Robba: Indice d'un opérateur différentiel $p$-adique IV. Cas des systèmes. Mesure de l'irrégularité dans un disque

Table of contents for this issue | Previous article | Next article

Philippe Robba
Indice d'un opérateur différentiel $p$-adique IV. Cas des systèmes. Mesure de l'irrégularité dans un disque
Annales de l'institut Fourier, 35 no. 2 (1985), p. 13-55, doi: 10.5802/aif.1008
Article PDF | Reviews MR 86j:12012 | Zbl 0548.12016 | 2 citations in Cedram

Résumé - Abstract

We are interested in showing that the differential operator of order $1, {d\over dx}+G$, where $G$ is a $k\times k$ matrix with rational coefficients, has an index in the space of functions analytic in a ball; we then wish to compute this index. In the case $k=1$, we show that this index exists (provided that the exponent of the differential operator in the ball is not a $p$-adic Liouville number) and we indicate how to compute this index. We can also show existence of index and compute this index when the system is equivalent to a triangular system. We give an interpretation of that index in term of the global irregularity of the differential operator in the ball.

Bibliography

[1] A. ADOLPHSON, An index theorem for p-adic differential operators, Trans. Amer. Math. Soc., 216 (1976), 279-293. MR 52 #8127 | Zbl 0297.47039
[2] Y. AMICE, Les nombres p-adiques, PUF collection Sup 14 (1975). MR 56 #5510 | Zbl 0313.12104
[3] F. BALDASSARRI, Differential modules and singular points of p-adic differential equations, Adv. in Math., 44 (1982), 155-179. MR 84i:12021 | Zbl 0493.12030
[4] G. CHRISTOL, Décomposition des matrices en facteurs singuliers. Application aux équations différentielles, GEAU, 7-8 (1979-1981), n° 5, 17 p. Numdam | Zbl 0466.12016
[5] G. CHRISTOL, Un théorème de transfert pour les disques singuliers réguliers, Astérisque, 119-120 (1984), 151-168. MR 86h:12012 | Zbl 0553.12014
[6] G. CHRISTOL, Opérateurs différentiels p-adiques, Kingston university publication.
[7] P. DELIGNE, Équations différentielles à points singuliers réguliers, Lecture Notes in Math., 169, Springer Verlag, 1970. MR 54 #5232 | Zbl 0244.14004
[8] B. DWORK, On p-adic differential equations II, Annals of Math., 98 (1973), 366-376. MR 58 #27987b | Zbl 0304.14015
[9] B. DWORK, P. ROBBA, On ordinary linear p-adic differential equations, Trans. A.M.S., 231 (1977), 1-46. MR 56 #5562 | Zbl 0375.34010
[10] B. DWORK, P. ROBBA, Effective p-adic bounds for solutions of homogeneous linear differential equations, Trans. A.M.S., 259 (1980), 559-577. MR 81k:12022 | Zbl 0439.12016
[11] B. MALGRANGE, Sur les points singuliers des équations différentielles, L'enseignement mathématique, XX (1974), 147-176. MR 51 #4316 | Zbl 0299.34011
[12] P. ROBBA, On the index of p-adic differential operators I, Annals of Math., 101 (1975), 280-316. MR 51 #498 | Zbl 0316.12102
[13] P. ROBBA, On the index of p-adic differential operators II. Duke Math. J., 43 (1976), 19-31. Article | MR 52 #10739 | Zbl 0338.12108
[14] P. ROBBA, Index of p-adic differential operators III. Application to twisted exponential sums, Astérisque, 119-120 (1984), 191-266. MR 86e:12011 | Zbl 0548.12015