AGM aquariums and elliptic curves over arbitrary finite fields
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909360621355008 |
|---|---|
| author | Kayath, June Lane, Connor Neifeld, Ben Ni, Tianyu Xue, Hui |
| author_facet | Kayath, June Lane, Connor Neifeld, Ben Ni, Tianyu Xue, Hui |
| contents | In this paper, we define a version of the arithmetic-geometric mean (AGM) function for arbitrary finite fields $\mathbb{F}_q$, and study the resulting AGM graph with points $(a,b) \in \mathbb{F}_q \times \mathbb{F}_q$ and directed edges between points $(a,b)$, $(\frac{a+b}{2},\sqrt{ab})$ and $(a,b)$, $(\frac{a+b}{2},-\sqrt{ab})$. The points in this graph are naturally associated to elliptic curves over $\mathbb{F}_q$ in Legendre normal form, with the AGM function defining a 2-isogeny between the associated curves. We use this correspondence to prove several results on the structure, size, and multiplicity of the connected components in the AGM graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17969 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | AGM aquariums and elliptic curves over arbitrary finite fields Kayath, June Lane, Connor Neifeld, Ben Ni, Tianyu Xue, Hui Number Theory Algebraic Geometry 14H52, 11G20 In this paper, we define a version of the arithmetic-geometric mean (AGM) function for arbitrary finite fields $\mathbb{F}_q$, and study the resulting AGM graph with points $(a,b) \in \mathbb{F}_q \times \mathbb{F}_q$ and directed edges between points $(a,b)$, $(\frac{a+b}{2},\sqrt{ab})$ and $(a,b)$, $(\frac{a+b}{2},-\sqrt{ab})$. The points in this graph are naturally associated to elliptic curves over $\mathbb{F}_q$ in Legendre normal form, with the AGM function defining a 2-isogeny between the associated curves. We use this correspondence to prove several results on the structure, size, and multiplicity of the connected components in the AGM graph. |
| title | AGM aquariums and elliptic curves over arbitrary finite fields |
| topic | Number Theory Algebraic Geometry 14H52, 11G20 |
| url | https://arxiv.org/abs/2410.17969 |