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Continuity of the radius of convergence of differential equations on p-adic analytic curves

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This paper deals with connections on non-archimedean, especially p-adic, analytic curves, in the sense of Berkovich. The curves must be compact but the connections are allowed to have a finite number of meromorphic singularities on them. For any choice of a semistable formal model of the curve, we define a geometric, intrinsic notion of normalized radius of convergence of a full set of local solutions as a function on the curve, with values in (0, 1]. For a sufficiently refined choice of the semistable model, we prove continuity, logarithmic concavity and logarithmic piece-wise linearity of that function. We introduce and characterize Robba connections, that is connections whose sheaf of solutions is constant on any open disk contained in the curve, precisely as it happens in the classical case.

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References

  1. André, Y.: Period mappings and differential equations. In: From ℂ to ℂ p , MSJ Memoirs 12. Tôhoku-Hokkaidô Lectures in Arithmetic Geometry, With appendices by F. Kato and N. Tsuzuki. Mathematical Society of Japan, Tokyo (2003). viii+246

    Google Scholar 

  2. André, Y.: Sur la conjecture des p-courbures de Grothendieck-Katz et un problème de Dwork. Geometric Aspects of Dwork Theory, vol. I, pp. 55–112. de Gruyter, Berlin (2004)

    Google Scholar 

  3. Baker, M.H.: An introduction to Berkovich analytic spaces and non-archimedean potential theory on curves in p-adic geometry. In: Savitt, D., Thakur, D.S. (eds.) Lectures from the 2007 Arizona Winter School. University Lecture Series, vol. 45, pp. 123–173. American Mathematical Society, Providence (2008)

    Google Scholar 

  4. Baker, M.H., Rumely, R.: Potential Theory on the Berkovich Projective Line. Book in preparation, available online at www.math.uga.edu/rr/NewBerkBook.pdf

  5. Baldassarri, F.: Differential modules and singular points of p-adic differential equations. Adv. Math. 44(2), 155–179 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  6. Baldassarri, F.: Formal models of analytic curves over a non-archimedean field (in preparation)

  7. Baldassarri, F.: Convergence polygons of differential equations on p-adic analytic curves (in preparation)

  8. Baldassarri, F., Di Vizio, L.: Continuity of the radius of convergence of p-adic differential equations on Berkovich analytic spaces. arXiv:0709.2008v3 [math.NT]

  9. Berkovich, V.G.: Spectral Theory and Analytic Geometry over Non-Archimedean Fields. Mathematical Surveys and Monographs, vol. 33. American Mathematical Society, Providence (1990)

    MATH  Google Scholar 

  10. Berkovich, V.G.: Étale cohomology for non-Archimedean analytic spaces. Inst. Hat. Études Sci., Publ. Math. 78, 5–161 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  11. Berkovich, V.G.: Vanishing cycles for formal schemes. Invent. Math. 115(3), 539–571 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  12. Berkovich, V.G.: Smooth p-adic analytic spaces are locally contractible. Invent. Math. 137, 1–83 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Berkovich, V.G.: Smooth p-adic analytic spaces are locally contractible. II. In: Geometric Aspects of Dwork Theory. Vols. I, II, pp. 293–370. de Gruyter, Berlin (2004)

    Google Scholar 

  14. Berkovich, V.G.: Integration of One-Forms on p-adic Analytic Spaces. Annals of Mathematics Studies, vol. 162. Princeton University Press, Princeton (2007)

    Google Scholar 

  15. Bosch, S.: Lectures on Formal and Rigid Geometry. Preprintreihe des Mathematischen Instituts, vol. 378. Westfälische Wilhelms-Universität, Münster (2005)

    Google Scholar 

  16. Bosch, S., Lütkebohmert, W.: Stable reduction and uniformization of Abelian varieties. I. Math. Ann. 270(3), 349–379 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  17. Bosch, S., Lütkebohmert, W.: Formal and rigid geometry. I. Rigid spaces. Math. Ann. 295, 291–317 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean Analysis. Grundlehren der Mathematischen Wissenschaften, vol. 261. Springer, Berlin (1984)

    MATH  Google Scholar 

  19. Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models. Ergebnisse der Mathematik, Series 3, vol. 21. Springer, Berlin (1990)

    MATH  Google Scholar 

  20. Christol, G.: Rayons des solutions de l’équation de Dwork. Manuscript (2009)

  21. Christol, G., Dwork, B.: Modules différentiels sur des couronnes. Univ. Grenoble. Annal. Inst. Fourier 44(3), 663–701 (1994)

    MATH  MathSciNet  Google Scholar 

  22. Christol, G., Mebkhout, Z.: Sur le théorème de l’indice des équations différentielles p-adiques. III. Ann. Math. 151(2), 385–457 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Christol, G., Mebkhout, Z.: Équations différentielles p-adiques et coefficients p-adiques sur les courbes. In: Cohomologies p-adiques et applications arithmétiques, II, vol. 279, pp. 125–183. SMF (2002)

  24. de Jong, A.J.: Étale fundamental groups of non-Archimedean analytic spaces. Compos. Math. 97, 89–118 (1995)

    MATH  Google Scholar 

  25. de Jong, A.J.: Families of curves and alterations. Ann. Inst. Fourier 47(2), 599–621 (1997)

    MATH  MathSciNet  Google Scholar 

  26. Dieudonné, J., Grothendieck, A.: Éléments de géométrie algébrique—chapitre IV, partie 3. Publ. Math. IHES 28, 5–255 (1966)

    Google Scholar 

  27. Dwork, B.: On the Tate constant. Comput. Math. 61, 43–59 (1987)

    MATH  MathSciNet  Google Scholar 

  28. Dwork, B., Gerotto, G., Sullivan, F.J.: An Introduction to G-Functions. Annals of Mathematics Studies, vol. 133. Princeton University Press, Princeton (1994)

    Google Scholar 

  29. Elkik, R.: Solutions d’équations à coefficients dans un anneau hensélien. Ann. Sci. ENS 6(4), 553–603 (1973)

    MATH  MathSciNet  Google Scholar 

  30. Engelking, R.: General Topology, 2nd edn. Sigma Series in Pure Mathematics, vol. 6. Heldermann Verlag, Berlin (1989)

    MATH  Google Scholar 

  31. Fresnel, J., Matignon, M.: Sur les espaces analytiques quasi-compacts de dimension 1 sur un corps valué complet ultramétrique. Ann. Mat. Pura Appl. 145(4), 159–210 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  32. Gachet, F.: Structure fuchsienne pour des modules différentiels sur une polycouronne ultramétrique. Rend. Sem. Mat. Univ. Padova 102, 157–218 (1999)

    MATH  MathSciNet  Google Scholar 

  33. Grauert, H., Remmert, R.: Über die Methode der diskret bewerteten Ringe in der nicht-archimedischen Analysis. Invent. Math. 2, 87–133 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  34. Kedlaya, K.S.: Local monodromy for p-adic differential equations: an overview. Int. J. Number Theory 1, 109–154 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  35. Kedlaya, K.S.: p-Adic Differential Equations. Cambridge Studies in Advanced Mathematics, vol. 125. Cambridge Univ. Press, Cambridge (2010)

    MATH  Google Scholar 

  36. Lütkebohmert, W.: Formal-algebraic and rigid-analytic geometry. Math. Ann. 286, 341–371 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  37. Pons, É.: Modules différentiels non solubles. Rayons de convergence et indices. Rend. Sem. Mat. Univ. Padova 103, 21–45 (2000)

    MATH  MathSciNet  Google Scholar 

  38. Pulita, A.: Rank one solvable p-adic differential equations and finite Abelian characters via Lubin-Tate groups. Math. Ann. 337(3), 489–555 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  39. Robba, P., Christol, G.: Équations différentielles p-adiques. Applications aux sommes exponentielles. Actualités Mathématiques. Hermann, Paris (1994)

    Google Scholar 

  40. Temkin, M.: On local properties of non-Archimedean analytic spaces. Math. Ann. 318(3), 585–607 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  41. Temkin, M.: Stable modification of relative curves. arXiv:0707.3953v2 [math.AG]

  42. Young, P.T.: Radii of convergence and index for p-adic differential operators. Trans. Am. Math. Soc. 333, 769–785 (1992)

    Article  MATH  Google Scholar 

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  1. Dipartimento di matematica pura e applicata, Università di Padova, Via Trieste, 63, 35121, Padova, Italy

    Francesco Baldassarri

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  1. Francesco Baldassarri
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Correspondence to Francesco Baldassarri.

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Baldassarri, F. Continuity of the radius of convergence of differential equations on p-adic analytic curves. Invent. math. 182, 513–584 (2010). https://doi.org/10.1007/s00222-010-0266-7

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