Geometrically simple counterexamples to a local-global principle for quadratic twists
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arXiv
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| Format: | Preprint |
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2025
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| author | Ambrosi, Emiliano Coppola, Nirvana Fité, Francesc |
| author_facet | Ambrosi, Emiliano Coppola, Nirvana Fité, Francesc |
| contents | Two abelian varieties $A$ and $B$ over a number field $K$ are said to be strongly locally quadratic twists if they are quadratic twists at every completion of $K$. While it was known that this does not imply that $A$ and $B$ are quadratic twists over $K$, the only known counterexamples (necessarily of dimension $\geq 4$) are not geometrically simple. We show that, for every prime $p\equiv 13 \pmod{24}$, there exists a pair of geometrically simple abelian varieties of dimension $p-1$ over $\mathbb{Q}$ that are strongly locally quadratic twists but not quadratic twists. The proof is based on Galois cohomology computations and class field theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04803 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometrically simple counterexamples to a local-global principle for quadratic twists Ambrosi, Emiliano Coppola, Nirvana Fité, Francesc Number Theory Algebraic Geometry 11G10, 11G35, 14G20, 14G25 Two abelian varieties $A$ and $B$ over a number field $K$ are said to be strongly locally quadratic twists if they are quadratic twists at every completion of $K$. While it was known that this does not imply that $A$ and $B$ are quadratic twists over $K$, the only known counterexamples (necessarily of dimension $\geq 4$) are not geometrically simple. We show that, for every prime $p\equiv 13 \pmod{24}$, there exists a pair of geometrically simple abelian varieties of dimension $p-1$ over $\mathbb{Q}$ that are strongly locally quadratic twists but not quadratic twists. The proof is based on Galois cohomology computations and class field theory. |
| title | Geometrically simple counterexamples to a local-global principle for quadratic twists |
| topic | Number Theory Algebraic Geometry 11G10, 11G35, 14G20, 14G25 |
| url | https://arxiv.org/abs/2501.04803 |