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.
Similar content being viewed by others
References
Bending P.: Curves of genus 2 with \({\sqrt{2}}\) multiplication. arXiv:math/9911273
Besser A.: Elliptic fibrations of K3 surfaces and QM Kummer surfaces. Math Z 228(2), 283–308 (1998)
Besser A.: CM cycles over Shimura curves. J Algebr. Geom. 4(4), 659–691 (1995)
Besser, A., Livne, R.: Universal Kummer families over Shimura curves, preprint (2010)
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)
Fricke, R.: Die Elliptischen Funktionen und ihre Anwendungen, zweiter teil, Teubner (1922)
Griffiths P., Harris J.: On Cayley’s explicit solution to Poncelet’s porism. Enseign Math(2) 24(1–2), 31–40 (1978)
Gruenewald, D.: Explicit algorithms for Humbert surfaces. Thesis, University of Sydney (2009). http://echidna.maths.usyd.edu.au/~davidg/
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)
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)
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)
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)
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)
Abhinav Kumar, K3 surfaces associated with curves of genus two. Int. Math. Res. Not. IMRN 2008, no. 6, Art. ID rnm165
Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235–265 (1997)
Wolfram Research, Inc., Mathematica, Version 7.0, Champaign, IL (2008)
Mestre J.F.: Courbes hyperelliptiques à à multiplications réelles. CR. Acad. Sci. Paris Ser. I Math. 307, 721–724 (1988)
Mestre J.F.: Courbes hyperelliptiques à à multiplications réelles. Progr. Math. 89, 193–208 (1991)
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
Runge B.: Endomorphism rings of abelian surfaces and projective models of their moduli spaces. Tohoku Math J (2) 51(3), 283–303 (1999)
SAGE Mathematics Software, Version 4.6. http://www.sagemath.org/
Sakai, Y.: Construction of genus two curves with real multiplication by Poncelet’s theorem. Dissertation, Waseda University (2010)
Sakai Y.: Poncelet’s theorem and hyperelliptic curve with real multiplication of Δ = 5. J. Ramanujan Math. Soc. 24(2), 143–170 (2009)
Shepherd-Barron N.I., Taylor R.: mod 2 and mod 5 icosahedral representations. J. Am. Math. Soc. 10(2), 283–298 (1997)
Shimura G.: On analytic families of polarized abelian varieties and automorphic functions. Ann. Math. (2) 78, 149–192 (1963)
Shimura G.: Abelian varieties with complex multiplication and modular functions. Princeton Mathematical Series, 46. Princeton University Press, Princeton, NJ (1998)
The PARI Group, PARI/GP, version 2.5.0, Bordeaux (2011) (http://pari.math.u-bordeaux.fr/)
Tautz W., Top J., Verberkmoes A.: Explicit hyperelliptic curves with real multiplication and permutation polynomials. Can. J. Math. 43(5), 1055–1064 (1991)
van der Geer G.: Hilbert modular surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 16. Springer, Berlin (1988)
Weil, A.: Variétés abéliennes et courbes algébriques, Actualités Sci. Ind., no. 1064 = Publ. Inst. Math. Univ. Strasbourg 8 (1946). Hermann, Paris (1948)
Weil, A.: Remarques sur un mémoire d’Hermite, Arch. Math. 5 (1954), 197–202 = Œ uvres (collected papers), vol. 2, pp. 111–116
Weil, A.: Euler and the Jacobians of elliptic curves, Arithmetic and geometry, vol. I, pp. 353–359, Progr. Math., 35, Birkhuser, Boston, MA (1983)
Wilson, J.: Curves of genus 2 with real multiplication by a square root of 5. Dissertation, University of Oxford, Oxford (1998)
Author information
Authors and Affiliations
Corresponding author
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
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13398-012-0079-1