Nontrivial Sha in the Jacobian of an Infinite Family of Curves of Genus 2.
We give an infinite family of curves of genus 2 whose Jacobians have non-trivial members of the Tate-Shafarevich group for descent via Richelot isogeny. We prove this by performing a descent via Richelot isogeny and a complete 2-descent on the isogenous Jacobian. We also give an explicit model of an...
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| Format: | Journal article |
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2009
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| _version_ | 1797093113340100608 |
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| author | Arnth-Jensen, A Flynn, E |
| author_facet | Arnth-Jensen, A Flynn, E |
| author_sort | Arnth-Jensen, A |
| collection | OXFORD |
| description | We give an infinite family of curves of genus 2 whose Jacobians have non-trivial members of the Tate-Shafarevich group for descent via Richelot isogeny. We prove this by performing a descent via Richelot isogeny and a complete 2-descent on the isogenous Jacobian. We also give an explicit model of an associated family of surfaces which violate the Hasse principle. |
| first_indexed | 2024-03-07T03:55:40Z |
| format | Journal article |
| id | oxford-uuid:c2cb56c2-1d31-47c7-80d6-daecefe49a1e |
| institution | University of Oxford |
| last_indexed | 2024-03-07T03:55:40Z |
| publishDate | 2009 |
| record_format | dspace |
| spelling | oxford-uuid:c2cb56c2-1d31-47c7-80d6-daecefe49a1e2022-03-27T06:11:29ZNontrivial Sha in the Jacobian of an Infinite Family of Curves of Genus 2.Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:c2cb56c2-1d31-47c7-80d6-daecefe49a1eMathematical Institute - ePrints2009Arnth-Jensen, AFlynn, EWe give an infinite family of curves of genus 2 whose Jacobians have non-trivial members of the Tate-Shafarevich group for descent via Richelot isogeny. We prove this by performing a descent via Richelot isogeny and a complete 2-descent on the isogenous Jacobian. We also give an explicit model of an associated family of surfaces which violate the Hasse principle. |
| spellingShingle | Arnth-Jensen, A Flynn, E Nontrivial Sha in the Jacobian of an Infinite Family of Curves of Genus 2. |
| title | Nontrivial Sha in the Jacobian of an Infinite Family of Curves of Genus 2. |
| title_full | Nontrivial Sha in the Jacobian of an Infinite Family of Curves of Genus 2. |
| title_fullStr | Nontrivial Sha in the Jacobian of an Infinite Family of Curves of Genus 2. |
| title_full_unstemmed | Nontrivial Sha in the Jacobian of an Infinite Family of Curves of Genus 2. |
| title_short | Nontrivial Sha in the Jacobian of an Infinite Family of Curves of Genus 2. |
| title_sort | nontrivial sha in the jacobian of an infinite family of curves of genus 2 |
| work_keys_str_mv | AT arnthjensena nontrivialshainthejacobianofaninfinitefamilyofcurvesofgenus2 AT flynne nontrivialshainthejacobianofaninfinitefamilyofcurvesofgenus2 |