Rational points on modular curves via maps to elliptic curves with rank zero
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_ | 1866916041969369088 |
|---|---|
| author | Mayle, Jacob Rouse, Jeremy |
| author_facet | Mayle, Jacob Rouse, Jeremy |
| contents | A fundamental problem in arithmetic geometry is to determine the image of the mod $N$ Galois representation for all elliptic curves over $\mathbb{Q}$ and integers $N \geq 1$. For a given subgroup $G \le \mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$, there is a modular curve $X_G$ whose rational points parametrize elliptic curves for which the image of the mod $N$ Galois representation is contained in $G$. If $X_G$ admits a map to an elliptic curve $E/\mathbb{Q}$ for which $E(\mathbb{Q})$ has rank $0$, then its rational points can be effectively determined, provided that a map $X_G \to E$ is known. In this article, we give a method for constructing such maps. Using this method, together with existing methods and results, we systematically determine the rational points of $X_G$ for more than $99\%$ of modular curves of level at most $70$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17202 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rational points on modular curves via maps to elliptic curves with rank zero Mayle, Jacob Rouse, Jeremy Number Theory Primary 11G18, 11F80, Secondary 11G05, 14G05 A fundamental problem in arithmetic geometry is to determine the image of the mod $N$ Galois representation for all elliptic curves over $\mathbb{Q}$ and integers $N \geq 1$. For a given subgroup $G \le \mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$, there is a modular curve $X_G$ whose rational points parametrize elliptic curves for which the image of the mod $N$ Galois representation is contained in $G$. If $X_G$ admits a map to an elliptic curve $E/\mathbb{Q}$ for which $E(\mathbb{Q})$ has rank $0$, then its rational points can be effectively determined, provided that a map $X_G \to E$ is known. In this article, we give a method for constructing such maps. Using this method, together with existing methods and results, we systematically determine the rational points of $X_G$ for more than $99\%$ of modular curves of level at most $70$. |
| title | Rational points on modular curves via maps to elliptic curves with rank zero |
| topic | Number Theory Primary 11G18, 11F80, Secondary 11G05, 14G05 |
| url | https://arxiv.org/abs/2601.17202 |