Mathematical Notes

, Volume 71, Issue 5–6, pp 604–616

Integrality of Power Expansions Related to Hypergeometric Series

  • V. V. Zudilin
Article

Abstract

In the present paper, we study the arithmetic properties of power expansions related to generalized hypergeometric differential equations and series. Defining the series f(z),g(z) in powers of z so that f(z) and \(f\left( z \right)\log z + g\left( z \right)\) satisfy a hypergeometric equation under a special choice of parameters, we prove that the series \(q\left( z \right) = ze^{{{g\left( {Cz} \right)} \mathord{\left/ {\vphantom {{g\left( {Cz} \right)} {f\left( {Cz} \right)}}} \right. \kern-\nulldelimiterspace} {f\left( {Cz} \right)}}}\) in powers of z and its inversion z(q) in powers of q have integer coefficients (here the constant C depends on the parameters of the hypergeometric equation). The existence of an integral expansion z(q) for differential equations of second and third order is a classical result; for orders higher than 3 some partial results were recently established by Lian and Yau. In our proof we generalize the scheme of their arguments by using Dwork's p-adic technique.

integral power expansion hypergeometric series linear differential equation Calabi--Yau manifold mirror map 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    G. Pólya and G. Szegó, Aufgaben und Lehrsätze aus der Analysis, 3d ed., Springer-Verlag, Berlin-Göttingen-Heidelberg-New York, 1964.Google Scholar
  2. 2.
    D. R. Morrison, “Picard-Fuchs equations and mirror maps for hypersurfaces,” in: Essays on Mirror Manifolds (S.-T. Yau, editor), International Press, Hong Kong, 1992, pp. 241–264; in:Mirror Symmetry, I (S.-T. Yau, editor), vol. 9, IP Stud. Adv. Math., Amer. Math. Soc., Providence, R.I., 1998, pp. 185–199.Google Scholar
  3. 3.
    V. V. Batyrev and D. van Straten, “Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties,” Comm. Math. Phys., 168 (1995), no. 3, 493–533.Google Scholar
  4. 4.
    B. H. Lian and S.-T. Yau, “Mirror maps, modular relations and hypergeometric series I,” in: E-print http://arXiv.org/abs/hep-th/9507151; “Integrality of certain exponential series,” in: Lectures in Algebra and Geometry (M.-C. Kang, editor), Proceedings of the International Conference on Algebra and Geometry, National Taiwan University (Taipei, Taiwan, December 26–30, 1995), International Press, Cambridge, MA, 1998, pp. 215–227.Google Scholar
  5. 5.
    B. Dwork, “On p-adic differential equations IV. Generalized hypergeometric functions as p-adic analytic functions in one variable,” Ann. Sci. École Norm. Sup. (4 ), 6 (1973), no. 3, 295–315.Google Scholar
  6. 6.
    N. M. Katz, “Algebraic solutions of differential equations (p-curvature and the Hodge fibration),” Invent. Math., 18 (1972), no. 1/2, 1–118.Google Scholar
  7. 7.
    D. R. Morrison, “Mathematical aspects of mirror symmetry,” in: Complex Algebraic Geometry (J. Kollár, editor), vol. 3, Lectures of a Summer Program (Park City, UT, 1993), IAS/Park City Math. Ser., Amer. Math. Soc., Providence, R.I., 1997, pp. 267–340.Google Scholar
  8. 8.
    W. Zudilin, “Number theory casting a look at the mirror,” in: E-print http://arXiv.org/abs/math/ 0008237 (preprint, 2000; submitted for publication).Google Scholar
  9. 9.
    N. Kobliz, p-Adic Numbers, p-Adic Analysis, and Zeta-Functions, Springer-Verlag, Heidelberg, 1977.Google Scholar
  10. 10.
    Y. André, G-Functions and Geometry, vol. E13, Aspects Math. (A Publication of the Max-Planck-Institut für Mathematik, Bonn), Vieweg, Braunschweig, 1989.Google Scholar
  11. 11.
    S. Lang, Cyclotomic Fields, I, II, Combined 2nd edition, vol. 121, Graduate Texts in Math., Springer-Verlag, New York, 1990.Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • V. V. Zudilin
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityRussia

Personalised recommendations