Generalised height pairings and the Albanese kernel
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908963440689152 |
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| author | Dogra, Netan |
| author_facet | Dogra, Netan |
| contents | The Chabauty--Coleman--Kim method in depth two describes the rational points on a curve in terms of a generalisation of Nekovář's $p$-adic height pairing which replaces $\mathbb{G}_m$ with a higher Chow group. It is unclear both what the domain of definition of this pairing is, and how to compute it. This paper explores the relevance of the Beilinson--Bloch conjectures to this problem. In particular, it is shown that if $X$ is a smooth projective curve and the Albanese kernel of $X\times X$ is torsion, then there is an algorithm to compute the generalised height pairing on a pair of rational points on the Jacobian. This leads to the consideration of certain `motivic refinements' of the nonabelian cohomology varieties which arise in nonabelian Chabauty. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10111 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalised height pairings and the Albanese kernel Dogra, Netan Number Theory Algebraic Geometry The Chabauty--Coleman--Kim method in depth two describes the rational points on a curve in terms of a generalisation of Nekovář's $p$-adic height pairing which replaces $\mathbb{G}_m$ with a higher Chow group. It is unclear both what the domain of definition of this pairing is, and how to compute it. This paper explores the relevance of the Beilinson--Bloch conjectures to this problem. In particular, it is shown that if $X$ is a smooth projective curve and the Albanese kernel of $X\times X$ is torsion, then there is an algorithm to compute the generalised height pairing on a pair of rational points on the Jacobian. This leads to the consideration of certain `motivic refinements' of the nonabelian cohomology varieties which arise in nonabelian Chabauty. |
| title | Generalised height pairings and the Albanese kernel |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2507.10111 |