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Zeta functions of alternate mirror Calabi–Yau families

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  • Published: 26 September 2018
  • volume 228, pages 665–705 (2018)
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Zeta functions of alternate mirror Calabi–Yau families
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  • Charles F. Doran1,
  • Tyler L. Kelly2,
  • Adriana Salerno3,
  • Steven Sperber4,
  • John Voight5 &
  • …
  • Ursula Whitcher6 

  • 630 Accesses

  • 8 Citations

  • 3 Altmetric

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Abstract

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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References

  1. A. Adolphson and S. Sperber, Exponential sums and Newton polyhedra: cohomology and estimates, Annals of Mathematics 130 (1989), 367–406.

    Article  MathSciNet  Google Scholar 

  2. A. Adolphson and S. Sperber, On the zeta function of a projective complete intersection, Illinois Journal of Mathematics 52 (2008), 389–417.

    MathSciNet  MATH  Google Scholar 

  3. A. Adolphson and S. Sperber, Distinguished-root formulas for generalized Calabi–Yau hypersurfaces, Algebra & Number Theory 11 (2017), 1317–1356.

    Article  MathSciNet  Google Scholar 

  4. M. Aldi and A. Peruničić, p-adic Berglund–Hübsch duality, Advances in Theoretical and Mathematical Physiocs 19 (2015), 1115–1139.

    MATH  Google Scholar 

  5. M. Artebani, S. Boissière and A. Sarti, The Berglund–Hübsch–Chiodo–Ruan mirror symmetry for K3 surfaces, Journal de Mathématiques Pures et Appliquées 102 (2014), 758–781.

    Article  MathSciNet  Google Scholar 

  6. P. Berglund and T. Hübsch, A generalized construction of mirror manifolds, Nuclear Physics. B 393 (1993), 377–391.

    Article  MathSciNet  Google Scholar 

  7. F. Beukers, Hypergeometric functions in one variable, Notes, 2008, available at https://www.staff.science.uu.nl/~beuke106/springschool99.pdf.

    Google Scholar 

  8. F. Beukers, H. Cohen and A. Mellit, Finite hypergeometric functions, Pure and Applied Mathematics Quarterly 11 (2015), 559–589.

    Article  MathSciNet  Google Scholar 

  9. G. Bini and A. Garbagnati, Quotients of the Dwork pencil, Journal of Geometry and Physics 75 (2014), 173–198.

    Article  MathSciNet  Google Scholar 

  10. G. Bini, B. van Geemen and T. L. Kelly, Mirror quintics, discrete symmetries and Shioda maps, Journal of Algebraic Geometry 21 (2012), 401–412.

    Article  MathSciNet  Google Scholar 

  11. W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, Journal of Symbolic Computation 24 (1997), 235–265.

    Article  MathSciNet  Google Scholar 

  12. P. Candelas and X. de la Ossa, The Zeta-function of a p-adic manifold, Dwork theory for physicists, Communications in Number Theory and Physics 1 (2007), 479–512.

    Article  Google Scholar 

  13. P. Candelas, X. C. de la Ossa, P. S. Green and L. Parkes, A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory, Nuclear Physics. B 359 (1991), 21–74.

    Article  MathSciNet  Google Scholar 

  14. P. Candelas, X. de la Ossa and F. Rodriguez Villegas, Calabi–Yau manifolds over finite fields, I, arXiv:hep-th/0012233v1.

  15. P. Candelas, X. de la Ossa snd F. Rodriguez-Villegas, Calabi–Yau manifolds over finite fields II, in Calabi–Yau Varieties and Mirror Symmetry (Toronto, ON, 2001), Fields Institute Communications, Vol. 38, American Mathematical Society, Providence, RI, 2003, pp. 121–157.

    MATH  Google Scholar 

  16. F. Charles, The Tate conjecture for K3 surfaces over finite fields, Inventiones Mathematicae 194 (2013), 119–145.

    Article  MathSciNet  Google Scholar 

  17. F. Charles, On the Picard number of K3 surfaces over number fields, Algebra & Number Theory 8 (2014), 1–17.

    Article  MathSciNet  Google Scholar 

  18. P. L. del Angel and S. Müller-Stach, Picard–Fuchs equations, integrable systems and higher algebraic K-theory, in Calabi–Yau Varieties and Mirror Symmetry (Toronto, ON, 2001), Fields Institute Communications, Vol. 38, American Mathematical Society, Providence, RI, 2003, pp. 43–55.

    MATH  Google Scholar 

  19. I. Dolgachev, Weighted projective varieties, in Group Actions and Vector Fields (Vancouver, BC, 1981), Lecture Notes in Mathematics, Vol. 956, Springer, Berlin, 1982, pp. 34–71.

    MATH  Google Scholar 

  20. C. F. Doran and R. S. Garavuso, Hori–Vafa mirror periods, Picard–Fuchs equations, and Berglund–Hübsch–Krawitz duality, Journal of High Energy Physics (2011), 10, Art. No. 128.

    MATH  Google Scholar 

  21. C. F. Doran, B. Greene and S. Judes, Families of Quintic Calabi–Yau 3-folds with Discrete Symmetries, Communications in Mathematical Physics 280 (2008), 675–725.

    Article  MathSciNet  Google Scholar 

  22. C. F. Doran, T. L. Kelly, A. Salerno, S. Sperber, J. Voight and U. Whitcher, Hypergeometric properties of symmetric K3 quartic pencils, preprint.

  23. B. Dwork, A deformation theory for the zeta function of a hypersurface, in Proceedings of the International Congress of Mathematicians (Stockholm, 1962), Institute Mittag-Leffler, Djursholm, 1963, pp. 247–259.

    MATH  Google Scholar 

  24. B. Dwork, p-adic cycles, Institut des Hautes Études Scientifiques. Publications Mathématiques 37 (1969), 27–115.

    Article  Google Scholar 

  25. B. Dwork, On the uniqueness of Frobenius operator on differential equations, in Algebraic Number Theory, Advanced Studies in Pure Mathematics, Vol. 17, Academic Press, Boston, MA, 1989, pp. 89–96.

    Article  MathSciNet  Google Scholar 

  26. N. D. Elkies and M. Schütt, K3 families of high Picard rank, unpublished notes.

  27. W. Ebeling, S. M. Gusein-Zade, Orbifold zeta functions for dual invertible polynomials, Proceedings of the Edinburgh Mathematical Society 60 (2017), 99–106.

    Article  MathSciNet  Google Scholar 

  28. L. Fu and D. Wan, Mirror congruence for rational points on Calabi–Yau varieties, Asian Journal of Mathematics 10 (2006), 1–10.

    Article  MathSciNet  Google Scholar 

  29. S. Gährs, Picard–Fuchs equations of special one-parameter families of invertible polynomials, Ph.D. thesis, Gottfried Wilhelm Leibniz Universität Hannover, 2011, arXiv:1109.3462.

    MATH  Google Scholar 

  30. S. Gährs, Picard–Fuchs equations of special one-parameter families of invertible polynomials, in Arithmetic and Geometry of K3 surfaces and Calabi–Yau Threefolds, Fields Institute Communications, Vol. 67, Springer, New York, 2013, pp. 285–310.

    Article  MathSciNet  Google Scholar 

  31. B. R. Greene and M. Plesser, Duality in Calabi–Yau moduli space, Nuclear Physics. B 338 (1990), 15–37.

    Article  MathSciNet  Google Scholar 

  32. G. Harder and M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Mathematische Annalen 212 (1975), 215–248.

    Article  MathSciNet  Google Scholar 

  33. D. Huybrechts, Lectures on K3 Surfaces, Cambridge Studies in Advanced Mathematics, Vol. 158, Cambridge university Press, Cambridge, 2016.

  34. S. Kadir, The arithmetic of Calabi–Yau manifolds and mirror symmetry, Ph.D. thesis, University of Oxford, 2004, arXiv: hep-th/0409202.

    Google Scholar 

  35. S. Kadir, Arithmetic mirror symmetry for a two-parameter family of Calabi–Yau manifolds, in Mirror Symmetry. V, AMS/IP Studies in Advanced Mathematics, Vol. 38, American Mathematical Society, Providence, RI, 2006, pp. 35–86.

    MATH  Google Scholar 

  36. N. Katz, On the differential equations satisfied by period matrices, Institut des Hautes Études Scientifiques. Publications Mathématiques 35 (1968), 223–258.

    MathSciNet  Google Scholar 

  37. N. M. Katz, Algebraic solutions of differential equations (p-curvature and the Hodge filtration), Inventiones Mathematicae 18 (1972), 1–118.

    Article  MathSciNet  Google Scholar 

  38. N. M. Katz, Exponential Sums and Differential Equations, Annals of Mathematics Studies, Vol. 124, Princeton University Press, Princeton, NJ, 1990.

  39. N. M. Katz, Another look at the Dwork family, in Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin. Vol. II, Progress in Mathematics, Vol. 270, Birkhäuser Boston, Boston, MA, 2009, pp. 89–126.

    Google Scholar 

  40. W. Kim and K. Madapusi Pera, 2-adic integral canonical models, Forum of Mathematics, Sigma (2016), e28.

    MATH  Google Scholar 

  41. R. Kloosterman, Monomial deformations of Delsarte hypersurface, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications 13 (2017), paper no. 087.

    Google Scholar 

  42. M. Krawitz, FJRW rings and Landau–Ginzburg Mirror Symmetry, Ph.D. University of Michiga, 2010, arxiv: 0906.0796.

    Google Scholar 

  43. M. Kreuzer and H. Skarke, On the classification of quasihomogeneous functions, Communications in Mathematical Physics 150 (1992), 137–147.

    Article  MathSciNet  Google Scholar 

  44. S. Levy, ed., The Eightfold Way, Mathematical Sciences Research Institute Publications, Vol. 35, Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  45. C. Magyar and U. Whitcher, Strong arithmetic mirror symmetry and toric isogenies, in Higher Genus Curves in Mathematical Physics and Arithmetic Geometry, Contemporary Mathematics, Vol. 703, American Mathematical Society, Providence, RI, 2018, 117–129.

    Article  MathSciNet  Google Scholar 

  46. B. Mazur, Frobenius and the Hodge filtration, Annals of Mathematics 98 (1973), 58–95.

    Article  MathSciNet  Google Scholar 

  47. K. Miyatani, Monomial deformations of certain hypersurfaces and two hypergeometric functions, International Journal of Number Theory 11 (2015), 2405–2430.

    Article  MathSciNet  Google Scholar 

  48. S. Mukai, Finite groups of automorphisms and the Mathieu group, Inventiones Mathematicae 94 (1988), 183–221.

    Article  MathSciNet  Google Scholar 

  49. N. Narumiya and H. Shiga, The mirror map for a family of K3 surfaces induced from the simplest 3-dimensional reflexive polytope, Proceedings on Moonshine and related topics (Montréal, QC, 1999), CRM Proceedings & Lecture Notes, Vol. 30, American Mathematical Society, Providence, RI, 2001, pp. 139–161.

    MATH  Google Scholar 

  50. K. Oguiso and D.-Q. Zhang, The simple group of order 168 and K3 surfaces, in Complex Geometry (Göttingen, 2000), Springer, Berlin, 2002, pp. 165–184.

    Google Scholar 

  51. K. Madapusi Pera, The Tate conjecture for K3 surfaces in odd characteristic, Inventiones Mathematicae 201 (2015), 625–668.

    Article  MathSciNet  Google Scholar 

  52. C. Sabbah, Hypergeometric differential and q-difference equations, https://www.cmls.polytechnique.fr/perso/sabbah/exposes/sabbah_lisbonne05.pdf.

  53. T. Shioda, An explicit algorithm for computing the Picard number of certain algebraic surfaces, American Journal of Mathematics 108 (1986), 415–432.

    Article  MathSciNet  Google Scholar 

  54. S. Sperber and J. Voight, Computing zeta functions of nondegenerate hypersurfaces with few monomials, LMS Journal of Computation and Mathematics 16 (2013), 9–44.

    Article  MathSciNet  Google Scholar 

  55. R. van Luijk, K3 surfaces with Picard number one and infinitely many rational points, Algebra & Number Theory 1 (2007), 1–15.

    Article  MathSciNet  Google Scholar 

  56. D. Wan, Mirror symmetry for zeta functions, in Mirror Symmetry. V, AMS/IP Studies in Advanced Mathematics, Vol. 38, American Mathematical Society, Providence, RI, 2006, pp. 159–184.

    MATH  Google Scholar 

  57. J.-D. Yu, Variation of the unit root along the Dwork family of Calabi–Yau varieties, Mathematische Annalen 343 (2009), 53–78.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Alberta, Edmonton, AB, Canada

    Charles F. Doran

  2. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB, UK

    Tyler L. Kelly

  3. Department of Mathematics, Bates College, 3 Andrews Rd., Lewiston, ME, 04240, USA

    Adriana Salerno

  4. School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, MN, 55455, USA

    Steven Sperber

  5. Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH, 03755, USA

    John Voight

  6. Mathematical Reviews, 416 Fourth St, Ann Arbor, MI, 48103, USA

    Ursula Whitcher

Authors
  1. Charles F. Doran
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Corresponding author

Correspondence to Tyler L. Kelly.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://doi.org/creativecommons.org/licenses/by/4.0/), 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.

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Doran, C.F., Kelly, T.L., Salerno, A. et al. Zeta functions of alternate mirror Calabi–Yau families. Isr. J. Math. 228, 665–705 (2018). https://doi.org/10.1007/s11856-018-1783-0

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  • Received: 27 February 2017

  • Revised: 19 January 2018

  • Published: 26 September 2018

  • Issue Date: October 2018

  • DOI: https://doi.org/10.1007/s11856-018-1783-0

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