Israel Journal of Mathematics

, Volume 228, Issue 2, pp 665–705 | Cite as

Zeta functions of alternate mirror Calabi–Yau families

  • Charles F. Doran
  • Tyler L. KellyEmail author
  • Adriana Salerno
  • Steven Sperber
  • John Voight
  • Ursula Whitcher
Open Access


We prove that if two Calabi–Yau invertible pencils have the same dual weights, then they share a common factor in their zeta functions. By using Dwork cohomology, we demonstrate that this common factor is related to a hypergeometric Picard–Fuchs differential equation. The factor in the zeta function is defined over the rationals and has degree at least the order of the Picard–Fuchs equation. As an application, we relate several pencils of K3 surfaces to the Dwork pencil, obtaining new cases of arithmetic mirror symmetry.


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© The Hebrew University of Jerusalem 2018

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution and reproduction in any medium, provided the appropriate credit is given to the original authors and the source, and a link is provided to the Creative Commons license, indicating if changes were made.

Authors and Affiliations

  • Charles F. Doran
    • 1
  • Tyler L. Kelly
    • 2
    Email author
  • Adriana Salerno
    • 3
  • Steven Sperber
    • 4
  • John Voight
    • 5
  • Ursula Whitcher
    • 6
  1. 1.Department of MathematicsUniversity of AlbertaEdmontonCanada
  2. 2.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK
  3. 3.Department of MathematicsBates CollegeLewistonUSA
  4. 4.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  5. 5.Department of MathematicsDartmouth CollegeHanoverUSA
  6. 6.Mathematical ReviewsAnn ArborUSA

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