Two Lectures on the Arithmetic of K3 Surfaces

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

Abstract

In these lecture notes we review different aspects of the arithmetic of K3 surfaces. Topics include rational points, Picard number and Tate conjecture, zeta functions and modularity.

Key words

K3 surface Rational points Elliptic fibration Picard number Singular K3 surface Modular form Class group 

Mathematics Subject Classifications (2010)

14J28 11F03 11G05 11G15 11G25 11G35 14G05 14G15 14G25 14J10 14J27 

Notes

Acknowledgements

It is a great pleasure to thank the other organisers of the Fields workshop and particularly all the participants for creating such a stimulating atmosphere. Special thanks to the Fields Institute for the great hospitality and to the referee for his comments. These lecture notes were written down while the author enjoyed support from the ERC under StG 279723 (SURFARI) which is gratefully acknowledged.

References

  1. 1.
    T.G. Abbott, K. Kedlaya, D. Roe, Bounding Picard numbers of surfaces using p-adic cohomology, in Arithmetic, Geometry and Coding Theory (AGCT 2005). Séminaires et Congrès, vol. 21 (Societé Mathématique de France, Paris, 2009), pp. 125–159Google Scholar
  2. 2.
    N. Aoki, T. Shioda, Generators of the Néron–Severi group of a Fermat surface, in Arithmetic and Geometry, vol. I. Progress in Mathematics, vol. 35 (Birkhäuser, Boston, 1983), pp. 1–12Google Scholar
  3. 3.
    M. Artin, P. Swinnerton-Dyer, The Shafarevich–Tate conjecture for pencils of elliptic curves on K3 surfaces. Invent. Math. 20, 249–266 (1973)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    W. Barth, K. Hulek, C. Peters, A. van de Ven, Compact Complex Surfaces, 2nd edn. Erg. der Math. und ihrer Grenzgebiete, 3. Folge, Band 4 (Springer, Berlin, 2004)Google Scholar
  5. 5.
    F.A. Bogomolov, Y. Tschinkel, Density of rational points on elliptic K3 surfaces. Asian J. Math. 4(2), 351–368 (2000)MathSciNetMATHGoogle Scholar
  6. 6.
    F.A. Bogomolov, B. Hassett, Y. Tschinkel, Constructing rational curves on K3 surfaces. Duke Math. J. 157, 535–550 (2011)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    J.W.S. Cassels, A Diophantine equation over a function field. J. Aust. Math. Soc. Ser. A 25(4), 489–496 (1978)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    F. Charles, On the Picard number of K3 surfaces over number fields. Algebra Number Theor. Preprint (2011), arXiv: 1111.4117Google Scholar
  9. 9.
    D.A. Cox, Mordell Weil groups of elliptic curves over C(t) with p g = 0 or 1. Duke Math. J. 49(3), 677–689 (1982)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    N.D. Elkies, Three lectures on elliptic surfaces and curves of high rank. 840 Preprint (2007), arXiv: 0709.2908 841 211 844Google Scholar
  11. 11.
    N.D. Elkies, Shimura Curve Computations Via K3 Surfaces of NS -Rank at Least 19. ANTS VIII, 2008. Lecture Notes in Computer Science, vol. 5011 (Springer, Berlin, 2008), pp. 196–211Google Scholar
  12. 12.
    N.D. Elkies, M. Schütt, Modular forms and K3 surfaces. Preprint (2008), arXiv: 0809.0830Google Scholar
  13. 13.
    N.D. Elkies, M. Schütt, K3 families of high Picard rank, in preparationGoogle Scholar
  14. 14.
    J. Ellenberg, K3 surfaces over number fields with geometric Picard number one, in Arithmetic of Higher-dimensional Algebraic Varieties, Palo Alto, CA, 2002. Progress in Mathematics, vol. 226 (Birkhäuser, Boston, 2004), pp. 135–140Google Scholar
  15. 15.
    A.-S. Elsenhans, J. Jahnel, The Picard group of a K3 surface and its reduction modulo p. Algebra Number Theor. 5, 1027–1040 (2011)Google Scholar
  16. 16.
    G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73(3), 349–366 (1983)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    B. van Geemen, Some remarks on Brauer groups on K3 surfaces. Adv. Math. 197, 222–247 (2005)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    B. Hassett, A. Vàrilly-Alvarado, Failure of the Hasse principle on general K3 surfaces. J. Inst. Math. Jussieu. Preprint (2011), arXiv: 1110.1738Google Scholar
  19. 19.
    B. Hassett, A. Vàrilly-Alvarado, P. Varilly, Transcendental obstructions to weak approximation on general K3 surfaces. Adv. Math. 228, 1377–1404 (2011)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    K. Hulek, M. Schütt, Arithmetic of singular Enriques surfaces. Algebra Number Theor. 6(2), 195–230. Preprint (2012), arXiv: 1002.1598Google Scholar
  21. 21.
    H. Inose, Defining equations of singular K3 surfaces and a notion of isogeny, in Proceedings of the International Symposium on Algebraic Geometry, Kyoto University, Kyoto, 1977 (Kinokuniya Book Store, Tokyo, 1978), pp. 495–502Google Scholar
  22. 22.
    I. Karzhemanov, One construction of a K3 surface with the dense set of rational points. 865 Preprint (2011), arXiv: 1102.1873Google Scholar
  23. 23.
    R. Kloosterman, Elliptic K3 surfaces with geometric Mordell–Weil rank 15. Can. Math. Bull. 50(2), 215–226 (2007)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    K. Kodaira, On compact analytic surfaces I–III. Ann. Math. 71, 111–152 (1960); Ann. Math. 77, 563–626 (1963); Ann. Math. 78, 1–40 (1963)Google Scholar
  25. 25.
    M. Kuwata, Elliptic K3 surfaces with given Mordell–Weil rank. Comment. Math. Univ. St. Pauli 49(1), 91–100 (2000)MathSciNetMATHGoogle Scholar
  26. 26.
    J. Li, C. Liedtke, Rational curves on K3 surfaces. Invent. Math. 188(3), 713–727 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    M. Lieblich, D. Maulik, A. Snowden, Finiteness of K3 surfaces and the Tate conjecture. 874 Preprint (2011), arXiv: 1107.1221Google Scholar
  28. 28.
    R. Livné, Motivic orthogonal two-dimensional representations of Gal\((\bar{\mathbb{Q}}/\mathbb{Q})\). Isr. J. Math. 92, 149–156 (1995)MATHCrossRefGoogle Scholar
  29. 29.
    R. van Luijk, K3 surfaces with Picard number one and infinitely many rational points. Algebra Number Theor. 1(1), 1–15 (2007)MATHCrossRefGoogle Scholar
  30. 30.
    Y.I. Manin, Le groupe de Brauer-Grothendieck en géométrie diophantienne, in Actes du Congrès International des Mathématiciens, Nice, 1970. Tome 1 (Gauthier-Villars, Paris, 1971), pp. 401–411Google Scholar
  31. 31.
    J. Milne, On a conjecture of Artin and Tate. Ann. Math. 102, 517–533 (1975)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    M. Mizukami, Birational mappings from quartic surfaces to Kummer surfaces (in Japanese), Master’s Thesis, University of Tokyo, 1975Google Scholar
  33. 33.
    D.R. Morrison, On K3 surfaces with large Picard number. Invent. Math. 75(1), 105–121 (1984)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    H.-V. Niemeier, Definite quadratische Formen der Dimension 24 und Diskriminante 1. J. Number Theor. 5, 142–178 (1973)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    V.V. Nikulin, Finite groups of automorphisms of Kählerian K3 surfaces. Trudy Moskov. Mat. Obshch. 38, 75–137 (1979)MathSciNetMATHGoogle Scholar
  36. 36.
    V.V. Nikulin, Integral symmetric bilinear forms and some of their applications. Math. USSR Izv. 14(1), 103–167 (1980)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    K.-I. Nishiyama, The Jacobian fibrations on some K3 surfaces and their Mordell–Weil groups. Jpn. J. Math. 22, 293–347 (1996)MathSciNetMATHGoogle Scholar
  38. 38.
    N. Nygaard, A. Ogus, Tate’s conjecture for K3 surfaces of finite height. Ann. Math. 122, 461–507 (1985)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    K. Oguiso, On Jacobian fibrations on the Kummer surfaces of the product of nonisogenous elliptic curves. J. Math. Soc. Jpn. 41(4), 651–680 (1989)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    I.I. Piatetski-Shapiro, I.R. Shafarevich, Torelli’s theorem for algebraic surfaces of type K3. Izv. Akad. Nauk SSSR Ser. Mat. 35, 530–572 (1971)MathSciNetMATHGoogle Scholar
  41. 41.
    M. Raynaud, “p-torsion” du schéma de Picard. Astérisque 64, 87–148 (1979)Google Scholar
  42. 42.
    K. Ribet, Galois representations attached to eigenforms with Nebentypus, in Modular Functions of One Variable V, ed. by J.-P. Serre, D.B. Zagier, Bonn, 1976. Lecture Notes in Mathematics, vol. 601 (Springer, Berlin, 1977), pp. 17–52Google Scholar
  43. 43.
    M. Schütt, Fields of definition of singular K3 surfaces. Comm. Number Theor. Phys. 1(2), 307–321 (2007)MATHGoogle Scholar
  44. 44.
    M. Schütt, CM newforms with rational coefficients. Ramanujan J. 19, 187–205 (2009)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    M. Schütt, K3 surfaces of Picard rank 20 over \(\mathbb{Q}\). Algebra Number Theor. 4(3), 335–356 (2010)MATHCrossRefGoogle Scholar
  46. 46.
    M. Schütt, Sandwich theorems for Shioda–Inose structures. Izvestiya Mat. 77, 211–222 (2013)MATHCrossRefGoogle Scholar
  47. 47.
    M. Schütt, T. Shioda, R. van Luijk, Lines on the Fermat quintic surface. J. Number Theor. 130, 1939–1963 (2010)MATHCrossRefGoogle Scholar
  48. 48.
    E. Selmer, The Diophantine equation \(a{x}^{3} + b{y}^{3} + c{z}^{3} = 0\). Acta Math. 85, 203–362 (1951)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    T. Shioda, On the Picard number of a complex projective variety. Ann. Sci. École Norm. Sup.(4) 14(3), 303–321 (1981)Google Scholar
  50. 50.
    T. Shioda, On the Mordell–Weil lattices. Comment. Math. Univ. St. Pauli 39, 211–240 (1990)MathSciNetMATHGoogle Scholar
  51. 51.
    T. Shioda, Kummer sandwich theorem of certain elliptic K3 surfaces. Proc. Jpn. Acad. Ser. A 82, 137–140 (2006)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    T. Shioda, K3 surfaces and sphere packings. J. Math. Soc. Jpn. 60, 1083–1105 (2008)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    T. Shioda, H. Inose, in On Singular K3 Surfaces, ed. by W.L. Baily Jr., T. Shioda. Complex Analysis and Algebraic Geometry (Iwanami Shoten, Tokyo, 1977), pp. 119–136Google Scholar
  54. 54.
    T. Shioda, N. Mitani, Singular abelian surfaces and binary quadratic forms, in Classification of Algebraic Varieties and Compact Complex Manifolds. Lecture Notes in Mathematics, vol. 412 (Springer, Berlin, 1974), pp. 259–287Google Scholar
  55. 55.
    J.H. Silverman, Rational points on K3 surfaces: a new canonical height. Invent. Math. 105, 347–373 (1991)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    A. Skorobogatov, in Torsors and Rational Points. Cambridge Tracts in Mathematics, vol. 144 (Cambridge University Press, Cambridge, 2001)Google Scholar
  57. 57.
    H. Sterk, Finiteness results for algebraic K3 surfaces. Math. Z. 189(4), 507–513 (1985)MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    J. Tate, Algebraic cycles and poles of zeta functions, in Arithmetical Algebraic Geometry. Proceedings of Conference of Purdue University, 1963 (Harper & Row, New York, 1965), pp. 93–110Google Scholar
  59. 59.
    J. Tate, in On the Conjectures of Birch and Swinnerton-Dyer and a Geometric Analog, ed. by A. Grothendieck, N.H. Kuiper. Dix exposés sur la cohomologie des schemas (North-Holland, Amsterdam, 1968), pp. 189–214Google Scholar
  60. 60.
    J. Tate, Algorithm for determining the type of a singular fibre in an elliptic pencil, in Modular Functions of One Variable IV, Antwerpen, 1972. SLN, vol. 476 (Springer, Berlin, 1975), pp. 33–52Google Scholar
  61. 61.
    T. Terasoma, Complete intersections with middle Picard number 1 defined over \(\mathbb{Q}\). Math. Z. 189(2), 289–296 (1985)MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    P.J. Weinberger, Exponents of the class groups of complex quadratic fields. Acta Arith. 22, 117–124 (1973)MathSciNetMATHGoogle Scholar
  63. 63.
    A. Wiles, Modular elliptic curves and Fermat’s Last Theorem. Ann. Math. (2) 141(3), 443–551 (1995)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institut für Algebraische GeometrieLeibniz Universität HannoverHannoverGermany

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