Représentations galoisiennes et opérateurs de Bessel p-adiques  [ Galois representations and p-adic Bessel operators ]
Annales de l'Institut Fourier, Volume 52 (2002) no. 3, p. 779-808

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.

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 ((1/x))). 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.

DOI : https://doi.org/10.5802/aif.1901
Classification:  12H25,  11S15
Keywords: p-adic representations, Bessel equations, p-adic monodromy
@article{AIF_2002__52_3_779_0,
     author = {Andr\'e, Yves},
     title = {Repr\'esentations galoisiennes et op\'erateurs de Bessel $p$-adiques},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {3},
     year = {2002},
     pages = {779-808},
     doi = {10.5802/aif.1901},
     zbl = {1014.12007},
     mrnumber = {1907387},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2002__52_3_779_0}
}
André, Yves. Représentations galoisiennes et opérateurs de Bessel $p$-adiques. Annales de l'Institut Fourier, Volume 52 (2002) no. 3, pp. 779-808. doi : 10.5802/aif.1901. http://www.numdam.org/item/AIF_2002__52_3_779_0/

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