The $p$-adic monodromy theorem over algebraic-affinoid algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914470967640064 |
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| author | Mikami, Yutaro |
| author_facet | Mikami, Yutaro |
| contents | In the previous paper of the author, motivated by the categorical $p$-adic local Langlands correspondence, the author studied families of $G_K$-equivariant vector bundles over the Fargues-Fontaine curve parametrized by algebraic-affinoid $\mathbb{Q}_{p,\square}$-algebras (e.g., $\mathbb{Q}_p[T]$). In this paper, we study the $p$-adic Hodge theoretic properties of such families. More precisely, we define the notions of Hodge-Tate, de Rham, and semistable representations for such families, and then prove the $p$-adic monodromy theorem ("de Rham" implies "potentially semistable") in this setting. This is a generalization of the work of Berger-Colmez. As an application, we prove the classification of families of $G_K$-equivariant line bundles. While a similar classification was previously obtained in the previous paper under a certain freeness condition by relying on the results of Kedlaya--Pottharst--Xiao, the approach in the present paper removes this freeness condition and entirely bypasses their results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_12280 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $p$-adic monodromy theorem over algebraic-affinoid algebras Mikami, Yutaro Number Theory Algebraic Geometry In the previous paper of the author, motivated by the categorical $p$-adic local Langlands correspondence, the author studied families of $G_K$-equivariant vector bundles over the Fargues-Fontaine curve parametrized by algebraic-affinoid $\mathbb{Q}_{p,\square}$-algebras (e.g., $\mathbb{Q}_p[T]$). In this paper, we study the $p$-adic Hodge theoretic properties of such families. More precisely, we define the notions of Hodge-Tate, de Rham, and semistable representations for such families, and then prove the $p$-adic monodromy theorem ("de Rham" implies "potentially semistable") in this setting. This is a generalization of the work of Berger-Colmez. As an application, we prove the classification of families of $G_K$-equivariant line bundles. While a similar classification was previously obtained in the previous paper under a certain freeness condition by relying on the results of Kedlaya--Pottharst--Xiao, the approach in the present paper removes this freeness condition and entirely bypasses their results. |
| title | The $p$-adic monodromy theorem over algebraic-affinoid algebras |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2604.12280 |