The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians

Bruin, Nils ; Doerksen, Kevin

Canadian journal of mathematics, Tome 63 (2011) no. 5, pp. 992-1024

Résumé

In this paper we study genus 2 curves whose Jacobians admit a polarized (4, 4)-isogeny to a product of elliptic curves. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their 4-torsion, and we derive the relation their absolute invariants satisfy.As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus 2 curves, where previously only Richelot isogenies with kernels that are pointwise defined over the base field were considered.Our main tool is a Galois theoretic characterization of genus 2 curves admitting multiple Richelot isogenies.

Détail

DOI : 10.4153/CJM-2011-039-3

Mots-clés : 11G30, 14H40, genus 2 curves, isogenies, split Jacobians, elliptic curves

Bibliographie

[1] [1] Bolza, O., Ueber die Reduction hyperelliptischer Integrale erster Ordnung und erster Gattung auf elliptische durch eine Transformation vierten Grades. Math. Ann. 28(1886), no. 3, 447–456. Google Scholar

[2] [2] Bosch, S., Lütkebohmert, W., and Raynaud, M., Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21, Springer-Verlag, Berlin, 1990. Google Scholar

[3] [3] Bosma, W., Cannon, J., and Playoust, C., The Magma algebra system. I. The user language. Computational algebra and number theory (London, 1993), J. Symbolic Comput. 24(1997), no. 3–4, 235–265. doi:10.1006/jsco.1996.0125 Google Scholar

[4] [4] J.-B., Bost and J.-F., Mestre Moyenne arithmético-géométrique et périodes des courbes de genre 1 et 2. Gaz. Math. 38(1988), 36–64. Google Scholar

[5] [5] Bruin, N. and Doerksen, K., Electronic resources. http://www.cecm.sfu.ca/-nbruin/splitigusa. Google Scholar

[6] [6] Cassels, J. W. S. and Flynn, E. V., Prolegomena to a middlebrow arithmetic of curves of genus 2. London Mathematical Society Lecture Note Series, 230, Cambridge University Press, Cambridge, 1996. Google Scholar

[7] [7] Donagi, R. and R. Livné, The arithmetic-geometric mean and isogenies for curves of higher genus. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28(1999), no. 2, 323–339. Google Scholar

[8] [8] Frey, G., On elliptic curves with isomorphic torsion structures and corresponding curves of genus 2. In: Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong, 1993), Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 79–98. Google Scholar

[9] [9] Frey, G. and Kani, E., Curves of genus 2 covering elliptic curves and an arithmetical application. In: Arithmetic algebraic geometry (Texel, 1989), Progr. Math., 89, Birkhäuser Boston, Boston, MA, 1991, pp. 153–176. Google Scholar

[10] [10] Gaudry, P. and Schost, É., On the invariants of the quotients of the Jacobian of a curve of genus 2. In: Applied algebra, algebraic algorithms and error-correcting codes (Melbourne, 2001), Lecture Notes in Comput. Sci., 2227, Springer, Berlin, 2001, pp. 373–386. Google Scholar

[11] [11] Igusa, J.-I., Arithmetic variety of moduli for genus two. Ann. of Math. (2) 72(1960), 612–649. doi:10.2307/1970233 Google Scholar

[12] [12] Igusa, J.-I., On Siegel modular forms of genus two. Amer. J. Math. 84(1962), 175–200. doi:10.2307/2372812 Google Scholar

[13] [13] Kani, E., Elliptic curves on abelian surfaces. Manuscripta Math. 84(1994), no. 2, 199–223. doi:10.1007/BF02567454 Google Scholar

[14] [14] Kani, E., Hurwitz spaces of genus 2 covers of an elliptic curve. Collect. Math. 54(2003), no. 1, 1–51. Google Scholar

[15] [15] L.-C., Kappe and B.Warren, An elementary test for the Galois group of a quartic polynomial. Amer. Math. Monthly 96(1989), no. 2, 133–137. doi:10.2307/2323198 Google Scholar

[16] [16] Krazer, A., Lehrbuch der Thetafunktionen, Tuebner, 1903, http://hdl.handle.net/2027/miun.acq9458.0001.001. Google Scholar

[17] [17] Kuhn, R. M., Curves of genus 2 with split Jacobian. Trans. Amer. Math. Soc. 307(1988), no. 1, 41–49. Google Scholar

[18] [18] Magaard, K., Shaska, T., and H. Völklein, Genus 2 curves that admit a degree 5 map to an elliptic curve. Forum Math. 21(2009), no. 3, 547–566. doi:10.1515/FORUM.2009.027 Google Scholar

[19] [19] Milne, J. S., Abelian varieties. In: Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 103–150. Google Scholar

[20] [20] Murabayashi, N., The moduli space of curves of genus two covering elliptic curves. Manuscripta Math. 84(1994), no. 2, 125–133. doi:10.1007/BF02567449 Google Scholar

[21] [21] Shaska, T., Genus 2 curves with (3, 3)-split Jacobian and large automorphism group. In: Algorithmic number theory (Sydney, 2002), Lecture Notes in Comput. Sci., 2369, Springer, Berlin, 2002, pp. 205–218. Google Scholar

[22] [22] Shaska, T., Genus 2 fields with degree 3 elliptic subfields. Forum Math. 16(2004), no. 2, 263–280. doi:10.1515/form.2004.013 Google Scholar

[23] [23] Silverberg, A., Explicit families of elliptic curves with prescribed mod N representations. In: Modular forms and Fermat's last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 447–461. Google Scholar

[24] [24] Smith, B., Explicit endomorphisms and correspondences. Ph.D. thesis, University of Sydney, 2005, http://hdl.handle.net/2123/1066. Google Scholar