An effective Chabauty–Kim theorem
The Chabauty–Kim method allows one to find rational points on curves under certain technical conditions, generalising Chabauty’s proof of the Mordell conjecture for curves with Mordell–Weil rank less than their genus. We show how the Chabauty–Kim method, when these technical conditions are satisfied...
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| Published in | Compositio mathematica Vol. 155; no. 6; pp. 1057 - 1075 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
London, UK
London Mathematical Society
01.06.2019
Cambridge University Press |
| Subjects | |
| Online Access | Get full text |
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| Abstract | The Chabauty–Kim method allows one to find rational points on curves under certain technical conditions, generalising Chabauty’s proof of the Mordell conjecture for curves with Mordell–Weil rank less than their genus. We show how the Chabauty–Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a non-abelian generalisation of Coleman’s effective Chabauty theorem. |
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| AbstractList | The Chabauty–Kim method allows one to find rational points on curves under certain technical conditions, generalising Chabauty’s proof of the Mordell conjecture for curves with Mordell–Weil rank less than their genus. We show how the Chabauty–Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a non-abelian generalisation of Coleman’s effective Chabauty theorem. |
| Author | Dogra, Netan Balakrishnan, Jennifer S. |
| Author_xml | – sequence: 1 givenname: Jennifer S. surname: Balakrishnan fullname: Balakrishnan, Jennifer S. email: jbala@bu.edu organization: Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, MA 02215, USA email jbala@bu.edu – sequence: 2 givenname: Netan surname: Dogra fullname: Dogra, Netan email: dogra@maths.ox.ac.uk organization: Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK email dogra@maths.ox.ac.uk |
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| Cites_doi | 10.1090/S0894-0347-10-00665-X 10.2977/prims/1234361156 10.1090/S0025-5718-2015-02927-5 10.1215/00127094-3673558 10.1006/jnth.1995.1095 10.1007/978-1-4612-1112-9 10.1007/s00208-007-0151-x 10.1007/978-3-642-59141-9 10.4171/JEMS/857 10.1215/0023608X-2010-015 10.1215/S0012-7094-85-05240-8 10.1515/crelle-2014-0048 10.1007/s00222-004-0433-9 |
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| Title | An effective Chabauty–Kim theorem |
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