Représentations galoisiennes et opérateurs de Bessel $p$-adiques

André, Yves

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

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.

[AS] A. Adolphson; S. Sperber Twisted Kloosterman sums and p-adic Bessel functions, Amer. J. Math, Volume 106 (1984), pp. 549-591 | DOI | MR | Zbl

[B] P. Berthelot Cohomologie rigide et théorie de Dwork : le cas des sommes exponentielles (Astérisque), Volume No 119-120 (1984), pp. 17-49 | Numdam | Zbl

[Ba] F. Baldassarri Differential modules and singular points of p-adic differential equations, Advances in Math, Volume 44 (1982), pp. 155-179 | DOI | MR | Zbl

[Be] F. Beukers Differential Galois theory, From Number Theory to Physics, chap. 8, Springer-Verlag ed., 1995 | MR | Zbl

[Bo] N. Bourbaki Algèbre commutative, chapitres I, VII, Masson, 1985

[CD] G. Christol; B. Dwork Modules différentiels sur des couronnes, Ann. Inst. Fourier, Grenoble, Volume 44 (1994) no. 3, pp. 663-701 | DOI | Numdam | MR | Zbl

[CM2] G. Christol; Z. Mebkhout Sur le théorème de l'indice des équations différentielles p-adiques II, Ann. of Maths, Volume 146 (1997), pp. 345-410 | DOI | MR | Zbl

[CM3] G. Christol; Z. Mebkhout Sur le théorème de l'indice des équations différentielles p-adiques III, Ann. of Maths, Volume 151 (2000), pp. 385-457 | DOI | MR | Zbl

[Co] H.S.M. Coxeter Complex regular polytopes, Cambridge Univ. Press, 1991 | MR | Zbl

[Cr1] R. Crew $F$-isocrystals and $p$-adic representations, Algebraic Geometry - Bowdoin 1985 (Proc. Symp. Pure Math. (2)), Volume XLVI (1987), pp. 111-138 | Zbl

[Cr2] R. Crew Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve, Ann. Scient. Éc. Norm. Sup, Volume 31 (1998), pp. 717-763 | Numdam | MR | Zbl

[Cr3] R. Crew Canonical extensions, irregularities, and the Swan conductor, Math. Ann., Volume 316 (2000), pp. 19-37 | DOI | MR | Zbl

[D] B. Dwork Bessel functions as $p$-adic functions of their argument, Duke Math. J, Volume 41 (1974), pp. 711-738 | DOI | MR | Zbl

[F] J.M. Fontaine Représentations $p$-adiques des corps locaux, Grothendieck Festschrift II (Progress in Math.), Volume 87 (1990), pp. 249-309 | Zbl

[FH] W. Fulton; J. Harris Representation theory, GTM, 129, Springer, 1991 | MR | Zbl

[H] D. Husemöller Elliptic curves, GTM, 111, Springer, 1987 | MR | Zbl

[Hy] O. Hyodo A cohomological construction of Swan representations over the Witt ring II, Proc. Japan Acad, Volume 64A (1988), pp. 350-351 | MR | Zbl

[K1] N. Katz On the calculation of some differential Galois groups, Invent. Math, Volume 87 (1987), pp. 13-61 | DOI | MR | Zbl

[K2] N. Katz Local-to-global extensions of representations of fundamental groups, Ann. Inst. Fourier, Grenoble, Volume 36 (1986) no. 4, pp. 69-106 | DOI | Numdam | MR | Zbl

[K3] N. Katz Gauss sums, Kloosterman sums, and monodromy groups, Annals of Math. Studies, 116, Princeton, 1988 | MR | Zbl

[M1] S. Matsuda Local indices of $p$-adic differential operators corresponding to Artin-Schreier-Witt coverings, Duke Math. J, Volume 77 (1995), pp. 607-625 | DOI | MR | Zbl

[M2] S. Matsuda Katz correspondence for quasi-unipotent overconvergent isocrystals (1997) (manuscrit) | Zbl

[P] M. van der Put Galois theory of differential equations, algebraic groups and Lie algebras, J. Symbolic Computation, Volume 28 (1999), pp. 441-472 | DOI | MR | Zbl

[S1] J.P. Serre Sur la rationalité des représentations d'Artin, Ann. of Maths, Volume 72 (1960), pp. 406-420 | MR | Zbl

[S2] J.P. Serre Corps locaux, Hermann, 1968 | MR | Zbl

[T1] N. Tsuzuki The local index and the Swan conductor, Compos. Math, Volume 111 (1998), pp. 245-288 | DOI | MR | Zbl

[T2] N. Tsuzuki Slope filtration of quasi-unipotent overconvergent F-isocrystals, Ann. Inst. Fourier, Grenoble, Volume 48 (1998) no. 2, pp. 379-412 | DOI | Numdam | MR | Zbl

[T3] N. Tsuzuki Finite local monodromy of overconvergent unit-root F-isocrystal on a curve, Amer. J. Math, Volume 120 (1998), pp. 1165-1190 | DOI | MR | Zbl

[V] M.F. Vignéras Arithmétique des algèbres de quaternions, Lect. Notes, 800, Springer, 1980 | MR | Zbl

[WW] E. Whittaker; G. Watson A course of modern analysis, Cambridge Univ. Press, 1996 | MR