Bounds on the Chabauty-Kim Locus of Hyperbolic Curves
Conditionally on the Tate-Shafarevich and Bloch-Kato Conjectures, we give an explicit upper bound on the size of the $p$-adic Chabauty-Kim locus, and hence on the number of rational points, of a smooth projective curve $X/{\mathbb{Q}}$ of genus $g\geq 2$ in terms of $p$, $g$, the Mordell-Weil rank $...
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| Main Authors: | , , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
June 2024
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| In: |
International mathematics research notices
Year: 2024, Issue: 12, Pages: 9705-9727 |
| ISSN: | 1687-0247 |
| DOI: | 10.1093/imrn/rnae067 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1093/imrn/rnae067
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| Author Notes: | L. Alexander Betts, David Corwin and Marius Leonhardt |
| Summary: | Conditionally on the Tate-Shafarevich and Bloch-Kato Conjectures, we give an explicit upper bound on the size of the $p$-adic Chabauty-Kim locus, and hence on the number of rational points, of a smooth projective curve $X/{\mathbb{Q}}$ of genus $g\geq 2$ in terms of $p$, $g$, the Mordell-Weil rank $r$ of its Jacobian, and the reduction types of $X$ at bad primes. This is achieved using the effective Chabauty-Kim method, generalizing bounds found by Coleman and Balakrishnan-Dogra using the abelian and quadratic Chabauty methods. |
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| Item Description: | Online veröffentlicht: 18. April 2024 Gesehen am 22.07.2024 |
| Physical Description: | Online Resource |
| ISSN: | 1687-0247 |
| DOI: | 10.1093/imrn/rnae067 |