Skip to main content
SpringerLink
Log in

Isogenies of certain K3 surfaces of rank 18

  • Research
  • Published:
Research in the Mathematical Sciences Aims and scope Submit manuscript

Abstract

We construct geometric isogenies between three types of two-parameter families of K3 surfaces of Picard rank 18. One is the family of Kummer surfaces associated with Jacobians of genus-two curves admitting an elliptic involution, another is the family of Kummer surfaces associated with the product of two non-isogenous elliptic curves, and the third is the twisted Legendre pencil. The isogenies imply the existence of algebraic correspondences between these K3 surfaces and prove that the associated four-dimensional Galois representations are isomorphic. We also apply our result to several subfamilies of Picard rank 19. The result generalizes work of van Geemen and Top (Bull Lond Math Soc 38(2):209–223, 2006).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. By definition, \({\mathbb {H}}_2\) is the set of two-by-two symmetric matrices over \({\mathbb {C}}\) whose imaginary part is positive definite

  2. We corrected two minor typos in the statement of the main theorem.

  3. They were given explicitly in [14]

  4. All surfaces in Theorem 3.31 have been twisted by \((-1)\) compared to [60] This is due to a different sign choice in the definition of the twisted Legendre pencil in Eq. (3.2).

References

  1. Ahlgren, S., Ono, K., Penniston, D.: Zeta functions of an infinite family of \(K3\) surfaces. Am. J. Math. 124(2), 353–368 (2002)

    Article  MathSciNet  Google Scholar 

  2. Andrianov, A.N.: Quadratic Forms and Hecke Operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 286. Springer, Berlin (1987)

    Google Scholar 

  3. Artin, M.: The etale topology of schemes. In: Proceedings of the International Congress of Mathematicians (Moscow, 1966), Izdat, pp. 44– 56. “Mir”, Moscow (1968)

  4. Baker, H.F.: On a system of differential equations leading to periodic functions. Acta Math. 27(1), 135–156 (1903)

    Article  MathSciNet  Google Scholar 

  5. Barth, W.: Even Sets of Eight Rational Curves on a \(K3\)-surface, Complex Geometry (Göttingen, 2000), pp. 1–25. Springer, Berlin (2002)

    MATH  Google Scholar 

  6. Birkenhake, C., Lange, H.: Complex abelian varieties, Second, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302. Springer, Berlin (2004)

  7. Birkenhake, C., Wilhelm, H.: Humbert surfaces and the Kummer plane. Trans. Am. Math. Soc. 355(5), 1819–1841 (2003)

    Article  MathSciNet  Google Scholar 

  8. Bolza, O.: On binary sextics with linear transformations into themselves. Am. J. Math. 10(1), 47–70 (1887)

    Article  MathSciNet  Google Scholar 

  9. Bost, J.-B., Mestre, J.-F.: Moyenne arithmético-géométrique et périodes des courbes de genre \(1\) et \(2\). Gaz. Math. 38, 36–64 (1988)

    MATH  Google Scholar 

  10. Braeger, N., Malmendier, A., Sung, Y.: Kummer sandwiches and Greene-Plesser construction. J. Geom. Phys. 154, 103718, 18 (2020)

    Article  MathSciNet  Google Scholar 

  11. Cassels, J.W.S., Flynn, E.V.: Prolegomena to a middlebrow arithmetic of curves of genus \(2\). In: London Mathematical Society Lecture Note Series, vol. 230. Cambridge University Press, Cambridge (1996)

  12. Clingher, A., Doran, C.F.: Modular invariants for lattice polarized \(K3\) surfaces. Michigan Math. J. 55(2), 355–393 (2007)

    Article  MathSciNet  Google Scholar 

  13. Clingher, Adrian, Malmendier, Andreas: On the geometry of (1,2)-polarized Kummer surfaces, 201704. arXiv:1704.04884 [math.AG]

  14. Clingher, A., Malmendier, A.: Nikulin involutions and the CHL string. Commun. Math. Phys. 370(3), 959–994 (2019)

    Article  MathSciNet  Google Scholar 

  15. Clingher, A., Malmendier, A.: Normal forms for Kummer surfaces. Lond. Math. Soc. Lect. Note Ser. 2(459), 107–162 (2019)

    MATH  Google Scholar 

  16. Clingher, A., Malmendier, A., Shaska, T.: Geometry of Prym varieties for special bielliptic curves of genus three and five. arXiv:1901.09846 (2019)

  17. Clingher, A., Malmendier, A., Shaska, T.: On isogenies among certain abelian surfaces. Michigan Math. J. arXiv:1901.09846 (2019)

  18. Deligne, P.: La conjecture de Weil pour les surfaces \(K3\). Invent. Math. 15, 206–226 (1972)

    Article  MathSciNet  Google Scholar 

  19. Dolgachev, I., Ortland, D.: Point sets in projective spaces and theta functions, 1988, Astérisque, 165, 210 pp. (1989)

  20. Freitag, E.: Siegelsche Modulfunktionen, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 254. Springer, Berlin (1983)

    Google Scholar 

  21. Garbagnati, A.: Elliptic K3 surfaces with abelian and dihedral groups of symplectic automorphisms. Commun. Algebra 41(2), 583–616 (2013)

    Article  MathSciNet  Google Scholar 

  22. Gonzalez-Dorrego, M.R.: \((16,6)\) configurations and geometry of Kummer surfaces in \({ P}^3\). Mem. Am. Math. Soc. 107, 512, vi+101 (1994)

    MathSciNet  MATH  Google Scholar 

  23. Gritsenko, V.A., Nikulin, V.V.: Igusa modular forms and “the simplest” Lorentzian Kac-Moody algebras. Mat. Sb. 187(11), 27–66 (1996)

  24. Hecke, E.: Die eindeutige Bestimmung der Modulfunktionen \(q\)-ter Stufe durch algebraische Eigenschaften. Math. Ann. 111(1), 293–301 (1935)

    Article  MathSciNet  Google Scholar 

  25. Hoyt, W.L.: On surfaces associated with an indefinite ternary lattice, Number theory (New York, 1983–84), Lecture Notes in Math., 1135, pp. 197– 210. Springer, Berlin (1985)

  26. Hoyt, W.L.: Notes on elliptic \(K3\) surfaces, Number theory (New York, 1984–1985), Lecture Notes in Math., 1240, pp. 196– 213. Springer, Berlin (1987)

  27. Hoyt, W.L.: Elliptic fiberings of Kummer surfaces, Number theory (New York, 1985/1988), Lecture Notes in Math., 1383, pp. 89–110. Springer, Berlin (1989)

  28. Hoyt, W.L.: On twisted Legendre equations and Kummer surfaces, Theta functions—Bowdoin 1987, Part 1 (Brunswick, ME, 1987), Proc. Sympos. Pure Math., 49, pp. 695– 707. Amer. Math. Soc., Providence, RI (1989)

  29. Hoyt, W.L., Schwartz, C.F.: Yoshida surfaces with Picard number \(\rho \ge 17\), Proceedings on Moonshine and related topics (Montréal, QC, 1999), CRM Proc. Lecture Notes, 30, pp. 71–78. Amer. Math. Soc., Providence, RI (2001)

  30. Hudson, R.W.H.T.: Kummer’s quartic surface, Cambridge Mathematical Library, Cambridge University Press, Cambridge, With a foreword by W. Barth, Revised reprint of the 1905 original (1990)

  31. Humbert, G.: Sur les fonctionnes abéliennes singulières. I, II, III. J. Math. Pures Appl. serie 5, V, 233– 350 (1899); VI, 279– 386 (1900); VII, 97–123 (1901)

  32. Igusa, J.: On Siegel modular forms of genus two. Am. J. Math. 84, 175–200 (1962)

    Article  MathSciNet  Google Scholar 

  33. Igusa, J.: Modular forms and projective invariants. Am. J. Math. 89, 817–855 (1967)

    Article  MathSciNet  Google Scholar 

  34. Igusa, J.: On the ring of modular forms of degree two over \({ Z}\). Am. J. Math. 101(1), 149–183 (1979)

    Article  MathSciNet  Google Scholar 

  35. Inose, H.: On certain Kummer surfaces which can be realized as non-singular quartic surfaces in \(P^{3}\). J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23(3), 545–560 (1976)

    MathSciNet  MATH  Google Scholar 

  36. Kodaira, K.: On the structure of compact complex analytic surfaces. I, II. Proc. Nat. Acad. Sci. U.S.A. 50, 218–221 (1963); ibid. 51 (1963), 1100–1104

  37. Kumar, A.: Elliptic fibrations on a generic Jacobian Kummer surface. J. Algebraic Geom. 23(4), 599–667 (2014)

    Article  MathSciNet  Google Scholar 

  38. Kuwata, M., Shioda, T.: Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface, Algebraic geometry in East Asia—Hanoi 2005, Adv. Stud. Pure Math., 50, Math. Soc. Japan, Tokyo, pp. 177– 215 (2008)

  39. Malmendier, A., Shaska, T.: The Satake sextic in F-theory. J. Geom. Phys. 120, 290–305 (2017)

    Article  MathSciNet  Google Scholar 

  40. Malmendier, A.: Kummer surfaces associated with Seiberg-Witten curves. J. Geom. Phys. 62(1), 107–123 (2012)

    Article  MathSciNet  Google Scholar 

  41. Malmendier, A., Morrison, D.R.: K3 surfaces, modular forms, and non-geometric heterotic compactifications. Lett. Math. Phys. 105(8), 1085–1118 (2015)

    Article  MathSciNet  Google Scholar 

  42. Malmendier, A., Ono, K.: Moonshine for \(M_{24}\) and Donaldson invariants of \(\mathbb{CP}^2\). Commun. Number Theory Phys. 6(4), 759–770 (2012)

    Article  MathSciNet  Google Scholar 

  43. Malmendier, A., Shaska, T.: A universal genus-two curve from Siegel modular forms. SIGMA Symmetry Integrability Geom. Methods Appl., 13, Paper No. 089, 17 (2017)

  44. Malmendier, A., Sung, Y.: Counting rational points on Kummer surfaces. Res. Number Theory, 5, 3, Paper No. 27, 23 (2019)

  45. McDonald, J.H.: A problem in the reduction of hyperelliptic integrals. Trans. Am. Math. Soc. 7(4), 578–587 (1906)

    Article  MathSciNet  Google Scholar 

  46. Mehran, A.: Double covers of Kummer surfaces. Manuscr. Math. 123(2), 205–235 (2007)

    Article  MathSciNet  Google Scholar 

  47. Mestre, J.-F.: Construction de courbes de genre \(2\) à partir de leurs modules, Effective methods in algebraic geometry (Castiglioncello, 1990), Progr. Math., 94, pp. 313–334. Birkhäuser Boston, Boston, MA (1991)

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

    Article  MathSciNet  Google Scholar 

  49. Mumford, D.: Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, Published for the Tata Institute of Fundamental Research, Bombay, by Hindustan Book Agency, New Delhi, 5, With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second (1974) edition (2008)

  50. Nikulin, V.V.: Kummer surfaces. Izv. Akad. Nauk SSSR Ser. Mat. 39(2), 278–293, 471 (1975)

    MathSciNet  Google Scholar 

  51. Oguiso, K.: On Jacobian fibrations on the Kummer surfaces of the product of nonisogenous elliptic curves. J. Math. Soc. Japan 41(4), 651–680 (1989)

    Article  MathSciNet  Google Scholar 

  52. Pringsheim, A.: Zur Transformation zweiten Grades der hyperelliptischen Functionen erster Ordnung. Math. Ann. 9(4), 445–475 (1876)

    Article  MathSciNet  Google Scholar 

  53. Richelot, F. J.: De transformatione integralium Abelianorum primi ordinis commentatio. Caput secundum. De computatione integralium Abelianorum primi ordinis. J. Reine Angew. Math. 16, 285–341 (1837)

  54. Serre, J.-P.: Rational points on curves over finite fields, notes by F. Gouvea of lectures at. Harvard University (1985)

  55. Shafarevitch, E.R.: Le théorème de Torelli pour les surfaces algébriques de type \(K3\), bookActes du Congrès International des Mathématiciens (Nice, 1970). Tome 1, 413–417 (1971)

    MathSciNet  Google Scholar 

  56. Shioda, T.: Kummer sandwich theorem of certain elliptic \(K3\) surfaces. Proc. Jpn. Acad. Ser. A Math. Sci. 82(8), 137–140 (2006)

    MathSciNet  MATH  Google Scholar 

  57. Shioda, T.: Classical Kummer surfaces and Mordell-Weil lattices, Algebraic geometry, Contemp. Math., 422, pp. 213–221. Amer. Math. Soc., Providence, RI (2007)

  58. Totaro, B.: Recent progress on the Tate conjecture. Bull. Am. Math. Soc. (N.S.) 54(4), 575–590 (2017)

    Article  MathSciNet  Google Scholar 

  59. van der Geer, G.: On the geometry of a Siegel modular threefold. Math. Ann. 260(3), 317–350 (1982)

    Article  MathSciNet  Google Scholar 

  60. van Geemen, B., Top, J.: An isogeny of \(K3\) surfaces. Bull. Lond. Math. Soc. 38(2), 209–223 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their thoughtful comments and effort toward improving the manuscript. The authors also thank Dr. Muhammad Arjumand Masood for some helpful discussions about Jacobian elliptic functions during an early stage of this project. N.B. and S.S. would like to acknowledge the support from the Office of Research and Graduate Studies at Utah State University. A.C. acknowledges support from a UMSL Mid-Career Research Grant. A.M. acknowledges support from the Simons Foundation through Grant No. 202367.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Utah State University, Logan, UT, 84322, USA

    Noah Braeger & Shantel Spatig

  2. Department of Mathematics and Statistics, University of Missouri, St. Louis, MO, 63121, USA

    Adrian Clingher

  3. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Andreas Malmendier

Authors
  1. Noah Braeger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Adrian Clingher
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andreas Malmendier
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shantel Spatig
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andreas Malmendier.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Braeger, N., Clingher, A., Malmendier, A. et al. Isogenies of certain K3 surfaces of rank 18. Res Math Sci 8, 57 (2021). https://doi.org/10.1007/s40687-021-00293-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40687-021-00293-0

Keywords

Mathematics Subject Classification

Search

Navigation