Hasse-Arf filtrations in 𝑝-adic analytic geometry

Abstract

We develop a theory of local Fourier transforms for abelian sheaves on the étale site of a p p -adic punctured disc, and we prove a principle of stationary phase linking these local Fourier transforms to the global Fourier transform that was introduced in one of our earlier works. We use this theory to study the local monodromy of abelian sheaves on the étale site of a p p -adic punctured disc, and in particular we exhibit a natural slope decomposition for (germs of) such sheaves, with properties that are wholly analogous to those of the slope decomposition for representation of the Galois group of a local field. We conclude with an application to the study of the so-called cohomological epsilon factor of a locally constant abelian sheaf on the étale site of an affine curve defined over a p p -adic field: namely, we exhibit a decomposition of such factor as a tensor product of “local epsilon factors” that depend only on the local monodromies of the abelian sheaf.