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Unipotent monodromy and arithmetic \({\mathcal {D}}\)-modules

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In the framework of Berthelot’s theory of arithmetic \({\mathcal {D}}\)-modules, we introduce the notion of arithmetic \({\mathcal {D}}\)-modules having potentially unipotent monodromy. For example, from Kedlaya’s semistable reduction theorem, the overconvergent isocrystals with Frobenius structure have potentially unipotent monodromy. We construct some coefficients stable under Grothendieck’s six operations, containing overconvergent isocrystals with Frobenius structure and whose objects have potentially unipotent monodromy. On the other hand, we introduce the notion of arithmetic \({\mathcal {D}}\)-modules having quasi-unipotent monodromy. These objects are overholonomic, contain the isocrystals having potentially unipotent monodromy and are stable under Grothendieck’s six operations and under base change.

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References

  1. Abe, T.: Langlands correspondence for isocrystals and existence of crystalline companion for curves (2013). arXiv:1310.0528

  2. Abe, T., Caro, D.: Theory of weights in \(p\)-adic cohomology (2013). arXiv:1303.0662

  3. Berthelot, P.: \(\cal{D}\)-modules arithmétiques. I. Opérateurs différentiels de niveau fini. Ann. Sci. École Norm. Sup. (4) 29(2), 185–272 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berthelot, P.: Introduction à la théorie arithmétique des \(\cal{D}\)-modules. Astérisque 279, 1–80 (2002). Cohomologies \(p\)-adiques et applications arithmétiques, II

  5. Caro, D.: \(\cal{D}\)-modules arithmétiques surcohérents. Application aux fonctions L. Ann. Inst. Fourier (Grenoble) 54(6), 1943–1996 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caro, D.: Overconvergent F-isocrystals and differential overcoherence. Invent. Math. 170(3), 507–539 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caro, D.: \(\cal{D}\)-modules arithmétiques surholonomes. Ann. Sci. École Norm. Sup. (4) 42(1), 141–192 (2009)

    Article  MATH  Google Scholar 

  8. Caro, D.: Overconvergent log-isocrystals and holonomy. (Log-isocristaux surconvergents et holonomie). Compos. Math. 145(6), 1465–1503 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caro, D.: Pleine fidélité sans structure de Frobenius et isocristaux partiellement surconvergents. Math. Ann. 349, 747–805 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Caro, D.: Sur la préservation de la cohérence par image inverse extraordinaire par une immersion fermée. Mathematics, ArXiv e-prints arXiv:1207.0714 (2012)

  11. Caro, D.: Sur la préservation de la surconvergence par l’image directe d’un morphisme propre et lisse. Ann. Sci. Éc. Norm. Supér. (4) 48(1), 131–169 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Caro, D.: Sur la stabilité par produit tensoriel de complexes de \(\cal{D}\)-modules arithmétiques. Manuscr. Math. 147(1–2), 1–41 (2015)

    Article  MATH  Google Scholar 

  13. Caro, D.: La surcohérence entraîne l’holonomie. Bull. Soc. Math. France 144(3), 429–475 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Caro, D.: Systéms inductifs cohérents de \(\cal{D}\)-modules arithmétiques logarithmiques, stabilité par opérations cohomologiques. Doc. Math. 21, 1515–1606 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Caro, D., Tsuzuki, N.: Overholonomicity of overconvergent \(F\)-isocrystals over smooth varieties. Ann. Math. (2) 176(2), 747–813 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. de Jong, A.J.: Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. 83, 51–93 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kedlaya, K.S.: Semistable reduction for overconvergent \(F\)-isocrystals. I. Unipotence and logarithmic extensions. Compos. Math. 143(5), 1164–1212 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kedlaya, K.S.: Semistable reduction for overconvergent \(F\)-isocrystals. II. A valuation-theoretic approach. Compos. Math. 144(3), 657–672 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kedlaya, K.S.: Semistable reduction for overconvergent \(F\)-isocrystals. III: local semistable reduction at monomial valuations. Compos. Math. 145(1), 143–172 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kedlaya, K.S.: Semistable reduction for overconvergent \(F\)-isocrystals, IV: local semistable reduction at nonmonomial valuations. Compos. Math. 147(2), 467–523 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Milne, J.S.: Étale Cohomology. Princeton University Press, Princeton (1980)

    MATH  Google Scholar 

  22. Revêtements étales et groupe fondamental (SGA 1). Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3. Société Mathématique de France, Paris, 2003. Séminaire de géométrie algébrique du Bois Marie 1960–1961. [Geometric Algebra Seminar of Bois Marie 1960–1961], Directed by A. Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin]

  23. Shiho, A.: On logarithmic extension of overconvergent isocrystals. Math. Ann. 348(2), 467–512 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Virrion, A.: Trace et dualité relative pour les \(\cal{D}\)-modules arithmétiques. In: Adolphson, A., Baldassarri, F., Berthelot, P., Katz, N.M., Loeser, F. (eds.) Geometric Aspects of Dwork Theory, vol. I, II, pp. 1039–1112. Walter de Gruyter GmbH & Co. KG, Berlin (2004)

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  1. Universite de Caen Normandie, Caen, France

    Daniel Caro

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  1. Daniel Caro
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Correspondence to Daniel Caro.

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L’auteur a bénéficié du soutien de l’Institut Universitaire de France.

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Caro, D. Unipotent monodromy and arithmetic \({\mathcal {D}}\)-modules. manuscripta math. 156, 81–115 (2018). https://doi.org/10.1007/s00229-017-0959-y

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