On the distribution of shapes of totally real multiquadratic number fields
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911508332544000 |
|---|---|
| author | Jakhar, Anuj Ray, Anwesh |
| author_facet | Jakhar, Anuj Ray, Anwesh |
| contents | The shape of a number field $K$ of degree $m$ 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 $\mathscr{S}_{m-1} = \mathrm{GL}_{m-1}(\mathbb Z)\backslash \mathrm{GL}_{m-1}(\mathbb R)/\mathrm{GO}_{m-1}(\mathbb{R})$. The double quotient space is equipped with a natural measure $μ$ which is induced from the Haar measure on $\mathrm{GL}_{m-1}(\mathbb R)$. We study the distribution of shapes of totally real multiquadratic number fields of degree $m:=2^n$ in which $2$ is unramified. We show that the distribution is governed by the restriction of $μ$ to a certain torus orbit in $\mathscr{S}_{m-1}$. Our result resolves a conjecture of Haidar. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11443 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the distribution of shapes of totally real multiquadratic number fields Jakhar, Anuj Ray, Anwesh Number Theory 11R29, 11R45, 11R56 The shape of a number field $K$ of degree $m$ 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 $\mathscr{S}_{m-1} = \mathrm{GL}_{m-1}(\mathbb Z)\backslash \mathrm{GL}_{m-1}(\mathbb R)/\mathrm{GO}_{m-1}(\mathbb{R})$. The double quotient space is equipped with a natural measure $μ$ which is induced from the Haar measure on $\mathrm{GL}_{m-1}(\mathbb R)$. We study the distribution of shapes of totally real multiquadratic number fields of degree $m:=2^n$ in which $2$ is unramified. We show that the distribution is governed by the restriction of $μ$ to a certain torus orbit in $\mathscr{S}_{m-1}$. Our result resolves a conjecture of Haidar. |
| title | On the distribution of shapes of totally real multiquadratic number fields |
| topic | Number Theory 11R29, 11R45, 11R56 |
| url | https://arxiv.org/abs/2603.11443 |