Représentations galoisiennes et opérateurs de Bessel $p$-adiques

André, Yves

Représentations galoisiennes et opérateurs de Bessel

p

-adiques [ Galois representations and

p

-adic Bessel operators ]

André, Yves

Annales de l'Institut Fourier, Volume 52 (2002) no. 3, p. 779-808

Abstract
Résumé

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.

MR 1907387 | Zbl 1014.12007 | 2 citations in Numdam

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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