On the Kodaira types of elliptic curves with potentially good supersingular reduction
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913376412631040 |
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| author | Wang, Haiyang |
| author_facet | Wang, Haiyang |
| contents | Let $\mathcal{O}_K$ be a Henselian discrete valuation domain with field of fractions $K$. Assume that $\mathcal{O}_K$ has algebraically closed residue field $k$. Let $E/K$ be an elliptic curve with additive reduction. The semi-stable reduction theorem asserts that there exists a minimal extension $L/K$ such that the base change $E_L/L$ has semi-stable reduction.
It is natural to wonder whether specific properties of the semi-stable reduction and of the extension $L/K$ impose restrictions on what types of Kodaira type the special fiber of $E/K$ may have. In this paper we study the restrictions imposed on the reduction type when the extension $L/K$ is wildly ramified of degree $2$, and the curve $E/K$ has potentially good supersingular reduction. We also analyze the possible reduction types of two isogenous elliptic curves with these properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01985 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Kodaira types of elliptic curves with potentially good supersingular reduction Wang, Haiyang Number Theory Algebraic Geometry Let $\mathcal{O}_K$ be a Henselian discrete valuation domain with field of fractions $K$. Assume that $\mathcal{O}_K$ has algebraically closed residue field $k$. Let $E/K$ be an elliptic curve with additive reduction. The semi-stable reduction theorem asserts that there exists a minimal extension $L/K$ such that the base change $E_L/L$ has semi-stable reduction. It is natural to wonder whether specific properties of the semi-stable reduction and of the extension $L/K$ impose restrictions on what types of Kodaira type the special fiber of $E/K$ may have. In this paper we study the restrictions imposed on the reduction type when the extension $L/K$ is wildly ramified of degree $2$, and the curve $E/K$ has potentially good supersingular reduction. We also analyze the possible reduction types of two isogenous elliptic curves with these properties. |
| title | On the Kodaira types of elliptic curves with potentially good supersingular reduction |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2406.01985 |