On the distribution of shapes of pure quartic number fields

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 $n$ n , the shape is a point in the space of shapes script upper S n minus 1 $\mathcal{S}_{n-1}$ 𝒮 n − 1 , which is the double quotient GLn minus 1 left parenthesis double struck upper Z right parenthesis backslash GLn minus 1 left parenthesis double struck upper R right parenthesis slash GOn minus 1 left parenthesis double struck upper R right parenthesis $\operatorname{GL}_{n-1}(\mathbb{Z}) \backslash \operatorname{GL}_{n-1}(\mathbb{R}) / \operatorname{GO}_{n-1}(\mathbb{R})$ GL n − 1 ( ℤ ) \ GL n − 1 ( ℝ ) / GO n − 1 ( ℝ ) . In this paper, we investigate the distribution of shapes in the family of pure quartic fields Km equals double struck upper Q left parenthesis m 4 right parenthesis $K_m = \mathbb{Q}(\sqrt[4]{m})$ K m = ℚ ( m 4 ) . We prove that the shape of Km $K_m$ K m lies on one of 5 $5$