Sheaves of bounded $p$-adic logarithmic differential forms

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 ${\mathcal{M}}_{{\mathcal{O}}_{\dot{K}}}^0$ 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 ${\mathcal{M}}_{{\mathcal{O}}_{\dot{K}}}^0$. 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.

References in corpus (1)

• Integral structures in the $p$-adic holomorphic discrete series