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 &#8474; whose points parameterize genus 2 curves <em>C</em> with full √3-level structure on their Jacobian <em>J</em>. We...

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Những tác giả chính: Bruin, N, Flynn, EV, Shnidman, A
Định dạng: Journal article
Ngôn ngữ:English
Được phát hành: Springer 2023
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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 &#8474; 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>(&#8474;), 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>(&#8474;), as well as statistical results on the size of #<em>C<sub>d</sub></em>(&#8474;), as <em>d</em> varies through squarefree integers.</p>
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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 &#8474; 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>(&#8474;), 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>(&#8474;), as well as statistical results on the size of #<em>C<sub>d</sub></em>(&#8474;), 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