$p$-Adic Weight Spectral Sequences of Strictly Semi-stable Schemes over Formal Power Series Rings via Arithmetic $\mathcal{D}$-modules

TL;DR

This paper constructs a $p$-adic weight spectral sequence for strictly semi-stable schemes over formal power series rings using arithmetic $ ext{D}$-modules, linking rigid cohomology and nearby cycles.

Contribution

It introduces a new $p$-adic spectral sequence framework for semi-stable schemes, connecting rigid cohomology with nearby cycle conjectures.

Findings

Constructed the weight spectral sequence in $p$-adic cohomology.

Described $E_1$ terms via rigid cohomologies of components.

Conjectured the description of $E_ $ terms by nearby cycles.

Abstract

Let $k$ be a perfect field of characteristic $p > 0$. For a strictly semi-stable scheme over $k[[t]]$, we construct the weight spectral sequence in $p$-adic cohomology using the theory of arithmetic $\mathcal{D}$-modules, whose $E_1$ terms are described by rigid cohomologies of irreducible components of the closed fiber and whose $E_\infty$ terms are conjecturally described by the (unipotent) nearby cycle of Lazda-P\'{a}l's rigid cohomology over the bounded Robba ring. We also show its functoriality by pushforward and state the conjecture of its functoriality by pullback and dual.