Genus two curves with full √3-level structure and Tate-Shafarevich groups
<p>We give an explicit rational parameterization of the surface <strong><em>H</em><sub>3</sub></strong> over ℚ whose points parameterize genus 2 curves <em>C</em> with full √3-level structure on their Jacobian <em>J</em>. We...
| Autori principali: | , , |
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| Natura: | Journal article |
| Lingua: | English |
| Pubblicazione: |
Springer
2023
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| _version_ | 1826312948280197120 |
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| author | Bruin, N Flynn, EV Shnidman, A |
| author_facet | Bruin, N Flynn, EV Shnidman, A |
| author_sort | Bruin, N |
| collection | OXFORD |
| description | <p>We give an explicit rational parameterization of the surface <strong><em>H</em><sub>3</sub></strong> over ℚ whose points parameterize genus 2 curves <em>C</em> with full √3-level structure on their Jacobian <em>J</em>. We use this model to construct abelian surfaces <em>A</em> with the property that III (<em>A<sub>d</sub></em>) [3] ≠ 0 for a positive proportion of quadratic twists <em>A<sub>d</sub></em>. In fact, for 100% of <em>x</em>∈<strong><em>H</em><sub>3</sub></strong>(ℚ), this holds for the surface <em>A</em> = Jac(<em>C<sub>x</sub></em>)/⟨<em>P</em>⟩, where <em>P</em> is the marked point of order 3. Our methods also give an explicit bound on the average rank of <em>J<sub>d</sub></em>(ℚ), as well as statistical results on the size of #<em>C<sub>d</sub></em>(ℚ), as <em>d</em> varies through squarefree integers.</p> |
| first_indexed | 2024-03-07T07:52:44Z |
| format | Journal article |
| id | oxford-uuid:3f1e032d-3524-4283-bbdd-15c01d727a3a |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:04:55Z |
| publishDate | 2023 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:3f1e032d-3524-4283-bbdd-15c01d727a3a2024-05-20T09:52:04ZGenus two curves with full √3-level structure and Tate-Shafarevich groupsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:3f1e032d-3524-4283-bbdd-15c01d727a3aEnglishSymplectic ElementsSpringer2023Bruin, NFlynn, EVShnidman, A<p>We give an explicit rational parameterization of the surface <strong><em>H</em><sub>3</sub></strong> over ℚ whose points parameterize genus 2 curves <em>C</em> with full √3-level structure on their Jacobian <em>J</em>. We use this model to construct abelian surfaces <em>A</em> with the property that III (<em>A<sub>d</sub></em>) [3] ≠ 0 for a positive proportion of quadratic twists <em>A<sub>d</sub></em>. In fact, for 100% of <em>x</em>∈<strong><em>H</em><sub>3</sub></strong>(ℚ), this holds for the surface <em>A</em> = Jac(<em>C<sub>x</sub></em>)/⟨<em>P</em>⟩, where <em>P</em> is the marked point of order 3. Our methods also give an explicit bound on the average rank of <em>J<sub>d</sub></em>(ℚ), as well as statistical results on the size of #<em>C<sub>d</sub></em>(ℚ), as <em>d</em> varies through squarefree integers.</p> |
| spellingShingle | Bruin, N Flynn, EV Shnidman, A Genus two curves with full √3-level structure and Tate-Shafarevich groups |
| title | Genus two curves with full √3-level structure and Tate-Shafarevich groups |
| title_full | Genus two curves with full √3-level structure and Tate-Shafarevich groups |
| title_fullStr | Genus two curves with full √3-level structure and Tate-Shafarevich groups |
| title_full_unstemmed | Genus two curves with full √3-level structure and Tate-Shafarevich groups |
| title_short | Genus two curves with full √3-level structure and Tate-Shafarevich groups |
| title_sort | genus two curves with full √3 level structure and tate shafarevich groups |
| work_keys_str_mv | AT bruinn genustwocurveswithfull3levelstructureandtateshafarevichgroups AT flynnev genustwocurveswithfull3levelstructureandtateshafarevichgroups AT shnidmana genustwocurveswithfull3levelstructureandtateshafarevichgroups |