Classifications of Elliptic Fibrations of a Singular K3 Surface

  • Marie José Bertin
  • Alice Garbagnati
  • Ruthi Hortsch
  • Odile Lecacheux
  • Makiko Mase
  • Cecília Salgado
  • Ursula Whitcher
Part of the Association for Women in Mathematics Series book series (AWMS, volume 2)


We classify, up to automorphisms, the elliptic fibrations on the singular K3 surface X whose transcendental lattice is isometric to \(\langle 6\rangle \oplus \langle 2\rangle\).

MSC 2010:

Primary 14J28 14J27 Secondary 11G05 11G42 14J33 



We thank the organizers and all those who supported our project for their efficiency, their tenacity and expertise. The authors of the paper have enjoyed the hospitality of CIRM at Luminy, which helped to initiate a very fruitful collaboration, gathering from all over the world junior and senior women, bringing their skill, experience, and knowledge from geometry and number theory. Our gratitude goes also to the referee for pertinent remarks and helpful comments.

A.G is supported by FIRB 2012 “Moduli Spaces and Their Applications” and by PRIN 2010–2011 “Geometria delle varietà algebriche.” C.S is supported by FAPERJ (grant E26/112.422/2012). U.W. thanks the NSF-AWM Travel Grant Program for supporting her visit to CIRM.


  1. Atkin, A.O., Morain, F.: Finding suitable curves for the elliptic curve method of factorization. Math. Comput. 60, 399–405 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Beauville, A.: Les familles stables de courbes elliptiques sur \(\mathbb{P}^{1}\) admettant quatre fibres singulières. C. R. Acad. Sci. Paris Sér. I Math. 294, 657–660 (1982)MathSciNetzbMATHGoogle Scholar
  3. Bertin, M.J.: Mesure de Mahler et série L d’une surface K3 singulière. Actes de la Conférence: Fonctions L et Arithmétique. Publ. Math. Besan. Actes de la conférence Algèbre Théorie Nbr., Lab. Math. Besançon, pp. 5–28 (2010)Google Scholar
  4. Bertin, M.J., Lecacheux, O.: Elliptic fibrations on the modular surface associated to \(\Gamma _{1}(8)\). In: Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds. Fields Institute Communications, vol. 67, pp. 153–199. Springer, New York (2013)Google Scholar
  5. Braun, A.P., Kimura, Y., Watari, T.: On the classification of elliptic fibrations modulo isomorphism on K3 surfaces with large Picard number, Math. AG; High Energy Physics (2013) [arXiv:1312.4421]Google Scholar
  6. Cassels, J.W.S.: Lectures on Elliptic Curves. London Mathematical Society Student Texts, vol. 24. Cambridge University Press, Cambridge (1991)Google Scholar
  7. Comparin, P., Garbagnati, A.: Van Geemen-Sarti involutions and elliptic fibrations on K3 surfaces double cover of \(\mathbb{P}^{2}\). J. Math. Soc. Jpn. 66, 479–522 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Couveignes, J.-M., Edixhoven, S.: Computational Aspects of Modular Forms and Galois Representations, Annals of Math Studies 176. Princeton University Press, PrincetonGoogle Scholar
  9. Cox, D., Katz, S.: Mirror Symmetry and Algebraic Geometry. American Mathematical Society, Providence (1999)CrossRefzbMATHGoogle Scholar
  10. Elkies, N.D.: Three Lectures on Elliptic Surfaces and Curves of High Rank. Lecture notes, Oberwolfach (2007)Google Scholar
  11. Elkies, N., Schütt, M.: Genus 1 fibrations on the supersingular K3 surface in characteristic 2 with Artin invariant 1. Asian J. Math. (2014) [arXiv:1207.1239]Google Scholar
  12. Garbagnati, A., Sarti, A.: Elliptic fibrations and symplectic automorphisms on K3 surfaces. Commun. Algebra 37, 3601–3631 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Karp, D., Lewis, J., Moore, D., Skjorshammer, D., Whitcher, U.: On a family of K3 surfaces with \(\mathcal{S}_{4}\) symmetry. In: Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds. Fields Institute Communications. Springer, New York (2013)Google Scholar
  14. Kloosterman, R.: Classification of all Jacobian elliptic fibrations on certain K3 surfaces. J. Math. Soc. Jpn. 58, 665–680 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kondo, S.: Algebraic K3 surfaces with finite automorphism group. Nagoya Math. J. 116, 1–15 (1989)MathSciNetzbMATHGoogle Scholar
  16. Kondo, S.: Automorphisms of algebraic K3 surfaces which act trivially on Picard groups. J. Math. Soc. Jpn. 44, 75–98 (1992)CrossRefzbMATHGoogle Scholar
  17. Kubert, D.S.: Universal bounds on the torsion of elliptic curves. Proc. Lond. Math. Soc. 33(3), 193–237 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Kumar, A.: Elliptic fibrations on a generic Jacobian Kummer surface. J. Algebraic Geom. [arXiv:1105.1715] 23, 599–667 (2014)Google Scholar
  19. Kreuzer, M., Skarke, H.: Classification of reflexive polyhedra in three dimensions. Adv. Theor. Math. Phys. 2, 853–871 (1998)MathSciNetzbMATHGoogle Scholar
  20. Kreuzer, M., Skarke, H.: Complete classification of reflexive polyhedra in four dimensions. Adv. Theor. Math. Phys. 4, 1209–1230 (2000)MathSciNetzbMATHGoogle Scholar
  21. Martinet, J.: Perfect Lattices in Euclidean Spaces, vol. 327. Springer, Berlin/Heidelberg (2002)Google Scholar
  22. Montgomery, P.L.: Speeding the Pollard and Elliptic Curve Methods of Factorization. Math. Comput. 48. (1987) 243–264.Google Scholar
  23. Nikulin, V.V.: Finite groups of automorphisms of Kählerian K3 surfaces. (Russian) Trudy Moskov. Mat. Obshch. 38, 75–137 (1979). English translation: Trans. Moscow Math. Soc. 38, 71–135 (1980)Google Scholar
  24. Nikulin, V.V.: Integral symmetric bilinear forms and some of their applications. Math. USSR Izv. 14, 103–167 (1980)CrossRefzbMATHGoogle Scholar
  25. Nishiyama, K.: The Jacobian fibrations on some K3 surfaces and their Mordell–Weil groups. Jpn. J. Math. (N.S.) 22, 293–347 (1996)Google Scholar
  26. Oguiso, K.: On Jacobian fibrations on the Kummer surfaces of the product of nonisogenous elliptic curves. J. Math. Soc. Jpn. 41, 651–680 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Schütt, M.: K3 surface with Picard rank 20 over \(\mathbb{Q}\). Algebra Number Theory 4, 335–356 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Schütt, M., Shioda, T.: Elliptic surfaces. In: Algebraic Geometry in East Asia–Seoul 2008. Advanced Studies in Pure Mathematics, vol. 60, pp. 51–160. The Mathematical Society of Japan, Tokyo (2010)Google Scholar
  29. Shimada, I., Zhang, D.-Q.: Classification of extremal elliptic K3 surfaces and fundamental groups of open K3 surfaces. Nagoya Math. J. 161, 23–54 (2001)MathSciNetzbMATHGoogle Scholar
  30. Shioda, T.: On the Mordell–Weil lattices. Commun. Math. Univ. St. Pauli 39, 211–240 (1990)MathSciNetzbMATHGoogle Scholar
  31. Shioda, T.: On elliptic modular surfaces J. Math. Soc. Jpn. 24(1), 20–59 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  32. Stein, W.A., et al.: Sage Mathematics Software (Version 6.1). The Sage Development Team. (2014)
  33. Sterk, H.: Finiteness results for algebraic K3 surfaces. Math. Z. 180, 507–513 (1985)MathSciNetCrossRefGoogle Scholar
  34. Verrill, H.: Root lattices and pencils of varieties. J. Math. Kyoto Univ. 36, 423–446 (1996)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Marie José Bertin
    • 1
  • Alice Garbagnati
    • 2
  • Ruthi Hortsch
    • 3
  • Odile Lecacheux
    • 1
  • Makiko Mase
    • 4
  • Cecília Salgado
    • 5
  • Ursula Whitcher
    • 6
  1. 1.Jussieu Institute of MathematicsPierre and Marie Curie UniversityParisFrance
  2. 2.Department of MathematicsUniversity of MilanMilanItaly
  3. 3.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  4. 4.Tokyo Metropolitan UniversityTokyoJapan
  5. 5.Institute for MathematicsFederal University of Rio de JaneiroRio de JaneiroBrazil
  6. 6.Department of MathematicsUniversity of Wisconsin-Eau ClaireEau ClaireUSA

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