Modular curves and bad reduction
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908952915083264 |
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| author | Logan, Adam McKinnon, David |
| author_facet | Logan, Adam McKinnon, David |
| contents | We prove results that imply, under various hypotheses, that every elliptic curve over a
number field $k$ corresponding to a point on a modular curve has bad reduction at a
certain prime $p$ of $\mathcal{O}_k$.
For example, every elliptic curve with a cyclic torsion subgroup of order 20 defined over
$\mathbb{Q}(\sqrt{-11})$ or $\mathbb{Q}(\sqrt{17})$ has bad reduction at all primes lying over $3$.
The proofs of these statements are quite different, since $3$ is split in $\mathbb{Q}(\sqrt{-11})$
and inert in $\mathbb{Q}(\sqrt{17})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09536 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Modular curves and bad reduction Logan, Adam McKinnon, David Number Theory Algebraic Geometry 11G18 (Primary) 14G35 (Secondary) We prove results that imply, under various hypotheses, that every elliptic curve over a number field $k$ corresponding to a point on a modular curve has bad reduction at a certain prime $p$ of $\mathcal{O}_k$. For example, every elliptic curve with a cyclic torsion subgroup of order 20 defined over $\mathbb{Q}(\sqrt{-11})$ or $\mathbb{Q}(\sqrt{17})$ has bad reduction at all primes lying over $3$. The proofs of these statements are quite different, since $3$ is split in $\mathbb{Q}(\sqrt{-11})$ and inert in $\mathbb{Q}(\sqrt{17})$. |
| title | Modular curves and bad reduction |
| topic | Number Theory Algebraic Geometry 11G18 (Primary) 14G35 (Secondary) |
| url | https://arxiv.org/abs/2604.09536 |