A specialisation theorem for Lang-Néron groups
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913536670695424 |
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| author | Kahn, Bruno Liu, Long |
| author_facet | Kahn, Bruno Liu, Long |
| contents | We show that, for a polarised smooth projective variety $B \hookrightarrow \mathbb{P}^n_k$ of dimension $\geq 2$ over an infinite field $k$ and an abelian variety $A$ over the function field of $B$, there exists a dense Zariski open set of smooth geometrically connected hyperplane sections $h$ of $B$ such that $A$ has good reduction at $h$ and the specialisation homomorphism of Lang-Néron groups at $h$ is injective (up to a finite $p$-group in positive characteristic $p$). This gives a positive answer to a conjecture of the first author, which is used to deduce a negative definiteness result on his refined height pairing. This also sheds a new light on Néron's specialisation theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06114 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A specialisation theorem for Lang-Néron groups Kahn, Bruno Liu, Long Algebraic Geometry Number Theory 11G99, 14K99 We show that, for a polarised smooth projective variety $B \hookrightarrow \mathbb{P}^n_k$ of dimension $\geq 2$ over an infinite field $k$ and an abelian variety $A$ over the function field of $B$, there exists a dense Zariski open set of smooth geometrically connected hyperplane sections $h$ of $B$ such that $A$ has good reduction at $h$ and the specialisation homomorphism of Lang-Néron groups at $h$ is injective (up to a finite $p$-group in positive characteristic $p$). This gives a positive answer to a conjecture of the first author, which is used to deduce a negative definiteness result on his refined height pairing. This also sheds a new light on Néron's specialisation theorem. |
| title | A specialisation theorem for Lang-Néron groups |
| topic | Algebraic Geometry Number Theory 11G99, 14K99 |
| url | https://arxiv.org/abs/2405.06114 |