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Elliptic Fibrations on the Modular Surface Associated to Γ 1(8)

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Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds

Part of the book series: Fields Institute Communications ((FIC,volume 67))

Abstract

We give all the elliptic fibrations of the K3 surface associated to the modular group Γ1(8).

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References

  1. W. Barth, K. Hulek, C. Peters, A. Van de Ven, Compact Complex Surfaces, 2nd edn. (Springer, Berlin, 2004)

    MATH  Google Scholar 

  2. A. Beauville, Les familles stables de courbes elliptiques sur \({\mathbb{P}}^{1}\) admettant quatre fibres singulières C. R. Math. Acad. Sci. (Paris) Sér. I Math. 294, 657–660 (1982)

    Google Scholar 

  3. M.-J. Bertin, Mahler’s measure and L-series of K3 hypersurfaces, in Mirror Symmetry V, Proceedings of the BIRS Workshop on Calabi-Yau Varieties and Mirror Symmetry, ed. by S.-T. Yau. Studies in Advanced Mathematics (American Mathematical Society, International Press (AMS/IP))

    Google Scholar 

  4. F. Beukers, H. Montanus, Explicit calculation of elliptic K3-surfaces and their Belyi-maps, in LMS Lecture Notes Series, vol. 352 (Cambridge University Press, Cambridge, 2008), pp. 33–51

    Google Scholar 

  5. N. Bourbaki, Groupes et algèbres de Lie, Chaps. 4–6 (Masson, Paris, 1981)

    Google Scholar 

  6. J.W.S. Cassels, Lectures on elliptic curves, in London Mathematical Society Student Texts, vol. 24 (Cambridge University Press, Cambridge, 1991)

    Book  Google Scholar 

  7. N.D. Elkies, Three lectures on elliptic surfaces and curves of high rank. Arxiv preprint arXiv:0709.2908v1 [math. NT], 18 Sep. 2007

    Google Scholar 

  8. N.D. Elkies, The maximal Mordell-Weil rank of an elliptic K3 surface over \(\mathbb{Q}(t)\), in Talk at the Conference on Birational Automorphisms of Compact Complex Manifold and Dynamical Systems at Nagoya University, 28 Aug 2007

    Google Scholar 

  9. N.D. Elkies, Mordell-Weil generators for singular Shioda-Inose surfaces over \(\mathbb{Q}\), http://www.math.harvard.edu/~elkies/K3 20SI.html

    Google Scholar 

  10. T. Harrache, Elkies and Kodaira O. Lecacheux, Etude des fibrations elliptiques d’une surface K3. J. Théor. Nombres Bordeaux 23(1), 183–207 (2011)

    Google Scholar 

  11. K. Kodaira, On compact analytic surfaces I-III. Ann. Math. 71, 111–152 (1960); 77, 563–626 (1963); 78, 1–40 (1963)

    Google Scholar 

  12. A. Kumar, Elliptic fibration on a generic Jacobian Kummer surface, Arxiv preprint arXiv:1105.1715v2 [math. AG], 5 Aug 2012

    Google Scholar 

  13. M. Kuwata, The canonical height and elliptic surfaces. J. Number Theor. 36(2), 201–211 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Martinet, Les réseaux parfaits des espaces euclidiens (Masson, Paris, 1996)

    Google Scholar 

  15. A. Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux. Inst. Hautes Études Sci. Publ. Math. 21, 5–128 (1964)

    Article  Google Scholar 

  16. H.-V. Niemeier, Definite quadratische Formen der Dimension 24 und Diskriminante 1. J. Number Theor. 5, 142–178 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  17. V. Nikulin, Integral symmetric bilinear forms and some of their applications. Math. USSR Izv. Math. 14, 103–167 (1980)

    Article  MATH  Google Scholar 

  18. K.-I. Nishiyama, The Jacobian fibrations on some K3 surfaces and their Mordell-Weil groups. Jpn. J. Math. 22, 293–347 (1996)

    MathSciNet  MATH  Google Scholar 

  19. C. Peters, J. Stienstra, A pencil of K3-surfaces related to Apéry’s recurrence for ζ(3) and Fermi surfaces for potential zero, in Arithmetic of Complex Manifolds, Erlangen, 1988, ed. by W.-P. Barth, H. Lange. Lecture Notes in Mathematics, vol. 1399 (Springer, Berlin, 1989), pp. 110–127

    Google Scholar 

  20. I.-I. Piatetski-Shapiro, I.-R. Shafarevich, Torelli’s theorem for algebraic surfaces of type K3. Math. USSR Izv. 35, 530–572 (1971)

    MATH  Google Scholar 

  21. M. Schütt, Elliptic fibrations of some extremal K3 surfaces. Rocky Mt. J. Math. 37(2), 609–652 (2007)

    Article  MATH  Google Scholar 

  22. M. Schütt, K3 surfaces with Picard rank 20 over \(\mathbb{Q}\). Algebra Number Theor. 4, 335–356 (2010)

    Article  MATH  Google Scholar 

  23. M. Schütt, T. Shioda, Elliptic surfaces, in Algebraic Geometry in East Asia - Seoul 2008. Advanced Studies in Pure Mathematics, vol. 60 (Mathematical Society of Japan, Tokyo, 2010), pp. 51–160

    Google Scholar 

  24. I. Shimada, D.Q. Zhang, Classification of extremal elliptic K3 surfaces and fundamental groups of open K3 surfaces. Nagoya Math. J. 161, 23–54 (2001)

    MathSciNet  MATH  Google Scholar 

  25. T. Shioda, On the Mordell-Weil lattices. Comment. Math. Univ. St. Paul. 39, 211–240 (1990)

    MathSciNet  MATH  Google Scholar 

  26. H. Sterk, Finiteness results for algebraic K3 surfaces. Math. Z. 189, 507–513 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Stienstra, F. Beukers, On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3 surfaces. Math. Ann. 271(2), 269–304 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  28. J. Tate, Algorithm for determining the type of a singular fibre in an elliptic pencil, in Modular Functions of One Variable IV, Antwerpen, 1972. Lecture Notes in Mathematics, vol. 476 (Springer, Berlin, 1975), pp. 33–52

    Google Scholar 

  29. N. Yui, in Arithmetic of Certain Calabi-Yau Varieties and Mirror Symmetry, ed. by B. Conrad and K. Rubin. It is a co-publication of the American Mathematical Society and IAS/Park City Mathematics Institute. vol. 9 (2001), pp. 509–569

    Google Scholar 

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Acknowledgements

 We are grateful to Matthias Schütt for his suggestion to attack the problem and his many helpful comments. Our thanks go also to the organizers for the invitation to the workshop “Arithmetic and Geometry of K3 surfaces and Calabi–Yau threefolds” at the Fields Institute in August 2011. And special thanks to Noriko Yui for the stimulating atmosphere and the great hospitality.

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Correspondence to M. J. Bertin .

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Bertin, M.J., Lecacheux, O. (2013). Elliptic Fibrations on the Modular Surface Associated to Γ 1(8). In: Laza, R., Schütt, M., Yui, N. (eds) Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds. Fields Institute Communications, vol 67. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6403-7_6

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