Nilpotent orbit theorem in $p$-adic Hodge theory

TL;DR

This paper establishes p-adic analogues of the orbit theorems in Hodge theory, demonstrating convergence of nilpotent orbits to semistable points and providing distance estimates using p-adic Fourier analysis.

Contribution

It introduces the nilpotent orbit theorem in p-adic Hodge theory, extending complex Hodge theory results to the p-adic setting with new convergence and distance estimates.

Findings

Nilpotent orbits in p-adic Hodge structures converge to semistable points.

Distance estimates between nilpotent orbits and twisted period maps are established.

The p-adic orbit theorems are analogous to classical complex Hodge theory results.

Abstract

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. \'etale) 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}].}