Hypergeometric decomposition of symmetric \(K3\) quartic pencils. (English) Zbl 1441.14128
This paper studies the hypergeometric functions associated to the five one-parameter deformations of Delsatre \(K3\) quartic surfaces. These families are labelled as \(F_4, F_1L_3, F_2L_2, L_2L_2\) and \(L_4\), and for each of them, there is a discrete group of symmetriies acting symplectically. The main theorem shows that hypergeometric functions are associated to this set of Delsarte hypersurface pencils in two ways as Picard-Fuchs differential equations and as traces of Frobenius.
The Picard-Fuchs differential equations are explicitly determined for these families. Further, the zeta functions and \(L\)-functions of these families are computed explicitly. The method is computational on finite Gauss sums, and finite hypergeometric sums, which give rise to hypergeometric motives.
The Picard-Fuchs differential equations are explicitly determined for these families. Further, the zeta functions and \(L\)-functions of these families are computed explicitly. The method is computational on finite Gauss sums, and finite hypergeometric sums, which give rise to hypergeometric motives.
Reviewer: Noriko Yui (Kingston)
MSC:
| 14J28 | \(K3\) surfaces and Enriques surfaces |
| 14Q10 | Computational aspects of algebraic surfaces |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
| 33D67 | Basic hypergeometric functions associated with root systems |
Keywords:
\(K3\) quartic pencils; zeta functions; arithmetic mirror symmetry for \(K3\) surfaces; hypergeometric functions over finite fields; Gamma functions; hypergeometric motivesSoftware:
MagmaReferences:
| [1] | Berndt, B.; Evans, R.; Williams, K., Gauss and Jacobi Sums (1998), New York: Wiley, New York · Zbl 0906.11001 |
| [2] | Beukers, F., Fields of definition of finite hypergeometric functions, Hypergeometric Motives and Calabi-Yau Differential Equations (2018), New York: Springer, New York |
| [3] | Beukers, F.; Cohen, H.; Mellit, A., Finite hypergeometric functions, Pure Appl. Math. Q., 11, 4, 559-589 (2015) · Zbl 1397.11162 · doi:10.4310/PAMQ.2015.v11.n4.a2 |
| [4] | Beukers, F.; Heckman, G., Monodromy for the hypergeometric function \(_nF_{n-1}\), Invent. Math., 95, 2, 325-354 (1989) · Zbl 0663.30044 · doi:10.1007/BF01393900 |
| [5] | Bini, G., Quotients of hypersurfaces in weighted projective space, Adv. Geom., 11, 4, 653-667 (2011) · Zbl 1235.14035 · doi:10.1515/advgeom.2011.029 |
| [6] | Bini, G.; Garbagnati, A., Quotients of the Dwork pencil, J. Geom. Phys., 75, 173-198 (2014) · Zbl 1314.14078 · doi:10.1016/j.geomphys.2013.10.001 |
| [7] | Bini, G.; van Geemen, B.; Kelly, TL, Mirror quintics, discrete symmetries and Shioda maps, J. Alg. Geom., 21, 401-412 (2012) · Zbl 1246.14054 · doi:10.1090/S1056-3911-2011-00544-4 |
| [8] | Bosma, W.; Cannon, J.; Playoust, C., The Magma algebra system. I. The user language, J. Symb. Comput., 24, 3-4, 235-265 (1997) · Zbl 0898.68039 · doi:10.1006/jsco.1996.0125 |
| [9] | Candelas, P., de la Ossa, X., Rodríguez Villegas, F.: Calabi-Yau manifolds over finite fields, I. arXiv:hep-th/0012233 (2000) · Zbl 1100.14032 |
| [10] | Candelas, P.; de la Ossa, X.; Rodríguez Villegas, F., Calabi-Yau manifolds over finite fields II, Calabi-Yau Varieties and Mirror Symmetry, 121-157 (2003), Providence: American Mathematical Society, Providence · Zbl 1100.14032 |
| [11] | Clemens, CH, A Scrapbook of Complex Curve Theory (2003), Providence: American Mathematical Society, Providence · Zbl 1030.14010 |
| [12] | Cohen, H., Number Theory Volume I: Tools and Diophantine Equations (2007), New York: Springer, New York · Zbl 1119.11001 |
| [13] | Cohen, H.: Counting points of Calabi-Yau Manifolds after P. X. de la Ossa, and F. Rodrigez-Villegas. unpublished notes, Candelas (2010) |
| [14] | Costa, E.; Tschinkel, Y., Variation of Néron-Severi ranks of reductions of K3 surfaces, Exp. Math., 23, 4, 475-481 (2014) · Zbl 1311.14039 · doi:10.1080/10586458.2014.947054 |
| [15] | Cox, DA; Katz, S., Mirror Symmetry and Algebraic Geometry (1999), Providence: American Mathematical Society, Providence · Zbl 0951.14026 |
| [16] | Delsarte, J.: Nombre de solutions des équations polynomiales sur un corps fini. Sém. Bourbaki 39-01 (1951) |
| [17] | Doran, CF; Greene, B.; Judes, S., Families of Quintic Calabi-Yau 3-folds with discrete symmetries, Commun. Math. Phys., 280, 675-725 (2008) · Zbl 1158.14034 · doi:10.1007/s00220-008-0473-x |
| [18] | Doran, CF; Kelly, TL; Salerno, A.; Sperber, S.; Voight, J.; Whitcher, U., Zeta functions of alternate mirror Calabi-Yau families, Isr. J. Math., 228, 2, 665-705 (2018) · Zbl 1403.14055 · doi:10.1007/s11856-018-1783-0 |
| [19] | Duan, L.: Galois action on Néron-Severi Group of Dwork surfaces. arXiv:1809.08693 (2018) · Zbl 1455.11092 |
| [20] | Dwork, B., \(p\)-adic cycles, Inst. Hautes Études Sci. Publ. Math., 37, 27-115 (1969) · Zbl 0284.14008 · doi:10.1007/BF02684886 |
| [21] | Elkies, N. D., Schütt, M.: K3 families of high Picard rank. http://www2.iag.uni-hannover.de/ schuett/K3-fam.pdf (2008). Accessed 14 December 2017 |
| [22] | Furtado Gomida, E., On the theorem of Artin-Weil, Soc. Mat. São Paulo, 4, 267-277 (1951) |
| [23] | Fuselier, J., Long, L., Ramakrishna, R., Swisher, H., Tu, F.-T.: Hypergeometric Functions over Finite Fields. arXiv:1510.02575 (2017) · Zbl 1443.11254 |
| [24] | Gährs, S., Picard-Fuchs equations of special one-parameter families of invertible polynomials, Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds, 285-310 (2013), New York: Springer, New York · Zbl 1302.14034 |
| [25] | Goodson, H., A complete hypergeometric point count formula for dwork hypersurfaces, J. Number Theory, 179, 142-171 (2017) · Zbl 1418.11157 · doi:10.1016/j.jnt.2017.03.018 |
| [26] | Goodson, H., Hypergeometric functions and relations to Dwork hypersurfaces, Int. J. Number Theory, 13, 2, 439-485 (2017) · Zbl 1419.11135 · doi:10.1142/S1793042117500269 |
| [27] | Greene, J., Hypergeometric functions over finite fields, Trans. Am. Math. Soc., 301, 1, 77-101 (1987) · Zbl 0629.12017 · doi:10.1090/S0002-9947-1987-0879564-8 |
| [28] | Griffiths, P., On the periods of certain rational integrals I, Ann. Math., 2, 90, 460-495 (1969) · Zbl 0215.08103 · doi:10.2307/1970746 |
| [29] | Igusa, J., Class number of a definite quaternion with prime discriminant, Proc. Nat. Acad. Sci. USA, 44, 312-314 (1958) · Zbl 0081.03601 · doi:10.1073/pnas.44.4.312 |
| [30] | Kadir, S.: The Arithmetic of Calabi-Yau manifolds and mirror symmetry. Oxford DPhil Thesis. arXiv:hep-th/0409202 (2004) |
| [31] | Katz, N., Exponential Sums and Differential Equations (1990), Princeton: Princeton University Press, Princeton · Zbl 0731.14008 |
| [32] | Katz, NM, Another look at the Dwork family, Algebra, Arithmetic, and Geometry: in Honor of Yu. I. Manin, 89-126 (2009), Boston: Birkhäuser, Boston · Zbl 1195.14015 |
| [33] | Kelly, TL, Berglund-Hübsch-Krawitz Mirrors via Shioda Maps, Adv. Theor. Math. Phys., 17, 6, 1425-1449 (2013) · Zbl 1316.14076 · doi:10.4310/ATMP.2013.v17.n6.a8 |
| [34] | Kloosterman, R., The zeta function of monomial deformations of Fermat hypersurfaces, Algebra Number Theory, 1, 4, 421-450 (2007) · Zbl 1166.14016 · doi:10.2140/ant.2007.1.421 |
| [35] | Kloosterman, R.: Group actions on rigid cohomology with finite support—Erratum to “The zeta function of monomial deformations of Fermat hypersurfaces”. http://www.math.unipd.it/ klooster/ANT_erratum.pdf. Accessed 14 December 2017 |
| [36] | Kloosterman, R., Zeta functions of monomial deformations of Delsarte hypersurfaces, SIGMA, 13, 87, 22 (2017) · Zbl 1432.14019 |
| [37] | Koblitz, N., The number of points on certain families of hypersurfaces over finite fields, Comput. Math., 48, 3-23 (1983) · Zbl 0509.14023 |
| [38] | McCarthy, D., On a supercongruence conjecture of Rodríguez-Villegas, Proc. Am. Math. Soc., 140, 2241-2254 (2012) · Zbl 1354.11030 · doi:10.1090/S0002-9939-2011-11087-6 |
| [39] | McCarthy, D., Transformations of well-poised hypergeometric functions over finite fields, Finite Fields Appl., 18, 6, 1133-1147 (2012) · Zbl 1276.11198 · doi:10.1016/j.ffa.2012.08.007 |
| [40] | McCarthy, D., The trace of Frobenius of elliptic curves and the \(p\)-adic gamma function, Pac. J. Math., 261, 1, 219-236 (2013) · Zbl 1296.11079 · doi:10.2140/pjm.2013.261.219 |
| [41] | McCarthy, D.: The number of \({\mathbb{F}}_p\)-points on Dwork hypersurfaces and hypergeometric functions. Res. Math. Sci. 4, Paper No. 4, 15 pp. (2017) · Zbl 1410.11082 |
| [42] | Miyatani, K., Monomial deformations of certain hypersurfaces and two hypergeometric functions, Int. J. Number Theory, 11, 8, 2405-2430 (2015) · Zbl 1332.14028 · doi:10.1142/S1793042115501122 |
| [43] | Naskręcki, B.: On a certain hypergeometric motive of weight 2 and rank 3. arXiv:1702.07738 (2017) |
| [44] | Naskręcki, B.: On realisations of weight 0 hypergeometric motives of small degrees. Preprint |
| [45] | Salerno, A.: Hypergeometric Functions in Arithmetic Geometry. PhD. Thesis, University of Texas at Austin (2009) |
| [46] | Salerno, A., Counting points over finite fields and hypergeometric functions, Funct. Approx. Comment. Math., 49, 1, 137-157 (2013) · Zbl 1295.11074 · doi:10.7169/facm/2013.49.1.9 |
| [47] | Shioda, T., An explicit algorithm for computing the Picard number of certain algebraic surfaces, Am. J. Math., 108, 2, 415-432 (1986) · Zbl 0602.14033 · doi:10.2307/2374678 |
| [48] | Slater, LJ, Generalized Hypergeometric Functions (1966), Cambridge: Cambridge University Press, Cambridge · Zbl 0135.28101 |
| [49] | Voight, J.: Supplementary code. https://www.math.dartmouth.edu/ jvoight/magma/fiveK3fam.zip (2018) |
| [50] | Weil, A., Numbers of solutions of equations in finite fields, Bull. Am. Math. Soc., 55, 497-508 (1949) · Zbl 0032.39402 · doi:10.1090/S0002-9904-1949-09219-4 |
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