Journal of the Mathematical Society of Japan

Apéry–Fermi pencil of $K3$-surfaces and 2-isogenies

Marie José BERTIN and Odile LECACHEUX

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Given a generic $K3$-surface $Y_k$ of the Apéry–Fermi pencil, we use the Kneser–Nishiyama technique to determine all its non isomorphic elliptic fibrations. These computations lead to determine those fibrations with 2-torsion sections T. We classify the fibrations such that the translation by T gives a Shioda–Inose structure. The other fibrations correspond to a $K3$-surface identified by its transcendental lattice. The same problem is solved for a singular member $Y_2$ of the family showing the differences with the generic case. In conclusion we put our results in the context of relations between 2-isogenies and isometries on the singular surfaces of the family.

Article information

Source
J. Math. Soc. Japan, Advance publication (2019), 39 pages.

Dates
Received: 26 May 2018
Revised: 17 October 2018
First available in Project Euclid: 16 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1571212902

Digital Object Identifier
doi:10.2969/jmsj/80638063

Subjects
Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 11G05: Elliptic curves over global fields [See also 14H52] 14J27: Elliptic surfaces 14J50: Automorphisms of surfaces and higher-dimensional varieties 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]

Keywords
elliptic fibrations of $K3$-surfaces Morrison–Nikulin involutions isogenies

Citation

BERTIN, Marie José; LECACHEUX, Odile. Apéry–Fermi pencil of $K3$-surfaces and 2-isogenies. J. Math. Soc. Japan, advance publication, 16 October 2019. doi:10.2969/jmsj/80638063. https://projecteuclid.org/euclid.jmsj/1571212902


Export citation

References

  • [1] A. O. L. Atkin and J. Lehner, Hecke operators on $\Gamma_0 (m)$, Math. Ann., 185 (1970), 134–160.
  • [2] A. Beauville, Les familles stables de courbes elliptiques sur $\mathbb{P}^1$ admettant 4 fibres singulières, C. R. Acad. Sci. Paris Sér. I Math., 294 (1982), 657–660.
  • [3] M. J. Bertin, A. Garbagnati, R. Hortsch, O. Lecacheux, M. Mase, C. Salgado and U. Whitcher, Classifications of elliptic fibrations of a singular $K3$-surface, In: Women in Numbers Europe: Research Directions in Number Theory, Assoc. Women Math. Ser., 2, Springer, 2015, 17–49.
  • [4] M. J. Bertin, A. Feaver, J. Fuselier, M. Lalìn and M. Manes, Mahler measure of some singular $K3$-surfaces, In: Women in Numbers 2: Research Directions in Number Theory, Contemp. Math., 606, Centre Rech. Math. Proc., Amer. Math. Soc., Providence, RI, 2013, 149–169.
  • [5] M. J. Bertin and O. Lecacheux, Elliptic fibrations on the modular surface associated to $\Gamma_1(8)$, In: Arithmetic and Geometry of $K3$-Surfaces and Calabi–Yau Threefolds, Fields Inst. Commun., 67, Springer, New York, 2013, 153–199.
  • [6] M. J. Bertin and O. Lecacheux, Automorphisms of certain Niemeier lattices and elliptic fibrations, Albanian J. Math., 11 (2017), 13–34.
  • [7] S. Boissière, A. Sarti and D. Veniani, On prime degree isogenies between $K3$-surfaces, Rend. Circ. Mat. Palermo Ser. 2, 66 (2017), 3–18.
  • [8] N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6, Masson, Paris, 1981.
  • [9] D. Boyd, Private communication.
  • [10] P. Comparin and A. Garbagnati, Van Geemen–Sarti involutions and elliptic fibrations on $K3$ surfaces double cover of $\mathbb P^2$, J. Math. Soc. Japan, 66 (2014), 479–522.
  • [11] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, second edition, Grundlehren der Mathematischen Wissenschaften, 290, Springer-Verlag, New York, 1993.
  • [12] I. V. Dolgachev, Mirror symmetry for lattice polarized $K3$-surfaces, J. Math. Sci., 81 (1996), 2599–2630.
  • [13] E. Dardanelli and B. van Geemen, Hessians and the moduli space of cubic surfaces, In: Algebraic Geometry, Korean-Japan conference in honor of I. Dolgachev's 60th birthday, 2004, Contemp.Math., 422, Amer. Math. Soc., Providence, RI, 2007, 17–36.
  • [14] N. Elkies, Private communication.
  • [15] N. Elkies and A. Kumar, $K3$-surfaces and equations for Hilbert modular surfaces, Algebra Number Theory, 8 (2014), 2297–2411.
  • [16] B. van Geemen and A. Sarti, Nikulin involutions on $K3$-surfaces, Math. Z., 255 (2007), 731–753.
  • [17] J. Keum, A note on elliptic $K3$-surfaces, Trans. Amer. Math. Soc., 352 (2000), 2077–2086.
  • [18] K. Koike, Elliptic $K3$-surfaces admitting a Shioda–Inose structure, Comment. Math. Univ. St. Pauli, 61 (2012), 77–86.
  • [19] M. Kuwata and T. Shioda, Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface, In: Algebraic Geometry in East Asia — Hanoi 2005, Adv. Stud. Pure Math., 50, Math. Soc. Japan, Tokyo, 2008, 177–215.
  • [20] M. Kuwata, Maple library elliptic surface calculator, http://c-faculty.chuo-u.ac.jp/$\sim$kuwata/2012-13/ESC.php.
  • [21] D. Morrison, On $K3$-surfaces with large Picard number, Invent. Math., 75 (1984), 105–121.
  • [22] N. Narumiya and H. Shiga, The mirror map for a family of $K3$-surfaces induced from the simplest 3-dimensional reflexive polytope, In: Proceedings on Moonshine and Related Topics, Montréal, QC, 1999, CRM Proc. Lecture Notes, 30, Amer. Math. Soc., Providence, RI, 2001, 139–161.
  • [23] V. V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSR Izv., 14 (1980), 103–167.
  • [24] V. V. Nikulin, Finite automorphism groups of Kahler surfaces of type K3, Proc. Moscow Math. Soc., 38 (1979), 75–137.
  • [25] K.-I. Nishiyama, The Jacobian fibrations on some $K3$-surfaces and their Mordell–Weil groups, Japan. J. Math., 22 (1996), 293–347.
  • [26] The PARI Group, Bordeaux GP/PARI version 2.7.3, 2015, http://pari.math.u-bordeaux.fr.
  • [27] C. Peters and J. Stienstra, A pencil of $K3$-surfaces related to Apéry's recurrence for $\zeta (3)$ and Fermi surfaces for potential zero, In: Arithmetic of Complex Manifolds, Proceedings of a conference, Erlangen, 1988, (eds. W.-P. Barth and H. Lange), Lecture Notes in Math., 1399, Springer, Berlin, 1989, 110–127.
  • [28] M. Schütt, Sandwich theorems for Shioda–Inose structures, Izvestiya Mat., 77 (2013), 211–222.
  • [29] M. Schütt and T. Shioda, Elliptic surfaces, In: Algebraic Geometry in East Asia — Seoul 2008, Adv. Stud. Pure Math., 60, Math. Soc. Japan, Tokyo, 2010, 51–160.
  • [30] I. Shimada, On elliptic $K3$-surfaces, Michigan Math. J., 47 (2000), 423–446. (See also version with the complete list on arXiv:math/0505140.)
  • [31] T. Shioda, Kummer sandwich theorem of certain elliptic $K3$-surfaces, Proc. Japan Acad. Ser. A Math. Sci., 82 (2006), 137–140.
  • [32] T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan, 24 (1972), 20–59.
  • [33] T. Shioda and H. Inose, On singular $K3$-surfaces, In: Complex Analysis and Algebraic Geometry, (eds. W. L. Baily Jr. and T. Shioda), Iwanami Shoten, Tokyo, 1977, 119–136.
  • [34] T. Shioda and N. Mitani, Singular abelian surfaces and binary quadratic forms, In: Classification of Algebraic Varieties and Compact Complex Manifolds, Lecture. Notes in Math., 412, Springer, Berlin, 1974, 259–287.
  • [35] I. Shimada and D. Q. Zhang, Classification of extremal elliptic $K3$-surfaces and fundamental groups of open $K3$-surfaces, Nagoya Math. J., 161 (2001), 23–54.
  • [36] J. Silverman, The Arithmetic of Elliptic Curves, Grad. Texts in Math., 106, Springer-Verlag, New York, 1986.