Alternate Mirror Families and Hypergeometric Motives

  • Charles F. Doran
  • Tyler L. Kelly
  • Adriana SalernoEmail author
  • Steven Sperber
  • John Voight
  • Ursula Whitcher
Part of the MATRIX Book Series book series (MXBS, volume 2)


Mirror symmetry predicts surprising geometric correspondences between distinct families of algebraic varieties. In some cases, these correspondences have arithmetic consequences. Among the arithmetic correspondences predicted by mirror symmetry are correspondences between point counts over finite fields, and more generally between factors of their Zeta functions. In particular, we will discuss our results on a common factor for Zeta functions of alternate families of invertible polynomials. We will also explore closed formulas for the point counts for our alternate mirror families of K3 surfaces and their relation to their Picard–Fuchs equations. Finally, we will discuss how all of this relates to hypergeometric motives. This report summarizes work from two papers.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors heartily thank Xenia de la Ossa for her input and many discussions about this project. They thank Simon Judes for sharing his expertise, Frits Beukers for numerous helpful discussions, and Edgar Costa for sharing his code for computing zeta functions. The authors would like to thank the American Institute of Mathematics and its SQuaRE program, the Banff International Research Station, SageMath, and the MATRIX Institute for facilitating their work together. Kelly acknowledges that this material is based upon work supported by the NSF under Award No. DMS-1401446 and the EPSRC under EP/N004922/1. Voight was supported by an NSF CAREER Award (DMS-1151047).


  1. 1.
    Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebraic Geom. 3(3), 493–535 (1994)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Berglund, P., Hübsch, T.: A generalized construction of mirror manifolds. Nuclear Phys. B 393(1–2), 377–391 (1993)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Candelas, P., de la Ossa, X.: The Zeta-Function of a p-adic Manifold, Dwork Theory for Physicists. arxiv:0705.2056v1 (2008)Google Scholar
  4. 4.
    Candelas, P., de la Ossa, X., Rodriguez Villegas, F.: Calabi–Yau manifolds over finite fields, I. arXiv:hep-th/0012233v1 (2000)Google Scholar
  5. 5.
    Candelas, P., de la Ossa, X., Rodriguez-Villegas, F.: Calabi–Yau manifolds over finite fields II. In: Calabi–Yau Varieties and Mirror Symmetry, Toronto, pp. 121–157, (2001). hep-th/0402133Google Scholar
  6. 6.
    Delsarte, J.: Nombre de solutions des équations polynomiales sur un corps fini. Sém. Bourbaki 39(1), 321–329 (1951)Google Scholar
  7. 7.
    Doran, C.F., Greene, B., Judes, S.: Families of quintic Calabi–Yau 3-folds with discrete symmetries. Commun. Math. Phys. 280, 675–725 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Doran, C.F., Kelly, T., Salerno, A., Sperber, S., Voight, J., Whitcher, U.: Zeta functions of alternate mirror Calabi–Yau families. Israel J. Math. 228(2), 665–705 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Doran, C.F., Kelly, T., Salerno, A., Sperber, S., Voight, J., Whitcher, U.: Hypergeometric decomposition of symmetric K3 quartic pencils. arXiv:1810.06254Google Scholar
  10. 10.
    Dwork, B.: On the rationality of the zeta function of an algebraic variety. Am. J. Math. 82(3), 631–648 (1960)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dwork, B.: p-adic cycles. Inst. Hautes Études Sci. Publ. Math. 37, 27–115 (1969)Google Scholar
  12. 12.
    Fu, L., Wan, D.: Mirror congruence for rational points on Calabi-Yau varieties. Asian J. Math. 10(1), 1–10 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Furtado Gomida, E.: On the theorem of Artin-Weil. Soc. Mat. São Paulo 4, 267–277 (1951)Google Scholar
  14. 14.
    Gährs, S.: Picard–Fuchs equations of special one-parameter families of invertible polynomials, Ph.D. thesis, Gottfried Wilhelm Leibniz Univ. Hannover, arXiv:1109.3462Google Scholar
  15. 15.
    Greene, B.R., Plesser, M.: Duality in Calabi-Yau moduli space. Nuclear Phys. B 338(1), 15–37 (1990)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kloosterman, R.: The zeta function of monomial deformations of Fermat hypersurfaces. In: Algebra Number Theory, vol.1, no. 4. Mathematical Science Publishers, Berkeley (2007)Google Scholar
  17. 17.
    Krawitz, M.: FJRW rings and Landau-Ginzburg Mirror Symmetry. arxiv:0906.0796 (2009)Google Scholar
  18. 18.
    Mukai, S.: Finite groups of automorphisms of K3 surfaces and the Mathieu group. Invent. Math. 94(1), 183–221 (1988)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Oguiso, K.: A characterization of the Fermat quartic K3 surface by means of finite symmetries. Compos. Math. 141(2), 404–424 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wan, D.: Mirror symmetry for zeta functions. In:Mirror Symmetry. V, vol. 38. AMS/IP Studies in Advanced Mathematics (2006)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Charles F. Doran
    • 1
  • Tyler L. Kelly
    • 2
  • Adriana Salerno
    • 3
    Email author
  • Steven Sperber
    • 4
  • John Voight
    • 5
  • Ursula Whitcher
    • 6
  1. 1.University of AlbertaDepartment of MathematicsEdmontonCanada
  2. 2.School of MathematicsUniversity of BirminghamBirminghamUK
  3. 3.Department of MathematicsBates CollegeLewistonUSA
  4. 4.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  5. 5.Department of MathematicsDartmouth CollegeHanoverUSA
  6. 6.Mathematical ReviewsAnn ArborUSA

Personalised recommendations