A uniform quantitative Manin-Mumford theorem for curves over function fields
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866911397379571712 |
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| author | Looper, Nicole Silverman, Joseph Wilms, Robert |
| author_facet | Looper, Nicole Silverman, Joseph Wilms, Robert |
| contents | We prove that any smooth projective geometrically connected non-isotrivial curve of genus $g\ge 2$ over a one-dimensional function field of any characteristic has at most $16g^2+32g+124$ torsion points for any Abel-Jacobi embedding of the curve into its Jacobian. The proof uses Zhang's admissible pairing on curves, the arithmetic Hodge index theorem over function fields, and the metrized graph analogue of Elkies' lower bound for the Green function. More generally, we prove an explicit Bogomolov-type result bounding the number of geometric points of small Néron-Tate height on the curve embedded into its Jacobian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_11593 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A uniform quantitative Manin-Mumford theorem for curves over function fields Looper, Nicole Silverman, Joseph Wilms, Robert Number Theory Algebraic Geometry 11G30, 11G50 We prove that any smooth projective geometrically connected non-isotrivial curve of genus $g\ge 2$ over a one-dimensional function field of any characteristic has at most $16g^2+32g+124$ torsion points for any Abel-Jacobi embedding of the curve into its Jacobian. The proof uses Zhang's admissible pairing on curves, the arithmetic Hodge index theorem over function fields, and the metrized graph analogue of Elkies' lower bound for the Green function. More generally, we prove an explicit Bogomolov-type result bounding the number of geometric points of small Néron-Tate height on the curve embedded into its Jacobian. |
| title | A uniform quantitative Manin-Mumford theorem for curves over function fields |
| topic | Number Theory Algebraic Geometry 11G30, 11G50 |
| url | https://arxiv.org/abs/2101.11593 |