André, Yves Représentations galoisiennes et opérateurs de Bessel \(p\)-adiques. (Galois representations and \(p\)-adic Bessel operators). (French) Zbl 1014.12007 Ann. Inst. Fourier 52, No. 3, 779-808 (2002). Let \(K\) be a finite extension of the \(p\)-adic field \(\mathbb Q_p\). Then the “Robba ring at infinity” \(\mathcal R\) for \(K\) in \(x\) is the ring of (bi-infinite) power series \(\sum a_nx^n\), \(a_n \in K\), which converge in an open \(p\)-adic annulus with inner limit \(1\), and \(\mathcal E \subset \mathcal R\) is the subring (actually Henselian field) of series with bounded coefficients. The finite unramified extensions \(\mathcal E^\prime\) of \(\mathcal E\) are of similar form, based on a finite unramified extension \(K^\prime \supset K\) and power series over \(K^\prime\) in \(y\), where \(y\) is algebraic over \(\mathcal E\) and transcendental over \(K^\prime\). The corresponding Robba ring \(\mathcal R^\prime\) is isomorphic to \(\mathcal R \otimes_{\mathcal E} \mathcal E^\prime\). Define a derivation \(D\) on \(\mathcal R\) by \(D=x\frac{d}{dx}\). The author calls a differential \(\mathcal R\) module \(M\) quasi-unipotent if for some finite unramified extension \(\mathcal E^\prime \supseteq \mathcal E\), \(M \otimes_{\mathcal R} \mathcal R^\prime\) is an iterated extension of trivial differential modules. A Frobenius map \(\phi: \mathcal R \to \mathcal R\) is a continuous ring homomorphism lifting the usual Frobenius on the residue field \(\mathbb F_q((x^{-1}))\) of \(\mathcal E\). A Frobenius structure on a differential \(\mathcal R\) module \(M\) is a differential isomorphism of \(M\) with \(M\) base changed by \(\phi\). The author considers \(p\)-adic Bessel operators (these have a \(p\)-adic index, but the author shows that the corresponding differential modules do not depend on it, up to isomorphism). He shows that this differential module has a Frobenius structure, is quasi-unipotent, and he explicitly determines the field \(K^\prime\) when \(K\) is an unramified quadratic extension of \(\mathbb Q_2\). Reviewer: Andy R.Magid (Norman) Cited in 1 ReviewCited in 3 Documents MSC: 12H25 \(p\)-adic differential equations 11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) 11S15 Ramification and extension theory Keywords:Robba ring at infinity; differential module; \(p\)-adic Bessel operator; Henselian field; Frobenius map × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] Twisted Kloosterman sums and p-adic Bessel functions, Amer. J. Math, 106, 549-591 (1984) · Zbl 0552.12010 · doi:10.2307/2374285 [2] Cohomologie rigide et théorie de Dwork : le cas des sommes exponentielles, No 119-120, 17-49 (1984) · Zbl 0577.14013 [3] Differential modules and singular points of p-adic differential equations, Advances in Math, 44, 155-179 (1982) · Zbl 0493.12030 · doi:10.1016/0001-8708(82)90004-4 [4] Differential Galois theory, chap. 8 (1995) · Zbl 0813.12001 [5] Algèbre commutative, chapitres I, VII (1985) [6] Modules différentiels sur des couronnes, Ann. Inst. Fourier, Grenoble, 44, 3, 663-701 (1994) · Zbl 0859.12004 · doi:10.5802/aif.1414 [7] Sur le théorème de l’indice des équations différentielles p-adiques II, Ann. of Maths, 146, 345-410 (1997) · Zbl 0929.12003 · doi:10.2307/2952465 [8] Sur le théorème de l’indice des équations différentielles p-adiques III, Ann. of Maths, 151, 385-457 (2000) · Zbl 1078.12500 · doi:10.2307/121041 [9] Complex regular polytopes (1991) · Zbl 0732.51002 [10] \(F\)-isocrystals and \(p\)-adic representations, Algebraic Geometry - Bowdoin 1985, XLVI, 111-138 (1987) · Zbl 0639.14011 [11] Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve, Ann. Scient. Éc. Norm. Sup, 31, 717-763 (1998) · Zbl 0943.14008 [12] Canonical extensions, irregularities, and the Swan conductor, Math. Ann., 316, 19-37 (2000) · Zbl 0969.14012 · doi:10.1007/s002080050002 [13] Bessel functions as \(p\)-adic functions of their argument, Duke Math. J, 41, 711-738 (1974) · Zbl 0302.14008 · doi:10.1215/S0012-7094-74-04176-3 [14] Représentations \(p\)-adiques des corps locaux, Grothendieck Festschrift II, 87, 249-309 (1990) · Zbl 0743.11066 [15] Representation theory, 129 (1991) · Zbl 0744.22001 [16] Elliptic curves, 111 (1987) · Zbl 0605.14032 [17] A cohomological construction of Swan representations over the Witt ring II, Proc. Japan Acad, 64A, 350-351 (1988) · Zbl 0699.14026 [18] On the calculation of some differential Galois groups, Invent. Math, 87, 13-61 (1987) · Zbl 0609.12025 · doi:10.1007/BF01389152 [19] Local-to-global extensions of representations of fundamental groups, Ann. Inst. Fourier, Grenoble, 36, 4, 69-106 (1986) · Zbl 0564.14013 · doi:10.5802/aif.1069 [20] Gauss sums, Kloosterman sums, and monodromy groups, 116 (1988) · Zbl 0675.14004 [21] Local indices of \(p\)-adic differential operators corresponding to Artin-Schreier-Witt coverings, Duke Math. J, 77, 607-625 (1995) · Zbl 0849.12013 · doi:10.1215/S0012-7094-95-07719-9 [22] Katz correspondence for quasi-unipotent overconvergent isocrystals (1997) · Zbl 1101.14021 [23] Galois theory of differential equations, algebraic groups and Lie algebras, J. Symbolic Computation, 28, 441-472 (1999) · Zbl 0997.12008 · doi:10.1006/jsco.1999.0310 [24] Sur la rationalité des représentations d’Artin, Ann. of Maths, 72, 406-420 (1960) · Zbl 0202.32803 [25] Corps locaux (1968) · Zbl 0137.02601 [26] The local index and the Swan conductor, Compos. Math, 111, 245-288 (1998) · Zbl 0926.12004 · doi:10.1023/A:1000243409360 [27] Slope filtration of quasi-unipotent overconvergent F-isocrystals, Ann. Inst. Fourier, Grenoble, 48, 2, 379-412 (1998) · Zbl 0907.14007 · doi:10.5802/aif.1622 [28] Finite local monodromy of overconvergent unit-root F-isocrystal on a curve, Amer. J. Math, 120, 1165-1190 (1998) · Zbl 0943.14007 · doi:10.1353/ajm.1998.0052 [29] Arithmétique des algèbres de quaternions, 800 (1980) · Zbl 0422.12008 [30] A course of modern analysis (1996) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.