Inscription, twistors, and $p$-adic periods
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910021331189760 |
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| author | Howe, Sean |
| author_facet | Howe, Sean |
| contents | We introduce the theory of inscribed $v$-sheaves, a differentiable extension of the theory of diamonds and $v$-sheaves with internal tangent bundles that are often relative inscribed Banach-Colmez spaces, then apply this theory to the study of $p$-adic periods. In particular, we construct natural inscribed versions of the Hodge and Hodge-Tate period maps and their lattice refinements for de Rham torsors, then compute the derivatives of these period maps in terms of classical structures in $p$-adic Hodge theory. These torsors include infinite level global Shimura varieties and infinite level local Shimura varieties, and for these spaces we also give another moduli-theoretic construction of the inscribed structure; the construction in the local Shimura case applies more generally to the non-minuscule moduli of mixed characterisic local shtukas with one leg. The key new ingredients in our study of inscribed structures on $p$-adic Lie group torsors over smooth rigid varieties over a $p$-adic field are the Liu-Zhu period map, a refinement of the Hodge period map whose derivative is the geometric Sen morphism/canonical Higgs field, and a closely related exact tensor functor from $\mathbb{Q}_p$-local systems to a category of twistor bundles on the relative thickened Fargues-Fontaine curve. These new structures are only visible after passing to the inscribed setting. We also discuss some possible implications of our computations in the vein of ``differential topology for diamonds." |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11589 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inscription, twistors, and $p$-adic periods Howe, Sean Number Theory Algebraic Geometry We introduce the theory of inscribed $v$-sheaves, a differentiable extension of the theory of diamonds and $v$-sheaves with internal tangent bundles that are often relative inscribed Banach-Colmez spaces, then apply this theory to the study of $p$-adic periods. In particular, we construct natural inscribed versions of the Hodge and Hodge-Tate period maps and their lattice refinements for de Rham torsors, then compute the derivatives of these period maps in terms of classical structures in $p$-adic Hodge theory. These torsors include infinite level global Shimura varieties and infinite level local Shimura varieties, and for these spaces we also give another moduli-theoretic construction of the inscribed structure; the construction in the local Shimura case applies more generally to the non-minuscule moduli of mixed characterisic local shtukas with one leg. The key new ingredients in our study of inscribed structures on $p$-adic Lie group torsors over smooth rigid varieties over a $p$-adic field are the Liu-Zhu period map, a refinement of the Hodge period map whose derivative is the geometric Sen morphism/canonical Higgs field, and a closely related exact tensor functor from $\mathbb{Q}_p$-local systems to a category of twistor bundles on the relative thickened Fargues-Fontaine curve. These new structures are only visible after passing to the inscribed setting. We also discuss some possible implications of our computations in the vein of ``differential topology for diamonds." |
| title | Inscription, twistors, and $p$-adic periods |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2508.11589 |