Représentations galoisiennes et opérateurs de Bessel $p$-adiques

Nous traitons des liens entre équations différentielles $p$-adiques et représentations $p$-adiques de corps locaux de caractéristique $p$, en nous concentrant sur le cas Bessel. Nous démontrons que toute équation de Bessel $p$-adique normalisée à la Dwork, sur une fine couronne au bord du disque à l’infini, se trivialise sur un certain revêtement étale de cette couronne (revêtement provenant d’une extension finie séparable de ${𝔽}_p\left((1/x)\right)$). Le cas difficile est $p=2$, et nous explicitons complètement le revêtement et la représentation galoisienne diadique correspondante en termes de la cohomologie cristalline d’une certaine courbe elliptique supersingulière sur ${𝔽}_4$. Cela élimine en particulier un contrexemple putatif de Mebkhout à la conjecture de monodromie locale $p$-adique de Crew.

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 ${𝔽}_p\left((1/x)\right)$. 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.

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