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A note on Berkovich spaces and \(p\)-adic differential equations. (Espaces de Berkovich et équation différentielles \(p\)-adiques. Une note.) (French) Zbl 0974.12014

At first glance, Berkovich spaces theory seems to be convenient for \( p\)-adic differential equations because it looks upon Dwork’s generic points as actual points. Indeed F. Baldassarri and L. Di Vizio wrote about three years ago a paper taking up this point of view for \( p\)-adic differential equations over varieties. As far as the reviewer knows, this paper is not yet available. Fortunately, the present paper makes explicit these ideas in the one dimensional case and gives it applications.
Firstly, the overconvergent decomposition theorem is generalized by adapting the original Dwork-Robba proof to the new context. Actually, working only on generic points enables to get round difficulties that come from roots of unknown polynomials over the constant field. Secondly it shows that, for isocrystals over some affine subset of the projective line, the overconvergent and convergent conditions are equivalent: this is a special case of continuity for the generic radius of convergence.

MSC:

12H25 \(p\)-adic differential equations
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References:

[1] F. Baldassarri - B. Chiarellotto , Algebraic versus rigid cohomology with logarithmic coefficients, in Barsotti Memorial Symposium , Perspectives in Math. , vol. 15 , Academic Press ( 1994 ), pp. 11 - 50 . MR 1307391 | Zbl 0833.14010 · Zbl 0833.14010
[2] F. Baldassarri - L. Di Vizio , On arithmetic size of linear differential equations , A paraître. Zbl 0998.12008 · Zbl 0998.12008 · doi:10.1006/jabr.2000.8750
[3] V. Berkovich , Spectral theory and analytic geometry over non-archimedean fields , Math. Surveys and Monographs , Number 33 , AMS 1990 . MR 1070709 | Zbl 0715.14013 · Zbl 0715.14013
[4] P. Berthelot , Cohomologie rigide et cohomologie rigide à support propre , Pre publication, Rennes . · Zbl 0515.14015
[5] S. Bosch - U. Guntzer - R. Remmert , Non archimedean analysis, Grundlehren der math . Wissenschaften 261 , Springer-Verlag 1984 . MR 746961 | Zbl 0539.14017 · Zbl 0539.14017
[6] B. Dwork - PH. Robba , On ordinary linear differential equations , T.A.M.S. , 231 ( 1977 ), pp. 1 - 46 . MR 447247 | Zbl 0375.34010 · Zbl 0375.34010 · doi:10.2307/1997866
[7] E. Pons , Modules différentiels non solubles. Rayons de convergence et Indices , Dans ce volume . Numdam | Zbl 1006.12007 · Zbl 1006.12007
[8] PH. Robba , On the index of p-adic differential operators I , Annals of Math. , 101 ( 1975 ), pp. 280 - 316 . MR 364243 | Zbl 0316.12102 · Zbl 0316.12102 · doi:10.2307/1970992
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