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Mathematische Zeitschrift

, Volume 269, Issue 1–2, pp 59–82 | Cite as

Cut-by-curves criterion for the log extendability of overconvergent isocrystals

  • Atsushi ShihoEmail author
Article

Abstract

In this paper, we prove a ‘cut-by-curves criterion’ for an overconvergent isocrystal on a smooth variety over a field of characteristic p > 0 to extend logarithmically to its smooth compactification whose complement is a simple normal crossing divisor, under certain assumption. This is a p-adic analogue of a version of cut-by-curves criterion for regular singularity of an integrable connection on a smooth variety over a field of characteristic 0. In the course of the proof, we also prove a kind of cut-by-curves criteria on solvability, highest ramification break and exponent of ∇-modules.

Keywords

Overconvergent isocrystal Solvability Highest ramification break Exponent 

Mathematics Subject Classification (2000)

12H25 

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References

  1. 1.
    André Y., Baldassarri F.: De Rham Cohomology of Differential Modules on Algebraic Varieties. Progress in Mathematics, vol. 189. Birkhäuser, Basel (2001)Google Scholar
  2. 2.
    Baldassarri Y., Chiarellotto B.: Formal and p-adic theory of differential systems with logarithmic singularities depending upon parameters. Duke Math. J. 72, 241–300 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Berger R., Kiehl R., Kunz E., Nastold H.-J.: Differentialrechnung in der Analytischen Geometrie. Lecture Notes in Mathematics, vol. 38. Springer, Berlin (1967)Google Scholar
  4. 4.
    Berthelot, P.: Cohomologie rigide et cohomologie rigide à supports propres première partie. Prépublication de l’IRMAR 96-03. http://perso.univ-rennes1.fr/pierre.berthelot/
  5. 5.
    Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean Analysis. Grundlehren der Mathematischen Wissenschaften, vol. 261. Springer, Berlin (1984)Google Scholar
  6. 6.
    Christol G., Mebkhout Z.: Équations différentielles p-adiques et coefficients p-adiques sur les courbes. Astérisque 279, 125–183 (2002)MathSciNetGoogle Scholar
  7. 7.
    Christol G., Dwork B.: Modules différentiels sur des couronnes. Ann. Inst. Fourier 44, 663–701 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Deligne, P.: Equations Différentielles à Points Singuliers Réguliers. Lecture Note in Mathematics, vol. 163. Springer, Berlin (1970)Google Scholar
  9. 9.
    Dwork B.: On exponents of p-adic differential modules. J. Reine Angew. Math. 484, 85–126 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Grothendieck A., Dieudonné J.: Éléments de géométrie algébrique IV. Publ. Math. IHES 32, 5–333 (1967)Google Scholar
  11. 11.
    Grauert, H., Remmert, R.: Analytische Stellenalgebren. Grundlehren der Mathematischen Wissenschaften, vol. 176. Springer, Berlin (1971)Google Scholar
  12. 12.
    Kedlaya K.S.: Semistable reduction for overconvergent F-isocrystals, I: unipotence and logarithmic extensions. Compos. Math. 143, 1164–1212 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Kedlaya K.S.: Swan conductors for p-adic differential modules, I: a local construction. Algebra Number Theory 1, 269–300 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kedlaya, K.S.: Swan conductors for p-adic differential modules, II: global variation. (2008). arXiv:0705.0031v3Google Scholar
  15. 15.
    Kedlaya, K.S.: p-Adic Differential Equations. http://math.mit.edu/~kedlaya/papers/index.shtml. Cambridge University Press, London (2010, to appear)
  16. 16.
    Robinson Z.: A note on global pth powers of rigid analytic functions. J. Number Theory 116, 474–482 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Shiho A.: Crystalline fundamental groups II—log convergent cohomology and rigid cohomology. J. Math. Sci. Univ. Tokyo 9, 1–163 (2002)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Shiho, A.: Relative log convergent cohomology and relative rigid cohomology I. (2008). arXiv:0707.1742v2Google Scholar
  19. 19.
    Shiho, A.: On logarithmic extension of overconvergent isocrystals. Math. Ann. (2010, to appear)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesUniversity of TokyoTokyoJapan

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