Geodesic
Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $X_\lambda$ and $X_\lambda'$ be monomial deformations of two Delsarte hypersurfaces in weighted projective spaces. In this paper we give a sufficient condition so that their zeta functions have a common factor. This generalises results by Doran, Kelly, Salerno, Sperber, Voight and Whitcher [arXiv:1612.09249], where they showed this for a particular monomial deformation of a Calabi–Yau invertible polynomial. It turns out that our factor can be of higher degree than the factor found in [arXiv:1612.09249].
Keywords: monomial deformation of Delsarte surfaces; zeta functions. 
@article{SIGMA_2017_13_a86,
     author = {Remke Kloosterman},
     title = {Zeta {Functions} of {Monomial} {Deformations} of {Delsarte} {Hypersurfaces}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2017},
     volume = {13},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a86/}
}
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Remke Kloosterman. Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a86/

[1] Baldassarri F., Chiarellotto B., “Algebraic versus rigid cohomology with logarithmic coefficients”, Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), Perspect. Math., 15, Academic Press, San Diego, CA, 1994, 11–50 | MR | Zbl

[2] Berthelot P., Géométrie rigide et cohomologie des variétés algébriques de caractéristique $p$ (Marseille, 1982), Study Group on Ultrametric Analysis, 3, no. J2, 9th year: 1981/82, Inst. Henri Poincaré, Paris, 1983, 18 pp. | MR

[3] Berthelot P., “Dualité de Poincaré et formule de Künneth en cohomologie rigide”, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 493–498 | DOI | MR | Zbl

[4] Bini G., “Quotients of hypersurfaces in weighted projective space”, Adv. Geom., 11 (2011), 653–667, arXiv: 0905.2099 | DOI | MR | Zbl

[5] Bini G., van Geemen B., Kelly T. L., “Mirror quintics, discrete symmetries and Shioda maps”, J. Algebraic Geom., 21 (2012), 401–412, arXiv: 0809.1791 | DOI | MR | Zbl

[6] Candelas P., de la Ossa X., Rodriguez-Villegas F., “Calabi–Yau manifolds over finite fields. II”, Calabi–Yau Varieties and Mirror Symmetry (Toronto, ON, 2001), Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003, 121–157, arXiv: hep-th/0402133 | MR | Zbl

[7] Chahal J., Meijer M., Top J., “Sections on certain {$j=0$} elliptic surfaces”, Comment. Math. Univ. St. Paul., 49 (2000), 79–89, arXiv: math.NT/9911274 | MR | Zbl

[8] Doran C. F., Kelly T. L., Salerno A., Sperber S., Voight J., Whitcher U., Zeta functions of alternate mirror Calabi–Yau families, arXiv: 1612.09249

[9] Dwork B., “$p$-adic cycles”, Inst. Hautes Études Sci. Publ. Math., 1969, 27–115 | DOI | MR | Zbl

[10] Gährs S., Picard–Fuchs equations of special one-parameter families of invertible polynomials, Ph.D. Thesis, der Gottfried Wilhelm Leibniz Universität Hannover, 2011, arXiv: 1109.3462 | MR

[11] Griffiths P. A., “On the periods of certain rational integrals. II”, Ann. of Math., 90 (1969), 496–541 | DOI | MR | Zbl

[12] Katz N. M., “On the differential equations satisfied by period matrices”, Inst. Hautes Études Sci. Publ. Math., 1968, 223–258 | DOI | MR

[13] Kelly T. L., “Berglund–Hübsch–Krawitz mirrors via Shioda maps”, Adv. Theor. Math. Phys., 17 (2013), 1425–1449, arXiv: 1304.3417 | DOI | MR | Zbl

[14] Kloosterman R., “Explicit sections on Kuwata's elliptic surfaces”, Comment. Math. Univ. St. Pauli, 54 (2005), 69–86, arXiv: math.AG/0502017 | MR | Zbl

[15] Kloosterman R., “The zeta function of monomial deformations of Fermat hypersurfaces”, Algebra Number Theory, 1 (2007), 421–450, arXiv: math.NT/0703120 | DOI | MR | Zbl

[16] Kloosterman R., Group actions on rigid cohomology with compact support — Erratum to “The zeta function of monomial deformations of Fermat hypersurfaces”, Preprint | MR

[17] Lauder A. G. B., “Deformation theory and the computation of zeta functions”, Proc. London Math. Soc., 88 (2004), 565–602 | DOI | MR | Zbl

[18] Pancratz S., Tuitman J., “Improvements to the deformation method for counting points on smooth projective hypersurfaces”, Found. Comput. Math., 15 (2015), 1413–1464, arXiv: 1307.1250 | DOI | MR | Zbl

[19] Scholten J., Mordell–Weil groups of elliptic surfaces and Galois representations, Ph.D. Thesis, Rijksuniversiteit Groningen, 2000

[20] Shioda T., “An explicit algorithm for computing the Picard number of certain algebraic surfaces”, Amer. J. Math., 108 (1986), 415–432 | DOI | MR | Zbl

[21] van der Put M., “The cohomology of Monsky and Washnitzer”, Mém. Soc. Math. France (N.S.), 1986, 33–59 | DOI | MR