On the distribution of shapes of sextic pure number fields
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917203778994176 |
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| author | Jakhar, Anuj Kalwaniya, Ravi Ray, Anwesh Roy, Bidisha |
| author_facet | Jakhar, Anuj Kalwaniya, Ravi Ray, Anwesh Roy, Bidisha |
| contents | The shape of a number field $K$ of degree $n$ 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. The shape is a point in the space of shapes $\mathscr{S}_{n-1}$, which is the double quotient $\mathrm{GL}_{n-1}(\mathbb{Z}) \backslash \mathrm{GL}_{n-1}(\mathbb{R}) / \mathrm{GO}_{n-1}(\mathbb{R})$. We investigate the distribution of shapes of pure sextic number fields $K=\mathbb{Q}(\sqrt[6]{m})$, ordered by absolute discriminant. Such fields are partitioned into $20$ distinct Types determined by local conditions at $2$ and $3$, and an explicit integral basis is given in each case. For each Type, the shape of $K$ admits an explicit description in terms of shape parameters. Fixing the sign of $m$ and a Type, we prove that the corresponding shapes are equidistributed along a translated torus orbit in the space of shapes. The limiting distribution is given by an explicit measure expressed as the product of a continuous measure and a discrete measure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_09411 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the distribution of shapes of sextic pure number fields Jakhar, Anuj Kalwaniya, Ravi Ray, Anwesh Roy, Bidisha Number Theory 11R45, 11E12, 11P21 The shape of a number field $K$ of degree $n$ 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. The shape is a point in the space of shapes $\mathscr{S}_{n-1}$, which is the double quotient $\mathrm{GL}_{n-1}(\mathbb{Z}) \backslash \mathrm{GL}_{n-1}(\mathbb{R}) / \mathrm{GO}_{n-1}(\mathbb{R})$. We investigate the distribution of shapes of pure sextic number fields $K=\mathbb{Q}(\sqrt[6]{m})$, ordered by absolute discriminant. Such fields are partitioned into $20$ distinct Types determined by local conditions at $2$ and $3$, and an explicit integral basis is given in each case. For each Type, the shape of $K$ admits an explicit description in terms of shape parameters. Fixing the sign of $m$ and a Type, we prove that the corresponding shapes are equidistributed along a translated torus orbit in the space of shapes. The limiting distribution is given by an explicit measure expressed as the product of a continuous measure and a discrete measure. |
| title | On the distribution of shapes of sextic pure number fields |
| topic | Number Theory 11R45, 11E12, 11P21 |
| url | https://arxiv.org/abs/2601.09411 |