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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| _version_ | 1826274354476875776 |
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| author | Rössler, D |
| author_facet | Rössler, D |
| author_sort | Rössler, D |
| collection | OXFORD |
| description | 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 $\operatorname{Jac}(C)$. |
| first_indexed | 2024-03-06T22:42:10Z |
| format | Journal article |
| id | oxford-uuid:5bf2024b-ed94-47b8-8824-79d3a4b05601 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:42:10Z |
| publishDate | 2010 |
| publisher | Institut de Mathématiques de Bordeaux |
| record_format | dspace |
| spelling | oxford-uuid:5bf2024b-ed94-47b8-8824-79d3a4b056012022-03-26T17:25:09ZA note on the ramification of torsion points lying on curves of genus at least twoJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:5bf2024b-ed94-47b8-8824-79d3a4b05601EnglishSymplectic Elements at OxfordInstitut de Mathématiques de Bordeaux2010Rössler, DLet $C$ 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 $\operatorname{Jac}(C)$. |
| spellingShingle | Rössler, D A note on the ramification of torsion points lying on curves of genus at least two |
| title | A note on the ramification of torsion points lying on curves of genus at least two |
| title_full | A note on the ramification of torsion points lying on curves of genus at least two |
| title_fullStr | A note on the ramification of torsion points lying on curves of genus at least two |
| title_full_unstemmed | A note on the ramification of torsion points lying on curves of genus at least two |
| title_short | A note on the ramification of torsion points lying on curves of genus at least two |
| title_sort | note on the ramification of torsion points lying on curves of genus at least two |
| work_keys_str_mv | AT rosslerd anoteontheramificationoftorsionpointslyingoncurvesofgenusatleasttwo AT rosslerd noteontheramificationoftorsionpointslyingoncurvesofgenusatleasttwo |