Sheaves of bounded $p$-adic logarithmic differential forms

TL;DR

This paper constructs and analyzes sheaves of bounded $p$-adic logarithmic differential forms on Drinfel'd symmetric spaces, providing new insights into their cohomology and confirming conjectures on $p$-adic Hodge decompositions in specific cases.

Contribution

It introduces a $G$-equivariant sheaf of lattices in $p$-adic differential forms and establishes criteria for conjectures on $p$-adic Hodge theory, proving them in certain instances.

Findings

Constructed $G$-equivariant sheaves of lattices in $p$-adic differential forms.

Derived criteria for conjectures on $p$-adic Hodge decompositions.

Proved the conjectures in specific cases for spaces uniformized by $X$.

Abstract

Let $K$ be a local field, $X$ the Drinfel'd symmetric space $X$ of dimension $d$ over $K$ and ${\mathfrak X}$ the natural formal ${\mathcal O}_K$-scheme underlying $X$; thus $G={\rm GL}\sb {d+1}(K)$ acts on $X$ and ${\mathfrak X}$. Given a $K$-rational $G$-representation $M$ we construct a $G$-equivariant subsheaf ${\mathcal M}^0_{{\mathcal O}_{\dot{K}}}$ of ${\mathcal O}_K$-lattices in the constant sheaf $M$ on ${\mathfrak X}$. We study the cohomology of sheaves of logarithmic differential forms on $X$ (or ${\mathfrak X}$) with coefficients in ${\mathcal M}^0_{{\mathcal O}_{\dot{K}}}$. In the second part we give general criteria for two conjectures of P. Schneider on $p$-adic Hodge decompositions of the cohomology of $p$-adic local systems on projective varieties uniformized by $X$. Applying the results of the first part we prove the conjectures in certain cases.