## Journal of the Mathematical Society of Japan

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

#### 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
Revised: 17 October 2018
First available in Project Euclid: 16 October 2019

https://projecteuclid.org/euclid.jmsj/1571212902

Digital Object Identifier
doi:10.2969/jmsj/80638063

#### 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

#### 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.