A note on the ramification of torsion points lying on curves of genus at least two
Let $C$ be a curve of genus $g\ge 2$ defined over the fraction field $K$ of a complete discrete valuation ring $R$ with algebraically closed residue field. Suppose that $\operatorname{char}(K)=0$ and that the characteristic $p$ of the residue field is not $2$. Suppose that the Jacobian $\operatornam...
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| Format: | Journal article |
| Language: | English |
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Institut de Mathématiques de Bordeaux
2010
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| Summary: | Let $C$ be a curve of genus $g\ge 2$ defined over the fraction field $K$ of a complete discrete valuation ring $R$ with algebraically closed residue field. Suppose that $\operatorname{char}(K)=0$ and that the characteristic $p$ of the residue field is not $2$. Suppose that the Jacobian $\operatorname{Jac}(C)$ has semi-stable reduction over $R$. Embed $C$ in $\operatorname{Jac}(C)$ using a $K$-rational point. We show that the coordinates of the torsion points lying on $C$ lie in the unique tamely ramified quadratic extension of the field generated over $K$ by the coordinates of the $p$-torsion points on $\operatorname{Jac}(C)$. |
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