Représentations galoisiennes et opérateurs de Bessel $p$-adiques

Abstract

We deal with the links between p-adic differential equations and p-adic representations of local fields of characteristic p, focusing on the Bessel case. We prove that every (normalized) p-adic Bessel equation, on a thin annulus at the boundary of the disk at infinity, becomes trivial over some finite etale covering of that annulus coming from a separable finie extension of 𝔽 p ((1/x)). The difficult case is p=2; we give explicit constructions of that covering and of the corresponding diadic Galois representation in terms of the crystalline cohomology of a certain superelliptic curve over 𝔽 4 . This dismisses in particular a surmized counterexample of Mebkhout to Crew’s p-adic local monodromy conjecture.