Admissible pairs and $p$-adic Hodge structures I: Transcendence of the de Rham lattice
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913117491953664 |
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| author | Howe, Sean Klevdal, Christian |
| author_facet | Howe, Sean Klevdal, Christian |
| contents | For an algebraically closed non-archimedean extension $C/\mathbb{Q}_p$, we define a Tannakian category of $p$-adic Hodge structures over $C$ that is a local, $p$-adic analog of the global, archimedean category of $\mathbb{Q}$-Hodge structures in complex geometry. In this setting the filtrations of classical Hodge theory must be enriched to lattices over a complete discrete valuation ring, Fontaine's integral de Rham period ring $B^+_\mathrm{dR}$, and a pure $p$-adic Hodge structure is then a $\mathbb{Q}_p$-vector space equipped with a $B^+_\mathrm{dR}$-lattice satisfying a natural condition analogous to the transversality of the complex Hodge filtration with its conjugate. We show $p$-adic Hodge structures are equivalent to a full subcategory of basic objects in the category of admissible pairs, a toy category of cohomological motives over $C$ that is equivalent to the isogeny category of rigidified Breuil-Kisin-Fargues modules and closely related to Fontaine's $p$-adic Hodge theory over $p$-adic subfields. As an application, we characterize basic admissible pairs with complex multiplication in terms of the transcendence of $p$-adic periods. This generalizes an earlier result for one-dimensional formal groups and is an unconditional, local, $p$-adic analog of a global, archimedean characterization of CM motives over $\mathbb{C}$ conditional on the standard conjectures, the Hodge conjecture, and the Grothendieck period conjecture (known unconditionally for abelian varieties by work Cohen and Shiga and Wolfart). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11065 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Admissible pairs and $p$-adic Hodge structures I: Transcendence of the de Rham lattice Howe, Sean Klevdal, Christian Number Theory Algebraic Geometry For an algebraically closed non-archimedean extension $C/\mathbb{Q}_p$, we define a Tannakian category of $p$-adic Hodge structures over $C$ that is a local, $p$-adic analog of the global, archimedean category of $\mathbb{Q}$-Hodge structures in complex geometry. In this setting the filtrations of classical Hodge theory must be enriched to lattices over a complete discrete valuation ring, Fontaine's integral de Rham period ring $B^+_\mathrm{dR}$, and a pure $p$-adic Hodge structure is then a $\mathbb{Q}_p$-vector space equipped with a $B^+_\mathrm{dR}$-lattice satisfying a natural condition analogous to the transversality of the complex Hodge filtration with its conjugate. We show $p$-adic Hodge structures are equivalent to a full subcategory of basic objects in the category of admissible pairs, a toy category of cohomological motives over $C$ that is equivalent to the isogeny category of rigidified Breuil-Kisin-Fargues modules and closely related to Fontaine's $p$-adic Hodge theory over $p$-adic subfields. As an application, we characterize basic admissible pairs with complex multiplication in terms of the transcendence of $p$-adic periods. This generalizes an earlier result for one-dimensional formal groups and is an unconditional, local, $p$-adic analog of a global, archimedean characterization of CM motives over $\mathbb{C}$ conditional on the standard conjectures, the Hodge conjecture, and the Grothendieck period conjecture (known unconditionally for abelian varieties by work Cohen and Shiga and Wolfart). |
| title | Admissible pairs and $p$-adic Hodge structures I: Transcendence of the de Rham lattice |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2308.11065 |