Elliptic fibrations of maximal rank on a~supersingular K3 surface

T. Shioda

Izvestiya. Mathematics , Tome 77 (2013) no. 3, pp. 571-580.

Résumé

We study a class of elliptic $\mathrm{K3}$ surfaces defined by an explicit Weierstrass equation to find elliptic fibrations of maximal rank on $\mathrm{K3}$ surface in positive characteristic. In particular, we show that the supersingular $\mathrm{K3}$ surface of Artin invariant 1 (unique by Ogus) admits at least one elliptic fibration with maximal rank 20 in every characteristic $p>7$, $p\ne 13$, and further that the number, say $N(p)$, of such elliptic fibrations (up to isomorphisms), is unbounded as $p\to\infty$; in fact, we prove that $\lim_{p\to\infty} N(p)/p^{2} \geqslant (1/12)^{2}$. Bibliography: 19 titles.

Détail

Keywords: $\mathrm{K3}$ surface, Mordell–Weil lattice
Mots-clés : Artin invariant.

Bibliographie

[1] Math. USSR-Izv., 5:3 (1971), 547–588 | DOI | MR | Zbl | Zbl

[2] T. Shioda and H. Inose, “On singular K3 surfaces”, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, 119–136 | MR | Zbl

[3] K. Kodaira, “On compact analytic surfaces. II”, Ann. of Math. (2), 77:3 (1963), 563–626 | DOI | MR | Zbl

[4] M. Artin, “Supersingular K3 surfaces”, Ann. Sci. École Norm. Sup. (4), 7 (1974), 543–567 | MR | Zbl

[5] A. Ogus, “Supersingular K3 crystals”, Journées de Géométrie Algébrique de Rennes, vol. II (Rennes 1978), Astérisque, 64, Soc. Math. France, Paris, 1979, 3–86 | MR | Zbl

[6] M. Kuwata, “Elliptic $K3$ surfaces with given Mordell–Weil rank”, Comment. Math. Univ. St. Paul., 49:1 (2000), 91–100 | MR | Zbl

[7] H. Inose, “On certain Kummer surfaces which can be realized as non-singular quartic surfaces in $\mathbf P^3$”, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23:3 (1976), 545–560 | MR | Zbl

[8] H. Inose, “Defining equations of singular K3 surfaces and a notion of isogeny”, Proceedings of the International Symposium on Algebraic Geometry (Kyoto 1977), Kinokuniya Book Store, Tokyo, 1977, 495–502 | MR | Zbl

[9] T. Shioda, “A note on K3 surfaces and sphere packings”, Proc. Japan Acad. Ser. A Math. Sci., 76:5 (2000), 68–72 | MR | Zbl

[10] T. Shioda, “Correspondence of elliptic curves and Mordell–Weil lattices of certain elliptic K3's.”, Algebraic cycles and motives, v. 2, London Math. Soc. Lecture Note Ser., 344, Cambridge Univ. Press, Cambridge, 2007, 319–339 | MR | Zbl

[11] T. Shioda, “The Mordell–Weil lattice of $y^2=x^3+t^5 - 1/t^5 -11 $”, Comment. Math. Univ. St. Pauli, 56:1 (2007), 45–70 | MR | Zbl

[12] T. Shioda, “K3 surfaces and sphere packings”, J. Math. Soc. Japan, 60:4 (2008), 1083–1105 | DOI | MR | Zbl

[13] T. Shioda, “On the Mordell–Weil lattices”, Comment. Math. Univ. St. Paul., 39:2 (1990), 211–240 | MR | Zbl

[14] T. Shioda, “An example of unirational surfaces in characteristic $p$”, Math. Ann., 211:3 (1974), 233–236 | DOI | MR | Zbl

[15] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren Math. Wiss., 290, Springer-Verlag, New York, 1988 | MR | Zbl

[16] K. Oguiso and T. Shioda, “The Mordell–Weil lattice of a rational elliptic surface”, Comment. Math. Univ. St. Paul., 40:1 (1991), 83–99 | MR | Zbl

[17] J. Igusa, “Class number of a definite quaternion with prime discriminant”, Proc. Nat. Acad. Sci. U.S.A., 44 (1958), 312–314 | DOI | MR | Zbl

[18] H. Ohashi, Integral sections of some elliptic surface via the binary Golay code, RIMS-Preprint, 2010 http://www.kurims.kyoto-u.ac.jp/~pioggia/docu/11.pdf

[19] T. Shioda, “Supersingular K3 surfaces with big Artin invariant”, J. Reine Angew. Math., 381 (1987), 205–210 | MR | Zbl