Painlevé VI equations in p-adic time

  • Yu. I. ManinEmail author
Research Articles


Using the description of Painlevé VI family of differential equations in terms of a universal elliptic curve, going back to R. Fuchs, we translate it into the realm of Buium’s p-adic Arithmetic Differential Equations, where the role of derivative is played by a version of Fermat quotient.


Painlevé VI p-adic derivations Hamiltonian 


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  1. 1.
    F. Amano and M. Morishita, “Arithmetic Milnor invariants and multiple power residue symbols in number fields,” (2014), [arXiv:1412.6894].Google Scholar
  2. 2.
    M. Barrett and A. Buium, “Curvature on the integers I,” (2015), [arXiv:1512.02525].zbMATHGoogle Scholar
  3. 3.
    M. V. Babich and D. A. Korotkin, “Self-dual SU(2)-invariant Einstein metrics and modular dependence of Theta-functions,” Lett.Math. Phys. 46, 323–337 (1998), [arXiv:gr-qc/9810025v2].MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. Buium, Arithmetic Differential Equations, Math. Surveys and Monographs, 118 (MS, Providence RI, 2005).CrossRefGoogle Scholar
  5. 5.
    A. Buium, “Differential characters of abelian varieties over p-adic fields,” Inv. Math. 122, 309–340 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. Buium, “Differential characters and characteristic polynomial of Frobenius,” J. reine u. angew. Math. 485, 209–219 (1997).MathSciNetzbMATHGoogle Scholar
  7. 7.
    A. Buium, “Curvature on the integers II,” (2015), [arXiv: 1512.02527].zbMATHGoogle Scholar
  8. 8.
    A. Buium and Yu. Manin, “Arithmetic differential equations of Painlevé VI type,” in Arithmetic and Geometry, ed. by L. Dieulefait et al., LMS Lecture Note Series 420, 114–138 (2015), [arXiv:1307.3841].CrossRefGoogle Scholar
  9. 9.
    P. Deligne and L. Illusie, “Relè vementmodulo p2 et dé composition du complexe de De Rham,” Inv.Math. 89, 247–270 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    L. R. Gurney, Elliptic Curves with Complex Multiplication and Structures, PhD Thesis (Australian Nat. Univ., 2015).Google Scholar
  11. 11.
    N. J. Hitchin, “Twistor spaces, Einstein metrics and isomonodromic deformations,” J. Diff. Geom. 42 (1), 30–112 (1995).MathSciNetzbMATHGoogle Scholar
  12. 12.
    D. Kaledin, “Cartier isomorphism for unital associative algebras,” (2015), [arXiv:1509.08049].Google Scholar
  13. 13.
    Yu. Manin, “Sixth Painlevé equation, universal elliptic curve, and mirror of P2,” in Geometry of Differential Equations, ed. by A. Khovanskii, A. Varchenko and V. Vassiliev, Amer.Math. Soc. Transl. 186, (2) 131–151 (1996), [arXiv:alg-geom/9605010].Google Scholar
  14. 14.
    Yu. Manin, “Numbers as functions,” p-Adic Numbers Ultrametric Anal. Appl. 5 (4), 313–325 (2013), [arXiv:1312.5160].MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yu. Manin and M. Marcolli, “Big Bang, Blow up, and modular curves: Algebraic geometry in cosmology,” Symm. Integ. Geom. Meth. Appl. SIGMA 10 (2014), paper 073, 20 pp; [arXiv:1402.2158].MathSciNetzbMATHGoogle Scholar
  16. 16.
    Yu. Manin and M. Marcolli, “Symbolic dynamics, modular curves, and Bianchi IX cosmologies,” (2015), [arXiv:1504.04005 [gr-qc]].Google Scholar
  17. 17.
    K. Takasaki, “Painlevé -Calogero correspondence revisited,” J.Math. Phys. 42 (3), 1443–1473 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    K. P. Tod, “Self-dual Einstein metrics from the Painlevé VI equation,” Phys. Lett. A 190, 221–224 (1994).MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Max-Planck-Institut für MathematikBonnGermany

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