Journal of the Mathematical Society of Japan

Jacobian fibrations on the singular $K3$ surface of discriminant 3

Kazuki UTSUMI

Full-text: Open access

Abstract

In this paper we give the Weierstrass equations and the generators of Mordell–Weil groups for Jacobian fibrations on the singular $K3$ surface of discriminant 3.

Article information

Source
J. Math. Soc. Japan, Volume 68, Number 3 (2016), 1133-1146.

Dates
First available in Project Euclid: 19 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1468956162

Digital Object Identifier
doi:10.2969/jmsj/06831133

Mathematical Reviews number (MathSciNet)
MR3523541

Zentralblatt MATH identifier
1354.14060

Subjects
Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 14J27: Elliptic surfaces 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]

Keywords
$K3$ surface elliptic surface elliptic curve

Citation

UTSUMI, Kazuki. Jacobian fibrations on the singular $K3$ surface of discriminant 3. J. Math. Soc. Japan 68 (2016), no. 3, 1133--1146. doi:10.2969/jmsj/06831133. https://projecteuclid.org/euclid.jmsj/1468956162


Export citation

References

  • S. Y. An, S. Y. Kim, D. C. Marshall, S. H. Marshall, W. G. McCallum and A. R. Perlis, Jacobians of genus one curves, J. Number Theory, 90 (2001), 304–315.
  • A. P. Braun, Y. Kimura and T. Watari, On the classification of elliptic fibrations modulo isomorphism on $K3$ surfaces with large Picard number, arXiv:1312.4421.
  • I. Connell, Addendum to a paper of K. Harada and M.-L. Lang, Some elliptic curves arising from the Leech lattice [J. Algebra, 125 (1989), 298–310], J. Algebra, 145 (1992), 463–467.
  • K. Kodaira, On compact analytic surfaces II, Ann. of Math., 77 (1963), 563–626.
  • A. Kumar, Elliptic fibrations on a generic Jacobian Kummer surface, J. Algebraic Geom., 23 (2014), 599–667.
  • M. Kuwata, Maple Library “Elliptic Surface Calculator”, http://c-faculty.chuo-u.ac.jp/~kuwata/2009-10/ESC.html.
  • M. Kuwata and T. Shioda, Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface, Algebraic geometry in East Asia-Hanoi, 2005, 177–215, Adv. Stud. Pure Math., 50, Math. Soc. Japan, Tokyo, 2008.
  • K. Nishiyama, The Jacobian fibrations on some $K3$ surfaces and their Mordell–Weil groups, Japan. J. Math. (N.S.), 22 (1996), 293–347.
  • T. Sengupta, Elliptic fibrations on supersingular $K3$ surface with Artin invariant 1 in characteristic 3, arXiv:1204.6478.
  • T. Shioda, On the Mordell–Weil lattices, Comment. Math. Univ. St. Pauli, 39 (1990), 211–240.
  • T. Shioda and H. Inose, On singular $K3$ surfaces, Complex analysis and algebraic geometry, 119–136. Iwanami Shoten, Tokyo, 1977.
  • K. Utsumi, Weierstrass equations for Jacobian fibrations on a certain $K3$ surface, Hiroshima Math. J., 42 (2012), 355–383.