Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.09825 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911954918965248 |
|---|---|
| author | Yoshida, Takumi |
| author_facet | Yoshida, Takumi |
| contents | A rational face cuboid is a cuboid that all of edges, two of three face diagonals and space diagonal have rational lengths. \[ E_{1,s}: y^2=x(x-(2s)^2)(x+(s^2-1)^2) \] for a rational number $s \neq 0, \pm 1$, and define $\tilde{A}$ consisting of all pairs of a rational number $s$ and a non-torsion rational point $(α, β) \in E_{1,s}(\mathbb{Q})$. We construct a surjective map from $\tilde{A}$ to the set $\mathscr{F}$ of equivalence classes of rational face cuboids, and prove that this map is a $32:1$-map. In this way, we show that the set $\mathscr{F}$ has infinite elements. Also, we prove that there are infinitely many $s \in \mathbb{Q} \setminus \{ 0,\pm 1 \}$ with $\mathrm{rank} E_{1,s} (\mathbb{Q})>0$. In this proof, we construct pairs of $s$ and $(α, β) \in E_{1,s} (\mathbb{Q})$ which are not parametric solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09825 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The relationship between face cuboids and elliptic curves Yoshida, Takumi Number Theory A rational face cuboid is a cuboid that all of edges, two of three face diagonals and space diagonal have rational lengths. \[ E_{1,s}: y^2=x(x-(2s)^2)(x+(s^2-1)^2) \] for a rational number $s \neq 0, \pm 1$, and define $\tilde{A}$ consisting of all pairs of a rational number $s$ and a non-torsion rational point $(α, β) \in E_{1,s}(\mathbb{Q})$. We construct a surjective map from $\tilde{A}$ to the set $\mathscr{F}$ of equivalence classes of rational face cuboids, and prove that this map is a $32:1$-map. In this way, we show that the set $\mathscr{F}$ has infinite elements. Also, we prove that there are infinitely many $s \in \mathbb{Q} \setminus \{ 0,\pm 1 \}$ with $\mathrm{rank} E_{1,s} (\mathbb{Q})>0$. In this proof, we construct pairs of $s$ and $(α, β) \in E_{1,s} (\mathbb{Q})$ which are not parametric solutions. |
| title | The relationship between face cuboids and elliptic curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/2407.09825 |