Isolated points on modular curves
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908907471896576 |
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| author | Terao, Kenji |
| author_facet | Terao, Kenji |
| contents | We study isolated points on the modular curves $X_{H}$, for $H$ a subgroup of $\operatorname{GL}_{2}(\mathbb{Z}/n \mathbb{Z})$ for some $n \geq 1$. In particular, we prove a single-sink theorem for such isolated points, which traces the existence of all such isolated points with the same $j$-invariant back to an isolated point on a single curve. Building on this result, we also present a uniform strategy for determining the isolated points on any family of modular curves. As an example, we use this strategy to classify the isolated points with rational $j$-invariant on all modular curves of level 7, as well as the modular curves $X_{0}(n)$, the latter assuming a conjecture on images of Galois representations of elliptic curves over $\mathbb{Q}$. Underpinning all of this, we develop a theory of isolated divisors on geometrically disconnected varieties, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13108 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Isolated points on modular curves Terao, Kenji Number Theory Algebraic Geometry We study isolated points on the modular curves $X_{H}$, for $H$ a subgroup of $\operatorname{GL}_{2}(\mathbb{Z}/n \mathbb{Z})$ for some $n \geq 1$. In particular, we prove a single-sink theorem for such isolated points, which traces the existence of all such isolated points with the same $j$-invariant back to an isolated point on a single curve. Building on this result, we also present a uniform strategy for determining the isolated points on any family of modular curves. As an example, we use this strategy to classify the isolated points with rational $j$-invariant on all modular curves of level 7, as well as the modular curves $X_{0}(n)$, the latter assuming a conjecture on images of Galois representations of elliptic curves over $\mathbb{Q}$. Underpinning all of this, we develop a theory of isolated divisors on geometrically disconnected varieties, which may be of independent interest. |
| title | Isolated points on modular curves |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2412.13108 |