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Universal Kummer Families Over Shimura Curves

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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 a number of examples of an isomorphism between two types of moduli problems. The first classifies elliptic surfaces over the projective line with five specified singular fibers, of which four are fixed and one gives the parameter; the second classifies K3 surfaces with a specified isogeny to an abelian surface with quaternionic multiplication.

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References

  1. W.P. Barth, K. Hulek, C.A.M. Peters, A. Van de Ven, Compact complex surfaces, in Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 4, 2nd edn. [Results in Mathematics and Related Areas, 3rd Series. A Series of Modern Surveys in Mathematics] (Springer, Berlin, 2004)

    Google Scholar 

  2. A. Besser, Universal families over shimura curves. Ph.D. thesis, Tel Aviv University, 1993

    Google Scholar 

  3. A. Besser, CM cyles over Shimura curves. J. Algebr. Geom. 4, 659–693 (1995)

    MathSciNet  MATH  Google Scholar 

  4. A. Besser, Elliptic fibrations of K3 surfaces and QM Kummer surfaces. Math. Z. 228(2), 283–308 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Besser, R. Livne, Picard-Fuchs equations of families of QM abelian surfaces. Preprint (2012), URL http://arxiv.org/abs/1202.2808

  6. F. Beukers, A note on the irrationality of ζ(2) and ζ(3). Bull. Lond. Math. Soc. 11(3), 268–272 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. F. Beukers, C.A.M. Peters, A family of K3 surfaces and ζ(3). J. Reine Angew. Math. 351, 42–54 (1984)

    MathSciNet  MATH  Google Scholar 

  8. A. Borel, Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 8(1), 1–33 (1981)

    Google Scholar 

  9. A.I. Borevich, I.R. Shafarevich, Number Theory (Academic, New York, 1966)

    Google Scholar 

  10. J.F. Boutot, H. Carayol, Uniformisation p-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfel’d. Astérisque 196–197 (1991); 7, 45–158 (1992); Courbes modulaires et courbes de Shimura, Orsay, 1987/1988

    Google Scholar 

  11. C.H. Clemens, A scrapbook of complex curve theory, in Graduate Studies in Mathematics, vol. 55, 2nd edn. (American Mathematical Society, Providence, 2003)

    Google Scholar 

  12. P. Deligne, Formes modulaires ét representations ℓ-adiques, in Seminaire Bourbaki, vol. 355. Lecture Notes in Mathematics, vol. 179 (Springer, Heidelberg, 1971), pp. 139–172

    Google Scholar 

  13. P. Deligne, Travaux de Shimura, in Séminaire Bourbaki, 23ème année (1970/1971), Exp. No. 389. Lecture Notes in Mathematics, vol. 244 (Springer, Berlin, 1971), pp. 123–165

    Google Scholar 

  14. J. Dieudonné, Sur les groupes classiques (Hermann, Paris, 1973). Troisième édition revue et corrigée, Publications de l’Institut de Mathématique de l’Université de Strasbourg, VI, Actualités Scientifiques et Industrielles, No. 1040

    Google Scholar 

  15. I.V. Dolgachev, Mirror symmetry for lattice polarized K3 surfaces. Algebraic geometry, 4. J. Math. Sci. 81(3), 2599–2630 (1996);

    Google Scholar 

  16. C.F. Doran, Picard-Fuchs uniformization and modularity of the mirror map. Comm. Math. Phys. 212(3), 625–647 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. N.D. Elkies, Shimura curve computations, in Algorithmic Number Theory, Portland, OR (Springer, Berlin, 1998), pp. 1–47

    Google Scholar 

  18. N.D. Elkies, Shimura curve computations via K3 surfaces of Néron-Severi rank at least 19, in Algorithmic Number Theory. Lecture Notes in Computer Science, vol. 5011 (Springer, Berlin, 2008), pp. 196–211

    Google Scholar 

  19. R. Hartshorne, in Algebraic Geometry. Graduate Texts in Mathematics, No. 52 (Springer, New York, 1977)

    Google Scholar 

  20. K. Hashimoto, N. Murabayashi, Shimura curves as intersections of Humbert surfaces and defining equations of QM-curves of genus two. Tohoku Math. J. (2) 47(2), 271–296 (1995)

    Google Scholar 

  21. S. Herfurtner, Elliptic surfaces with four singular fibers. Math. Ann. 291, 319–342 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. N. Katz, Travaux de Dwork, in Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 409. Lecture Notes in Mathematics, vol. 317 (Springer, Berlin, 1973), pp. 167–200

    Google Scholar 

  23. A.W. Knapp, Elliptic curves, in Mathematical Notes, vol. 40 (Princeton University Press, Princeton, 1992)

    Google Scholar 

  24. K. Kodaira, On compact analytic surfaces: II. Ann. Math. 77, 563–626 (1962)

    Article  MathSciNet  Google Scholar 

  25. D.R. Morrison, On K3 surfaces with large Picard number. Invent. Math. 75(1), 105–121 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  26. D. Mumford, On the equations defining abelian varieties, I. Invent. Math. 1, 287–354 (1966)

    Article  MathSciNet  Google Scholar 

  27. D. Mumford, On the equations defining abelian varieties, II. Invent. Math. 3, 75–135 (1967)

    Article  MathSciNet  Google Scholar 

  28. D. Mumford, On the equations defining abelian varieties, III. Invent. Math. 3, 215–244 (1967)

    Article  MathSciNet  Google Scholar 

  29. D. Mumford, Abelian Varieties (Oxford University Press, Oxford, 1970)

    MATH  Google Scholar 

  30. V. Nikulin, Integral symmetric bilinear forms and some of their applications. Izv. Akad. Nauk SSSR 43, 75–137 (1979)

    Google Scholar 

  31. V.V. Nikulin, Finite groups of automorphisms of Kählerian K3 surfaces. Trudy Moskov. Mat. Obshch. 38, 75–137 (1979)

    MathSciNet  MATH  Google Scholar 

  32. K. Oguiso, T. Shioda, The Mordell-Weil lattice of a rational elliptic surface. Comment. Math. Univ. St. Paul. 40(1), 83–99 (1991)

    MathSciNet  Google Scholar 

  33. I.I. Piatetski-Shapiro, I.R. Shafarevic, A Torelli theorem for algebraic surfaces of type K3. Izv. Akad. Nauk SSSR 35, 503–572 (1971)

    Google Scholar 

  34. J.P. Serre, A Course in Arithmetic (Springer, New York, 1973) [Translated from the French, Graduate Texts in Mathematics, No. 7]

    Book  MATH  Google Scholar 

  35. G. Shimura, On analytic families of polarized abelian varieties and automorphic functions. Ann. Math. (2) 78, 149–192 (1963)

    Google Scholar 

  36. T. Shioda, On elliptic modular surfaces. J. Math. Soc. Jpn. 4, 20–59 (1972)

    Article  MathSciNet  Google Scholar 

  37. T. Shioda, M. Schütt, An interesting elliptic surface over an elliptic curve. Proc. Jpn. Acad. Ser. A Math. Sci. 83(3), 40–45 (2007)

    Article  MATH  Google Scholar 

  38. J.P. Smith, Picard-Fuchs differential equations for families of K3 surfaces. Ph.D. thesis, University of Warwick, 2006. URL http://arxiv.org/abs/0705.3658

  39. G. van der Geer, On the geometry of a Siegel modular threefold. Math. Ann. 260, 317–350 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

A. Besser was partially supported by ISF grant. R. Livné was partially supported by an ISF grant.

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Authors and Affiliations

  1. Department of Mathematics, Ben Gurion University of the Negev, Be’er Sheva, 84105 02138, Israel

    Amnon Besser

  2. Institute of Mathematics, Hebrew University of Jerusalem Givat-Ram, Jerusalem, 91904, Israel

    Ron Livné

Authors
  1. Amnon Besser
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  2. Ron Livné
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Correspondence to Amnon Besser .

Editor information

Editors and Affiliations

  1. , Mathematics Department, Stony Brook University, N/A, Stony Brook, 11794-3651, New York, USA

    Radu Laza

  2. , Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Matthias Schütt

  3. , Department of Math & Stats, Queen's University, University Ave., Kingston, K7L 3N6, Ontario, Canada

    Noriko Yui

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Besser, A., Livné, R. (2013). Universal Kummer Families Over Shimura Curves. 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_7

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