Ramified periods and field of definition
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912312815779840 |
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| author | Ancona, Giuseppe Frăţilă, Dragoş Vezzani, Alberto |
| author_facet | Ancona, Giuseppe Frăţilă, Dragoş Vezzani, Alberto |
| contents | Let $L/K$ be an extension of number fields that is ramified above $p$. We give a new obstruction to the descent to $K$ of smooth projective varieties defined over $L$. The obstruction is a matrix of $p$-adic numbers that we call ``ramified periods'' arising from the comparison isomorphism between de Rham cohomology and crystalline cohomology. As an application, we give simple examples of hyperelliptic curves over $\mathbb{Q}(\sqrt p)$ that are isomorphic to their Galois conjugates but such that their Jacobians do not descend to $\mathbb{Q}$ even up to isogeny. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04484 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ramified periods and field of definition Ancona, Giuseppe Frăţilă, Dragoş Vezzani, Alberto Algebraic Geometry Number Theory Let $L/K$ be an extension of number fields that is ramified above $p$. We give a new obstruction to the descent to $K$ of smooth projective varieties defined over $L$. The obstruction is a matrix of $p$-adic numbers that we call ``ramified periods'' arising from the comparison isomorphism between de Rham cohomology and crystalline cohomology. As an application, we give simple examples of hyperelliptic curves over $\mathbb{Q}(\sqrt p)$ that are isomorphic to their Galois conjugates but such that their Jacobians do not descend to $\mathbb{Q}$ even up to isogeny. |
| title | Ramified periods and field of definition |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2504.04484 |