Swan conductors for p-adic differential modules, II: Global variation

TL;DR

This paper introduces a numerical invariant called the differential Swan conductor for p-adic differential modules on varieties over positive characteristic fields, exploring its properties and variations, especially for surfaces.

Contribution

It extends the concept of Swan conductors to a global setting for overconvergent isocrystals and p-adic representations, analyzing their variation properties.

Findings

Defined the differential Swan conductor for overconvergent isocrystals.

Analyzed the variational behavior of the invariant on surfaces.

Connected the invariant to p-adic and l-adic representations.

Abstract

Using a local construction from a previous paper, we exhibit a numerical invariant, the differential Swan conductor, for an isocrystal on a variety over a perfect field of positive characteristic overconvergent along a boundary divisor; this leads to an analogous construction for certain p-adic and l-adic representations of the etale fundamental group of a variety. We then demonstrate some variational properties of this definition for overconvergent isocrystals, paying special attention to the case of surfaces.