Nous traitons des liens entre équations différentielles -adiques et représentations -adiques de corps locaux de caractéristique , en nous concentrant sur le cas Bessel. Nous démontrons que toute équation de Bessel -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 ). Le cas difficile est , 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 . Cela élimine en particulier un contrexemple putatif de Mebkhout à la conjecture de monodromie locale -adique de Crew.
We deal with the links between -adic differential equations and -adic representations of local fields of characteristic , focusing on the Bessel case. We prove that every (normalized) -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 . The difficult case is ; 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 . This dismisses in particular a surmized counterexample of Mebkhout to Crew’s -adic local monodromy conjecture.
Classification : 12H25, 11S15
Mots clés : représentations -adiques, équations de Bessel, monodromie -adique
@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}, pages = {779--808}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {52}, number = {3}, year = {2002}, doi = {10.5802/aif.1901}, zbl = {1014.12007}, mrnumber = {1907387}, language = {fr}, url = {http://archive.numdam.org/articles/10.5802/aif.1901/} }
TY - JOUR AU - André, Yves TI - Représentations galoisiennes et opérateurs de Bessel $p$-adiques JO - Annales de l'Institut Fourier PY - 2002 DA - 2002/// SP - 779 EP - 808 VL - 52 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.1901/ UR - https://zbmath.org/?q=an%3A1014.12007 UR - https://www.ams.org/mathscinet-getitem?mr=1907387 UR - https://doi.org/10.5802/aif.1901 DO - 10.5802/aif.1901 LA - fr ID - AIF_2002__52_3_779_0 ER -
André, Yves. Représentations galoisiennes et opérateurs de Bessel $p$-adiques. Annales de l'Institut Fourier, Tome 52 (2002) no. 3, pp. 779-808. doi : 10.5802/aif.1901. http://archive.numdam.org/articles/10.5802/aif.1901/
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