$p$-adic Periods and Selmer Scheme Images
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908960405061632 |
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| author | Corwin, David Dan-Cohen, Ishai |
| author_facet | Corwin, David Dan-Cohen, Ishai |
| contents | The Chabauty--Kim method was developed with the aim of approaching effective Faltings', the problem of explicitly determining the finite set of rational points on a hyperbolic curve. This method has seen success with the more particular Quadratic Chabauty method, but this method still applies only to certain curves. Previous applications of Chabauty--Kim beyond the quadratic level, as pursued by the authors, by S. Wewers, and by others, use mixed Tate motives and the $p$-adic period map of Chatzistamatiou-Ünver to approach the particular hyperbolic curve $\mathbb{P}^1\setminus\{0,1,\infty\}$.
The main purpose of this article is to lay foundations for extending the above approach to more general hyperbolic curves, in particular by defining an analogous $p$-adic period map for more general categories of motives and their non-conjectural cousins such as systems of realizations and $p$-adic Galois representations. We use this to describe a general setup for non-abelian Chabauty for an arbitrary hyperbolic curve.
Our period map also connects the study of $p$-adic iterated integrals with Goncharov's theory of motivic iterated integrals, and allows us to investigate Goncharov's conjectures from a $p$-adic point of view. In particular, it suggests the possibility of evaluating syntomic regulators by writing elements of $K$-theory in terms of motivic iterated integrals. Lastly, it forms the basis for a certain generalization of the $p$-adic period conjecture of Yamashita for mixed Tate motives well-suited to applications in Chabauty--Kim theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16591 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $p$-adic Periods and Selmer Scheme Images Corwin, David Dan-Cohen, Ishai Number Theory Algebraic Geometry 19F27 (Primary) 11F80, 14C15, 14G05, 18M25, 14F30 (Secondary) The Chabauty--Kim method was developed with the aim of approaching effective Faltings', the problem of explicitly determining the finite set of rational points on a hyperbolic curve. This method has seen success with the more particular Quadratic Chabauty method, but this method still applies only to certain curves. Previous applications of Chabauty--Kim beyond the quadratic level, as pursued by the authors, by S. Wewers, and by others, use mixed Tate motives and the $p$-adic period map of Chatzistamatiou-Ünver to approach the particular hyperbolic curve $\mathbb{P}^1\setminus\{0,1,\infty\}$. The main purpose of this article is to lay foundations for extending the above approach to more general hyperbolic curves, in particular by defining an analogous $p$-adic period map for more general categories of motives and their non-conjectural cousins such as systems of realizations and $p$-adic Galois representations. We use this to describe a general setup for non-abelian Chabauty for an arbitrary hyperbolic curve. Our period map also connects the study of $p$-adic iterated integrals with Goncharov's theory of motivic iterated integrals, and allows us to investigate Goncharov's conjectures from a $p$-adic point of view. In particular, it suggests the possibility of evaluating syntomic regulators by writing elements of $K$-theory in terms of motivic iterated integrals. Lastly, it forms the basis for a certain generalization of the $p$-adic period conjecture of Yamashita for mixed Tate motives well-suited to applications in Chabauty--Kim theory. |
| title | $p$-adic Periods and Selmer Scheme Images |
| topic | Number Theory Algebraic Geometry 19F27 (Primary) 11F80, 14C15, 14G05, 18M25, 14F30 (Secondary) |
| url | https://arxiv.org/abs/2601.16591 |