The Chabauty-Kim method for relative completions
<p>In this thesis we develop a Chabauty--Kim theory for the relative completion of motivic fundamental groups, including Selmer stacks and moduli spaces of admissible torsors for the relative completion of the de Rham fundamental group. On one hand, this work generalizes results of Kim (and th...
| প্রধান লেখক: | |
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| অন্যান্য লেখক: | |
| বিন্যাস: | গবেষণাপত্র |
| ভাষা: | English |
| প্রকাশিত: |
2019
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| বিষয়গুলি: |
| _version_ | 1826295631612739584 |
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| author | Kantor, N |
| author2 | Kim, M |
| author_facet | Kim, M Kantor, N |
| author_sort | Kantor, N |
| collection | OXFORD |
| description | <p>In this thesis we develop a Chabauty--Kim theory for the relative completion of motivic fundamental groups, including Selmer stacks and moduli spaces of admissible torsors for the relative completion of the de Rham fundamental group. On one hand, this work generalizes results of Kim (and therefore Chabauty) in the unipotent case by adding a reductive quotient of the fundamental group. From this perspective, the addition of a reductive part allows one to apply Chabauty-type methods to fundamental groups with trivial unipotent completion, such as <em>SL</em><sub>2</sub>(ℤ).</p>
<p>On the other hand, the unipotent part provides a natural extension of the recent work of Lawrence and Venkatesh. We show that their concern with the centralizer of Frobenius goes away as one moves up the unipotent tower and away from the reductive world of flag varieties and the Gauss--Manin connection. One is tempted to hope that the relative completion will provide a unified proof of Mordell's conjecture that takes advantage of the two methods. Toward this end, we apply our work to the Legendre family on the projective line minus three points, a particular example where the method of Lawrence and Venkatesh fails.</p>
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| first_indexed | 2024-03-07T04:04:01Z |
| format | Thesis |
| id | oxford-uuid:c588766f-2182-4f28-a3b4-7f81c4e66868 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T04:04:01Z |
| publishDate | 2019 |
| record_format | dspace |
| spelling | oxford-uuid:c588766f-2182-4f28-a3b4-7f81c4e668682022-03-27T06:31:43ZThe Chabauty-Kim method for relative completionsThesishttp://purl.org/coar/resource_type/c_db06uuid:c588766f-2182-4f28-a3b4-7f81c4e66868MathematicsNumber theoryArithmetic geometryEnglishHyrax Deposit2019Kantor, NKim, M<p>In this thesis we develop a Chabauty--Kim theory for the relative completion of motivic fundamental groups, including Selmer stacks and moduli spaces of admissible torsors for the relative completion of the de Rham fundamental group. On one hand, this work generalizes results of Kim (and therefore Chabauty) in the unipotent case by adding a reductive quotient of the fundamental group. From this perspective, the addition of a reductive part allows one to apply Chabauty-type methods to fundamental groups with trivial unipotent completion, such as <em>SL</em><sub>2</sub>(ℤ).</p> <p>On the other hand, the unipotent part provides a natural extension of the recent work of Lawrence and Venkatesh. We show that their concern with the centralizer of Frobenius goes away as one moves up the unipotent tower and away from the reductive world of flag varieties and the Gauss--Manin connection. One is tempted to hope that the relative completion will provide a unified proof of Mordell's conjecture that takes advantage of the two methods. Toward this end, we apply our work to the Legendre family on the projective line minus three points, a particular example where the method of Lawrence and Venkatesh fails.</p> |
| spellingShingle | Mathematics Number theory Arithmetic geometry Kantor, N The Chabauty-Kim method for relative completions |
| title | The Chabauty-Kim method for relative completions |
| title_full | The Chabauty-Kim method for relative completions |
| title_fullStr | The Chabauty-Kim method for relative completions |
| title_full_unstemmed | The Chabauty-Kim method for relative completions |
| title_short | The Chabauty-Kim method for relative completions |
| title_sort | chabauty kim method for relative completions |
| topic | Mathematics Number theory Arithmetic geometry |
| work_keys_str_mv | AT kantorn thechabautykimmethodforrelativecompletions AT kantorn chabautykimmethodforrelativecompletions |