Skip to main content
SpringerLink
Log in

7-Gons and genus three hyperelliptic curves

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

In this paper, we will give a general but completely elementary description for hyperelliptic curves of genus three whose Jacobian varieties have endomorphisms by the real cyclotomic field \({{\mathbb{Q}} (\zeta_7 + \overline{\zeta}_7)}\). We study the algebraic correspondences on these curves which are lifts of algebraic correspondences on a conic in P 2 associated with Poncelet 7-gons. These correspondences induce endomorphisms \({\phi}\) on the Jacobians which satisfy \({\phi^3+\phi^2-2\phi-1=0}\). Moreover, we study Humbert’s modular equations which characterize the curves of genus three having these real multiplications.

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

Similar content being viewed by others

References

  1. Bending P.: Curves of genus 2 with \({\sqrt{2}}\) multiplication. arXiv:math/9911273

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Besser A.: CM cycles over Shimura curves. J Algebr. Geom. 4(4), 659–691 (1995)

    MathSciNet  MATH  Google Scholar 

  4. Besser, A., Livne, R.: Universal Kummer families over Shimura curves, preprint (2010)

  5. Elkies, N.: Shimura curve computations via K3 surfaces of Néron-Severi rank at least 19. In: van der Poorten, A.J., Stein, A. (eds.) Proceedings of ANTS-8, Lecture Notes in Computer Science 5011, pp. 196–211 (2008)

  6. Fricke, R.: Die Elliptischen Funktionen und ihre Anwendungen, zweiter teil, Teubner (1922)

  7. Griffiths P., Harris J.: On Cayley’s explicit solution to Poncelet’s porism. Enseign Math(2) 24(1–2), 31–40 (1978)

    MathSciNet  MATH  Google Scholar 

  8. Gruenewald, D.: Explicit algorithms for Humbert surfaces. Thesis, University of Sydney (2009). http://echidna.maths.usyd.edu.au/~davidg/

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Hashimoto K., Sakai Y.: Poncelet’s theorem and versal family of curves of genus two with \({\sqrt{2}}\)-multiplication. RIMS Kokyuroku Bessatsu B 12, 249–261 (2009)

    MathSciNet  Google Scholar 

  11. Hashimoto K., Sakai Y.: General form of Humbert’s modular equation for curves with real multiplication of Δ =  5. Proc. Japan Acad. Ser. A Math. Sci. 85, 171–176 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Humbert, G.: Sur les fonctions abeliennes singulieres, Œ uvres de G. Humbert 2, pub. par les soins de Pierre Humbert et de Gaston Julia, Paris, Gauthier-Villars, pp. 297–401 (1936)

  13. Khare C., Wintenberger J-P.: On Serre’s conjecture for 2-dimensional mod p representations of \({Gal(\overline{Q}/Q)}\) . Ann. Math. (2) 169(1), 229–253 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Abhinav Kumar, K3 surfaces associated with curves of genus two. Int. Math. Res. Not. IMRN 2008, no. 6, Art. ID rnm165

  15. Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wolfram Research, Inc., Mathematica, Version 7.0, Champaign, IL (2008)

  17. Mestre J.F.: Courbes hyperelliptiques à à multiplications réelles. CR. Acad. Sci. Paris Ser. I Math. 307, 721–724 (1988)

    MathSciNet  MATH  Google Scholar 

  18. Mestre J.F.: Courbes hyperelliptiques à à multiplications réelles. Progr. Math. 89, 193–208 (1991)

    MathSciNet  Google Scholar 

  19. Mumford, D.: Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5 Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London 1970

  20. Runge B.: Endomorphism rings of abelian surfaces and projective models of their moduli spaces. Tohoku Math J (2) 51(3), 283–303 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. SAGE Mathematics Software, Version 4.6. http://www.sagemath.org/

  22. Sakai, Y.: Construction of genus two curves with real multiplication by Poncelet’s theorem. Dissertation, Waseda University (2010)

  23. Sakai Y.: Poncelet’s theorem and hyperelliptic curve with real multiplication of Δ =  5. J. Ramanujan Math. Soc. 24(2), 143–170 (2009)

    MathSciNet  MATH  Google Scholar 

  24. Shepherd-Barron N.I., Taylor R.: mod 2 and mod 5 icosahedral representations. J. Am. Math. Soc. 10(2), 283–298 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Shimura G.: Abelian varieties with complex multiplication and modular functions. Princeton Mathematical Series, 46. Princeton University Press, Princeton, NJ (1998)

    Google Scholar 

  27. The PARI Group, PARI/GP, version 2.5.0, Bordeaux (2011) (http://pari.math.u-bordeaux.fr/)

  28. Tautz W., Top J., Verberkmoes A.: Explicit hyperelliptic curves with real multiplication and permutation polynomials. Can. J. Math. 43(5), 1055–1064 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  29. van der Geer G.: Hilbert modular surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 16. Springer, Berlin (1988)

    Google Scholar 

  30. Weil, A.: Variétés abéliennes et courbes algébriques, Actualités Sci. Ind., no. 1064 =  Publ. Inst. Math. Univ. Strasbourg 8 (1946). Hermann, Paris (1948)

  31. Weil, A.: Remarques sur un mémoire d’Hermite, Arch. Math. 5 (1954), 197–202 =  Œ uvres (collected papers), vol. 2, pp. 111–116

  32. Weil, A.: Euler and the Jacobians of elliptic curves, Arithmetic and geometry, vol. I, pp. 353–359, Progr. Math., 35, Birkhuser, Boston, MA (1983)

  33. Wilson, J.: Curves of genus 2 with real multiplication by a square root of 5. Dissertation, University of Oxford, Oxford (1998)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 0803, USA

    J. William Hoffman

  2. Department of Mathematics, Southeast Missouri State University, Cape Girardeau, MO, 63701, USA

    Haohao Wang

Authors
  1. J. William Hoffman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Haohao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. William Hoffman.

Additional information

This paper is dedicated with gratitude to Heisuke Hironaka, who was the first author’s Ph.D. advisor at Harvard.

J. W. Hoffman is supported in part by NSA grant 115-60-5102.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hoffman, J.W., Wang, H. 7-Gons and genus three hyperelliptic curves. RACSAM 107, 35–52 (2013). https://doi.org/10.1007/s13398-012-0079-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-012-0079-1

Keywords

Mathematics Subject Classification (2000)

Search

Navigation