-space, with $\mu_{\infty}$ Archimedean and $\mu_{\mathrm{sf}}$ a discrete local-density factor. The authors carry out a detailed Gram-matrix analysis, separating Type I and II cases, and employ a renormalization framework to show the limiting shape depends only on $(\lambda_1,\lambda_2)$. The final density result combines geometry-of-numbers counts with precise p-adic congruence conditions, yielding an explicit limiting distribution that reflects both global and local arithmetic structure of the family.
Abstract
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.