Hasse principle violation for algebraic families of del Pezzo surfaces of degree 4 and hyperelliptic curves of genus congruent to 1 modulo 4
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908388165681152 |
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| author | Huang, Kai Liang, Yongqi |
| author_facet | Huang, Kai Liang, Yongqi |
| contents | Let g be a positive integer congruent to 1 modulo 4 and K be an arbitrary number field. We construct infinitely many explicit one-parameter algebraic families of degree 4 del Pezzo surfaces and of genus g hyperelliptic curves such that each K-member of the families violates the Hasse principle. In particular, we obtain algebraic families of non-trivial 2-torsion elements in the Tate-Shafarevich group of elliptic curves over K. These Hasse principle violations are explained by the Brauer-Manin obstruction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_11204 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hasse principle violation for algebraic families of del Pezzo surfaces of degree 4 and hyperelliptic curves of genus congruent to 1 modulo 4 Huang, Kai Liang, Yongqi Number Theory Algebraic Geometry Let g be a positive integer congruent to 1 modulo 4 and K be an arbitrary number field. We construct infinitely many explicit one-parameter algebraic families of degree 4 del Pezzo surfaces and of genus g hyperelliptic curves such that each K-member of the families violates the Hasse principle. In particular, we obtain algebraic families of non-trivial 2-torsion elements in the Tate-Shafarevich group of elliptic curves over K. These Hasse principle violations are explained by the Brauer-Manin obstruction. |
| title | Hasse principle violation for algebraic families of del Pezzo surfaces of degree 4 and hyperelliptic curves of genus congruent to 1 modulo 4 |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2312.11204 |