On the distribution of shapes of pure quartic number fields

2025/06/30 by Das, Sudipa, Kala, Sushant, Mukhopadhyay, Arunabha +1
#11P21 (Secondary) #11R16 #11R45 (Primary) 11E12 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2506.23766

Abstract

The shape of a number field is a subtle arithmetic invariant arising from the geometry of numbers. It is defined as the equivalence class of the lattice of integers with respect to linear operations that are composites of rotations, reflections, and positive scalar dilations. For a number field of degree n, the shape is a point in the space of shapes \mathscrS_n-1, which is the double quotient GL_n-1(ℤ) \backslash GL_n-1(ℝ) / GO_n-1(ℝ). In this paper, we investigate the distribution of shapes in the family of pure quartic fields K_m = ℚ(√[4]m). We prove that the shape of K_m lies on one of ten explicitly described torus orbits in \mathscrS₃, determined by the sign and residue class of m \bmod 32. It is shown that the shape on a given torus orbit is completely determined by two parameters, one of which varies continuously, while the other takes values in a discrete set. As a result, the distribution of shapes in this family is governed by a product of a continuous and a discrete measure. Our results shed new light on a question posed by Manjul Bhargava and Piper H concerning the distribution of shapes in families of non-generic number fields of fixed degree. Notably, the limiting distribution in our case does not arise as the restriction of the natural measure on \mathscrS₃ induced by Haar measure on GL₃(ℝ).

Citations

• On the shapes of pure prime degree number fields
• The shapes of pure cubic fields

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