×

On overconvergent isocrystals and \(F\)-isocrystals of rank one. (English) Zbl 0934.12004

Let \(k\) be a field of characteristic \(p>0\) and \(K\) a complete non-archimedean valued field of characteristic zero with residue field \(k\). The purpose of the paper is to prove that any overconvergent isocrystal of rank 1, defined over \(K\) and living above a Zariski open subset \(Z={\mathbb P}^1_k\setminus X\) of \({\mathbb P}^1_k\) has a (strong) Frobenius structure if for every point of \(X\) the local exponents of the crystal lie in \({\mathbb Q}\cap {\mathbb Z}_p\).
The overconvergent rank 1 crystal translates into a differential operator \(L=\frac{d}{dx}+f(x)\) with \(f\in K(x)\), having a non trivial solution in the “generic disk” \(B(t,1^-)\). The existence of a strong Frobenius translates into: \(L\) is equivalent to \(\phi ^s(L)\) where \(\phi\) is a suitable Frobenius endomorphism of the field \(K(x)\) and \(s\geq 1\) an integer. An arithmetic study of the operator \(L\) yields a local Frobenius structure for (say) the singular point \(x=0\) (under the assumption on the exponents). For the other singular points there are also local Frobenius structures. The various local Frobenius structures are glued together, using a method of B. Dwork, to the required strong global Frobenius structure.

MSC:

12H25 \(p\)-adic differential equations
14F30 \(p\)-adic cohomology, crystalline cohomology
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] Baldassarri, F. and Chiarellotto, B. : ” Algebraic versus rigid cohomology with logarithmic coefficients ” to appear in Proceedings of the Barsotti Memorial Symposium. Abano Terme (Padova) June 1991, Perspectives in Mathematics , Academic Press. · Zbl 0833.14010
[2] Berthelot, P. : ” Cohomologie rigide et théorie de Dwork: le cas des sommes exponentielles ”, Astérisque 119-120 (1984) 17-50. · Zbl 0577.14013
[3] Berthelot, P. : ” Cohomologie rigide et cohomologie rigide à support propre ”, to appear in Astérisque. · Zbl 0515.14015
[4] Berthelot, P. : ”Géométrie rigide et cohomologie des variétés algébriques de caractéristiques p” , Soc. Math. de France, 2e Série, Mémoires n.23 (1986) 7-32. · Zbl 0606.14017 · doi:10.24033/msmf.326
[5] Christol, G. : ” Modules différentiels et équations différentielles p-adiques ”. Queen’s Papers in Pure and Applied Mathematics, n.66, Queen’s University, Kingston (Ontario) 1983. · Zbl 0589.12020
[6] Christol, G. : ” Un théorème de transfert pour les disques singulier réguliers ”, Astérisque 119-120 (1984) 151-168. · Zbl 0553.12014
[7] Christol, G. : ” Fonctions et éléments algébriques ”, Pacific Journal of Math. 125 (1986) 1-37. · Zbl 0591.12018 · doi:10.2140/pjm.1986.125.1
[8] Christol, G. : ”Solutions algébriques des équations différentielles p-adiques”, Sém. Delange-Pisot-Poitou 1981/82 (1983) 51-58. · Zbl 0525.12020
[9] Christol, G. : ”Systèmes différentiels linéaires p-adiques: structure de Frobenius faible” , Bull. Soc. Math. France 109 (1981) 83-122. · Zbl 0459.12020 · doi:10.24033/bsmf.1933
[10] Dwork, B. : ” On p-adic differential equations II ”, Annals of Math. 98 (1973) 366-376. · Zbl 0304.14015 · doi:10.2307/1970786
[11] Dwork, B. : ”p-adic cycles” , Pubbl. IHES 37 (1969), 27-116. · Zbl 0284.14008 · doi:10.1007/BF02684886
[12] Fresnel, J. and Van Der Put, M. : ”Géométrie analytique rigide et applications”, Progress in Math. 18, Birkhauser (1981). · Zbl 0479.14015
[13] Matsuda, S. : ” Local indices of p-adic differential equations corresponding to Artin-Schreier-Witt covering ”, Preprint 1993, Tokyo. · Zbl 0849.12013 · doi:10.1215/S0012-7094-95-07719-9
[14] Motzkin, E. : ” Décomposition d’élément analytique en facteurs singuliers ”, Ann. Inst. Fourier 27 (1977) 67-82. · Zbl 0336.12010 · doi:10.5802/aif.642
[15] Raynaud, M. : ” Géométrie analytique rigide ”, Bull. Soc. Math. France, Mémoire 39-40 (1974) 319-327. · Zbl 0299.14003
[16] Robba, P. : ” Indice d’un opérateur différentiel p-adique IV. Cas des systèmes ”, Ann. Inst. Fourier 35, 2 (1985) 13-55. · Zbl 0548.12016 · doi:10.5802/aif.1008
[17] Robba, P. : ” Une introduction naive aux cohomologies de Dwork ”, Soc. Math. de France, 2e Série, Mémoires n.23 (1986) 61-105. · Zbl 0623.14005 · doi:10.24033/msmf.325
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.