Nilpotent orbit theorem in \(p\)-adic Hodge theory

• Published in: arXiv.org 2024-02
• Main Authors: Rahmati, Mohammad Reza, Flores, Gerardo
• Format: Article
• Language: English
• Subjects: Asymptotic properties; Orbits; Stability criteria
• ISSN: 2331-8422

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}].}