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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2309.02374 |
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| _version_ | 1866913350669041664 |
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| author | Morgan, Adam |
| author_facet | Morgan, Adam |
| contents | Conditional on finiteness of relevant Shafarevich--Tate groups, Harpaz and Skorobogatov used Swinnerton-Dyer's descent-fibration method to establish the Hasse principle for Kummer varieties associated to a 2-covering of a principally polarised abelian variety under certain largeness assumptions on its mod 2 Galois image. Their method breaks down however when the Galois image is maximal, due to the possible failure of the Shafarevich--Tate group of quadratic twists of A to have square order. In this work we overcome this obstruction by combining second descent ideas in the spirit of Harpaz and Smith with results on the parity of 2-infinity Selmer ranks in quadratic twist families. This allows Swinnerton-Dyer's method to be successfully applied to K3 surfaces arising as quotients of 2-coverings of Jacobians of genus 2 curves with no rational Weierstrass points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02374 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hasse principle for Kummer varieties in the case of generic 2-torsion Morgan, Adam Number Theory 14G05, 11G10 Conditional on finiteness of relevant Shafarevich--Tate groups, Harpaz and Skorobogatov used Swinnerton-Dyer's descent-fibration method to establish the Hasse principle for Kummer varieties associated to a 2-covering of a principally polarised abelian variety under certain largeness assumptions on its mod 2 Galois image. Their method breaks down however when the Galois image is maximal, due to the possible failure of the Shafarevich--Tate group of quadratic twists of A to have square order. In this work we overcome this obstruction by combining second descent ideas in the spirit of Harpaz and Smith with results on the parity of 2-infinity Selmer ranks in quadratic twist families. This allows Swinnerton-Dyer's method to be successfully applied to K3 surfaces arising as quotients of 2-coverings of Jacobians of genus 2 curves with no rational Weierstrass points. |
| title | Hasse principle for Kummer varieties in the case of generic 2-torsion |
| topic | Number Theory 14G05, 11G10 |
| url | https://arxiv.org/abs/2309.02374 |