Arithmetic differential operators on a semistable model of ${\mathbb{P}}^1$

Abstract

In this paper we study sheaves of logarithmic arithmetic differential operators on a particular semistable model of the projective line. The main result here is that the first cohomology group of these sheaves is non-torsion. We also consider a refinement of the order filtration on the sheaf of level zero (before taking the p-adic completion). The associated graded sheaf, which we explicitly determine, explains to some extent the occurrence of the cohomology classes in degree one.

Added acknowledgement of support by the ANR program p-adic Hodge Theory and beyond (ThéHopaD)