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Semistable reduction for overconvergent F-isocrystals, III: Local semistable reduction at monomial valuations

Part of: (Co)homology theory

Published online by Cambridge University Press:  01 January 2009

Kiran S. Kedlaya
Kiran S. Kedlaya*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA (email: kedlaya@mit.edu)
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Abstract

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We resolve the local semistable reduction problem for overconvergent F-isocrystals at monomial valuations (Abhyankar valuations of height 1 and residue transcendence degree zero). We first introduce a higher-dimensional analogue of the generic radius of convergence for a p-adic differential module, which obeys a convexity property. We then combine this convexity property with a form of the p-adic local monodromy theorem for so-called fake annuli.

Keywords

rigid cohomologyoverconvergent isocrystalssemistable reductionmonomial valuations

MSC classification

Secondary: 14F30: $p$-adic cohomology, crystalline cohomology 14F40: de Rham cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 1 , January 2009 , pp. 143 - 172
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Amice, Y., Les nombres p-adiques (Presses Universitaires de Paris, France, 1975).Google Scholar
[2]André, Y., Filtrations de type Hasse–Arf et monodromie p-adique, Invent. Math. 148 (2002), 285–317.Google Scholar
[3]Berkovich, V. G., Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33 (American Mathmatical Society, Providence, RI, 1990), (translated by N. I. Koblitz).Google Scholar
[4]Christol, G. and Dwork, B., Modules différentiels sur des couronnes, Ann. Inst. Fourier (Grenoble) 44 (1994), 663–701.CrossRefGoogle Scholar
[5]Christol, G. and Mebkhout, Z., Sur le théorème de l’indice des équations différentielles p-adiques. III, Ann. of Math. (2) 151 (2000), 385–457.CrossRefGoogle Scholar
[6]Christol, G. and Robba, P., Équations différentielles p-adiques, applications aux sommes exponentielles, Actualités Mathématiques (Hermann, Paris, 1994).Google Scholar
[7]de Jong, A. J., Smoothness, semi-stability and alterations, Publ. Math. Inst. Hautes Études Sci. 83 (1996), 51–93.CrossRefGoogle Scholar
[8]Dwork, B., Gerotto, G. and Sullivan, F. J., An introduction to G-functions, Annals of Mathematical Studies, vol. 133 (Princeton University Press, Princeton, NJ, 1994).Google Scholar
[9]Dwork, B. and Robba, P., On ordinary linear p-adic differential equations, Trans. Amer. Math. Soc. 231 (1977), 1–46.Google Scholar
[10]Fujishige, S., Submodular functions and optimization, Annals of Discrete Mathematics, vol. 58 second edition (Elsevier, Amsterdam, 2005).Google Scholar
[11]Kedlaya, K. S., A p-adic local monodromy theorem, Ann. of Math. (2) 160 (2004), 93–184.CrossRefGoogle Scholar
[12]Kedlaya, K. S., Local monodromy for p-adic differential equations: an overview, Int. J. Number Theory 1 (2005), 109–154.CrossRefGoogle Scholar
[13]Kedlaya, K. S., Slope filtrations revisited, Doc. Math. 10 (2005), 447–525.CrossRefGoogle Scholar
[14]Kedlaya, K. S., The p-adic local monodromy theorem for fake annuli, Rend. Sem. Mat. Padova 118 (2007), 101–146.Google Scholar
[15]Kedlaya, K. S., Swan conductors for p-adic differential modules, I: A local construction, Alg. and Number Theory 1 (2007), 269–300.CrossRefGoogle Scholar
[16]Kedlaya, K. S., Semistable reduction for overconvergent F-isocrystals, I: Unipotence and logarithmic extensions, Compositio Math. 143 (2007), 1164–1212.CrossRefGoogle Scholar
[17]Kedlaya, K. S., Semistable reduction for overconvergent F-isocrystals, II: A valuation-theoretic approach, Compositio Math. 144 (2008), 657–672.CrossRefGoogle Scholar
[18]Kedlaya, K. S., Slope filtrations for relative Frobenius, Astérisque 319 (2008), 259–301.Google Scholar
[19]Kedlaya, K. S., p-adic differential equations (version of 15 December 08), Preprint available at http://math.mit.edu/∼kedlaya/papers/.Google Scholar
[20]Kempf, G., Knudsen, F. F., Mumford, D. and Saint-Donat, B., Toroidal embeddings. I, Lecture Notes in Mathematics, vol. 339 (Springer, Berlin, 1973).CrossRefGoogle Scholar
[21]Knaf, H. and Kuhlmann, F.-V., Abhyankar places admit local uniformization in any characteristic, Ann. Sci. École Norm. Sup. 38 (2005), 833–846.CrossRefGoogle Scholar
[22]Kuhlmann, F.-V., Places of algebraic function fields in arbitrary characteristic, Adv. Math. 188 (2004), 399–424.CrossRefGoogle Scholar
[23]Mebkhout, Z., Analogue p-adique du Théorème de Turrittin et le Théorème de la monodromie p-adique, Invent. Math. 148 (2002), 319–351.CrossRefGoogle Scholar
[24]Ore, O., Theory of non-commutative polynomials, Ann. of Math. (2) 34 (1933), 480–508.CrossRefGoogle Scholar
[25]Pólya, G. and Szegő, G., Problems and theorems in Analysis, part I, reprint of the 1978 edition (Springer, Berlin, 1998).Google Scholar
[26]Ribenboim, P., The theory of classical valuations (Springer, New York, 1999).CrossRefGoogle Scholar
[27]Robba, P., Lemmes de Hensel pour les opérateurs différentiels. Application a la réduction formelle des équations différentielles, Enseign. Math. (2) 26 (1980), 279–311.Google Scholar
[28]Rockafellar, R. T., Convex analysis (Princeton University Press, Princeton, NJ, 1970).CrossRefGoogle Scholar
[29]Sabbah, C., Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2, Astérisque 263 (2000).Google Scholar
[30]Tsuzuki, N., Slope filtration of quasi-unipotent overconvergent F-isocrystals, Ann. Inst. Fourier (Grenoble) 48 (1998), 379–412.CrossRefGoogle Scholar
[31]Tsuzuki, N., Morphisms of F-isocrystals and the finite monodromy theorem for unit-rootF-isocrystals, Duke Math. J. 111 (2002), 385–418.CrossRefGoogle Scholar
[32]Young, P. T., Radii of convergence and index for p-adic differential operators, Trans. Amer. Math. Soc. 333 (1992), 769–785.Google Scholar