Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866914919497072640 |
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| author | Obus, Andrew Srinivasan, Padmavathi |
| author_facet | Obus, Andrew Srinivasan, Padmavathi |
| contents | We prove an inequality between the conductor and the discriminant for all hyperelliptic curves defined over discretely valued fields $K$ with perfect residue field of characteristic not 2. Specifically, if such a curve is given by $y^2 = f(x)$ with $f(x) \in \mathcal{O}_K[x]$, and if $X$ is its minimal regular model over $\mathcal{O}_K$, then the negative of the Artin conductor of $X$ (and thus also the number of irreducible components of the special fiber of $X$) is bounded above by the valuation of disc$(f)$. There are no restrictions on genus of the curve or on the ramification of the splitting field of $f$. This generalizes earlier work of Ogg, Saito, Liu, and the second author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_02589 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic Obus, Andrew Srinivasan, Padmavathi Algebraic Geometry Number Theory 11G20, 14H25, 14J17 (Primary), 14B05 (Secondary) We prove an inequality between the conductor and the discriminant for all hyperelliptic curves defined over discretely valued fields $K$ with perfect residue field of characteristic not 2. Specifically, if such a curve is given by $y^2 = f(x)$ with $f(x) \in \mathcal{O}_K[x]$, and if $X$ is its minimal regular model over $\mathcal{O}_K$, then the negative of the Artin conductor of $X$ (and thus also the number of irreducible components of the special fiber of $X$) is bounded above by the valuation of disc$(f)$. There are no restrictions on genus of the curve or on the ramification of the splitting field of $f$. This generalizes earlier work of Ogg, Saito, Liu, and the second author. |
| title | Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic |
| topic | Algebraic Geometry Number Theory 11G20, 14H25, 14J17 (Primary), 14B05 (Secondary) |
| url | https://arxiv.org/abs/1910.02589 |