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.
A. Adolphson and S. Sperber, Exponential sums and Newton polyhedra: cohomology and estimates, Annals of Mathematics 130 (1989), 367–406.
A. Adolphson and S. Sperber, On the zeta function of a projective complete intersection, Illinois Journal of Mathematics 52 (2008), 389–417.
A. Adolphson and S. Sperber, Distinguished-root formulas for generalized Calabi–Yau hypersurfaces, Algebra & Number Theory 11 (2017), 1317–1356.
M. Aldi and A. Peruničić, p-adic Berglund–Hübsch duality, Advances in Theoretical and Mathematical Physiocs 19 (2015), 1115–1139.
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.
P. Berglund and T. Hübsch, A generalized construction of mirror manifolds, Nuclear Physics. B 393 (1993), 377–391.
F. Beukers, Hypergeometric functions in one variable, Notes, 2008, available at https://www.staff.science.uu.nl/~beuke106/springschool99.pdf.
F. Beukers, H. Cohen and A. Mellit, Finite hypergeometric functions, Pure and Applied Mathematics Quarterly 11 (2015), 559–589.
G. Bini and A. Garbagnati, Quotients of the Dwork pencil, Journal of Geometry and Physics 75 (2014), 173–198.
G. Bini, B. van Geemen and T. L. Kelly, Mirror quintics, discrete symmetries and Shioda maps, Journal of Algebraic Geometry 21 (2012), 401–412.
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, Journal of Symbolic Computation 24 (1997), 235–265.
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.
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.
P. Candelas, X. de la Ossa and F. Rodriguez Villegas, Calabi–Yau manifolds over finite fields, I, arXiv:hep-th/0012233v1.
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.
F. Charles, The Tate conjecture for K3 surfaces over finite fields, Inventiones Mathematicae 194 (2013), 119–145.
F. Charles, On the Picard number of K3 surfaces over number fields, Algebra & Number Theory 8 (2014), 1–17.
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.
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.
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.
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.
C. F. Doran, T. L. Kelly, A. Salerno, S. Sperber, J. Voight and U. Whitcher, Hypergeometric properties of symmetric K3 quartic pencils, preprint.
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.
B. Dwork, p-adic cycles, Institut des Hautes Études Scientifiques. Publications Mathématiques 37 (1969), 27–115.
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.
N. D. Elkies and M. Schütt, K3 families of high Picard rank, unpublished notes.
W. Ebeling, S. M. Gusein-Zade, Orbifold zeta functions for dual invertible polynomials, Proceedings of the Edinburgh Mathematical Society 60 (2017), 99–106.
L. Fu and D. Wan, Mirror congruence for rational points on Calabi–Yau varieties, Asian Journal of Mathematics 10 (2006), 1–10.
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.
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.
B. R. Greene and M. Plesser, Duality in Calabi–Yau moduli space, Nuclear Physics. B 338 (1990), 15–37.
G. Harder and M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Mathematische Annalen 212 (1975), 215–248.
D. Huybrechts, Lectures on K3 Surfaces, Cambridge Studies in Advanced Mathematics, Vol. 158, Cambridge university Press, Cambridge, 2016.
S. Kadir, The arithmetic of Calabi–Yau manifolds and mirror symmetry, Ph.D. thesis, University of Oxford, 2004, arXiv: hep-th/0409202.
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.
N. Katz, On the differential equations satisfied by period matrices, Institut des Hautes Études Scientifiques. Publications Mathématiques 35 (1968), 223–258.
N. M. Katz, Algebraic solutions of differential equations (p-curvature and the Hodge filtration), Inventiones Mathematicae 18 (1972), 1–118.
N. M. Katz, Exponential Sums and Differential Equations, Annals of Mathematics Studies, Vol. 124, Princeton University Press, Princeton, NJ, 1990.
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.
W. Kim and K. Madapusi Pera, 2-adic integral canonical models, Forum of Mathematics, Sigma (2016), e28.
R. Kloosterman, Monomial deformations of Delsarte hypersurface, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications 13 (2017), paper no. 087.
M. Krawitz, FJRW rings and Landau–Ginzburg Mirror Symmetry, Ph.D. University of Michiga, 2010, arxiv: 0906.0796.
M. Kreuzer and H. Skarke, On the classification of quasihomogeneous functions, Communications in Mathematical Physics 150 (1992), 137–147.
S. Levy, ed., The Eightfold Way, Mathematical Sciences Research Institute Publications, Vol. 35, Cambridge University Press, Cambridge, 1999.
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.
B. Mazur, Frobenius and the Hodge filtration, Annals of Mathematics 98 (1973), 58–95.
K. Miyatani, Monomial deformations of certain hypersurfaces and two hypergeometric functions, International Journal of Number Theory 11 (2015), 2405–2430.
S. Mukai, Finite groups of automorphisms and the Mathieu group, Inventiones Mathematicae 94 (1988), 183–221.
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.
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.
K. Madapusi Pera, The Tate conjecture for K3 surfaces in odd characteristic, Inventiones Mathematicae 201 (2015), 625–668.
C. Sabbah, Hypergeometric differential and q-difference equations, https://www.cmls.polytechnique.fr/perso/sabbah/exposes/sabbah_lisbonne05.pdf.
T. Shioda, An explicit algorithm for computing the Picard number of certain algebraic surfaces, American Journal of Mathematics 108 (1986), 415–432.
S. Sperber and J. Voight, Computing zeta functions of nondegenerate hypersurfaces with few monomials, LMS Journal of Computation and Mathematics 16 (2013), 9–44.
R. van Luijk, K3 surfaces with Picard number one and infinitely many rational points, Algebra & Number Theory 1 (2007), 1–15.
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.
J.-D. Yu, Variation of the unit root along the Dwork family of Calabi–Yau varieties, Mathematische Annalen 343 (2009), 53–78.
About this article
Cite this article
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