Skip to main content

Arithmetic properties of mirror maps associated with Gauss hypergeometric equations


We draw up the list of Gauss hypergeometric differential equations having maximal unipotent monodromy at \(0\) whose associated mirror map has, up to a simple rescaling, integral Taylor coefficients at \(0\). We also prove that these equations are characterized by much weaker integrality properties (of \(p\)-adic integrality for infinitely many primes \(p\) in suitable arithmetic progressions). It turns out that the mirror maps with the above integrality property have modular origins.

This is a preview of subscription content, access via your institution.


  1. André, Y.: \(G\)-fonctions et transcendance. J. Reine Angew. Math. 476, 95–125 (1996)

    MathSciNet  MATH  Google Scholar 

  2. Delaygue, E.: Intégralité des coefficients de taylor de racines d’applications miroir. Journal de Théorie des Nombres de Bordeaux, (2011)

  3. Delaygue, E.: Propriétés arithmétiques des applications miroir. Thèse, Grenoble., (2011)

  4. Delaygue, E.: Critère pour l’intégralité des coefficients de taylor des applications miroir. J. Reine Angew. Math. 662, 205–252 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Dwork, B.: On \(p\)-adic differential equations, IV. Generalized hypergeometric functions as \(p\)-adic analytic functions in one variable. Ann. Sci. École Norm. Sup. 6(4), 295–315 (1973)

    MathSciNet  MATH  Google Scholar 

  6. Katz, N.M.: Exponential Sums and Differential Equations, Volume 124 of Annals of Mathematics Studies. Princeton University Press, Princeton (1990)

    Google Scholar 

  7. Krattenthaler, C., Rivoal, T.: On the integrality of the Taylor coefficients of mirror maps II. Commun. Number Theory Phys. 3(3), 555–591 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Krattenthaler, C., Rivoal, T.: On the integrality of the Taylor coefficients of mirror maps. Duke Math. J. 151(2), 175–218 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  9. Krattenthaler, C., Rivoal, T.: Analytic properties of mirror maps. J. Aust. Math. Soc., (2011)

  10. Lian, B.H., Yau, S.-T.: Arithmetic properties of mirror map and quantum coupling. Comm. Math. Phys. 176(1), 163–191 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  11. Lian, B.H., Yau, S.-T.: Integrality of certain exponential series. In: Algebra and geometry (Taipei 1995), vol. 2 of Lectures for Algebraic Geometry, pp 215–227. International Press, Cambridge (1998)

  12. Lian, B.H., Yau, S.-T.: The \(n\)th root of the mirror map. In: Calabi-Yau varieties and mirror symmetry (Toronto 2001), vol. 38 of Fields Institute Communication, pp 195–199. American Mathematical Society Providence, RI (2003)

  13. Slater, L.J.: Generalized hypergeometric functions. Cambridge University Press, Cambridge (1966)

    MATH  Google Scholar 

  14. Zudilin, V.V.: On the integrality of power expansions related to hypergeometric series. Mat. Zametki 71(5), 662–676 (2002)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Julien Roques.

Additional information

Communicated by C. Krattenthaler.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Roques, J. Arithmetic properties of mirror maps associated with Gauss hypergeometric equations. Monatsh Math 171, 241–253 (2013).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Hypergeometric series and equations
  • Mirror maps

Mathematics Subject Classification (2000)

  • 33C05