Nilpotent orbit theorem in $p$-adic Hodge theory
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arXiv
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| Natura: | Preprint |
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2016
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| _version_ | 1866929231991144448 |
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| author | Rahmati, Mohammad Reza Flores, Gerardo |
| author_facet | Rahmati, Mohammad Reza Flores, Gerardo |
| contents | We state and prove three orbit theorems on the period domains for the $p$-adic Hodge structure analogous to the complex case. We shall consider the variation of de Rham (resp. étale) cohomology in a family of projective varieties $f:\mathfrak{X} \to S$ defined over a p-adic field. First, we show that any nilpotent orbit in the period domain of p-adic Hodge structures converges to a semistable point (filtration) in the period domain of the p-adic Hodge structure. Furthermore, the nilpotent orbits of the limit point are asymptotic to the twisted period map [Theorem \ref{thm:nilpotent-orbit}]. The orbit theorems come with some estimates of the distance between the nilpotent orbit and the twisted period map. The distance estimate in the p-adic nilpotent orbit theorem is given concerning the non-archimedean metric and is based on the p-adic Fourier analysis of Amice-Schneider. The result is analogous to the orbit theorems of W. Schmid [\cite{Sch}-1973] on complex Hodge structures. Our proof is based on a \textit{Geometric Invariant Theory} (GIT) criterion for semi-stability (Kempf-Ness theorem) and estimates from the (Amice-Schneider) p-adic Fourier theory. We also state the $SL_2$-orbit theorem in the p-adic case, [Theorem \ref{th:homomorphism}]. Finally, we explain how the nilpotent orbit theorem should be modified and stated for a variation of the mixed Hodge structure [Theorem \ref{thm:mixed-orbit}].} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1612_03367 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Nilpotent orbit theorem in $p$-adic Hodge theory Rahmati, Mohammad Reza Flores, Gerardo Number Theory We state and prove three orbit theorems on the period domains for the $p$-adic Hodge structure analogous to the complex case. We shall consider the variation of de Rham (resp. étale) cohomology in a family of projective varieties $f:\mathfrak{X} \to S$ defined over a p-adic field. First, we show that any nilpotent orbit in the period domain of p-adic Hodge structures converges to a semistable point (filtration) in the period domain of the p-adic Hodge structure. Furthermore, the nilpotent orbits of the limit point are asymptotic to the twisted period map [Theorem \ref{thm:nilpotent-orbit}]. The orbit theorems come with some estimates of the distance between the nilpotent orbit and the twisted period map. The distance estimate in the p-adic nilpotent orbit theorem is given concerning the non-archimedean metric and is based on the p-adic Fourier analysis of Amice-Schneider. The result is analogous to the orbit theorems of W. Schmid [\cite{Sch}-1973] on complex Hodge structures. Our proof is based on a \textit{Geometric Invariant Theory} (GIT) criterion for semi-stability (Kempf-Ness theorem) and estimates from the (Amice-Schneider) p-adic Fourier theory. We also state the $SL_2$-orbit theorem in the p-adic case, [Theorem \ref{th:homomorphism}]. Finally, we explain how the nilpotent orbit theorem should be modified and stated for a variation of the mixed Hodge structure [Theorem \ref{thm:mixed-orbit}].} |
| title | Nilpotent orbit theorem in $p$-adic Hodge theory |
| topic | Number Theory |
| url | https://arxiv.org/abs/1612.03367 |