Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-23T22:03:30.688Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  20 November 2018

Nils Bruin  and
Kevin Doerksen
Nils Bruin
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6email: [email protected]@sfu.ca
Kevin Doerksen
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6email: [email protected]@sfu.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

11G3014H40genus 2 curvesisogeniessplit Jacobianselliptic curves
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 5 , 18 October 2011 , pp. 992 - 1024
Copyright
Copyright © Canadian Mathematical Society 2011

References

[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] Bosch, S., Lütkebohmert, W., and Raynaud, M., Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21, Springer-Verlag, Berlin, 1990.Google Scholar
[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.1996.0125Google Scholar
[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] Bruin, N. and Doerksen, K., Electronic resources. http://www.cecm.sfu.ca/-nbruin/splitigusa.Google Scholar
[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] 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] 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] 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] 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] Igusa, J.-I., Arithmetic variety of moduli for genus two. Ann. of Math. (2) 72(1960), 612–649. doi:10.2307/1970233Google Scholar
[12] Igusa, J.-I., On Siegel modular forms of genus two. Amer. J. Math. 84(1962), 175–200. doi:10.2307/2372812Google Scholar
[13] Kani, E., Elliptic curves on abelian surfaces. Manuscripta Math. 84(1994), no. 2, 199–223. doi:10.1007/BF02567454Google Scholar
[14] Kani, E., Hurwitz spaces of genus 2 covers of an elliptic curve. Collect. Math. 54(2003), no. 1, 1–51.Google Scholar
[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:10.2307/2323198Google Scholar
[16] Krazer, A., Lehrbuch der Thetafunktionen, Tuebner, 1903, http://hdl.handle.net/2027/miun.acq9458.0001.001.Google Scholar
[17] Kuhn, R. M., Curves of genus 2 with split Jacobian. Trans. Amer. Math. Soc. 307(1988), no. 1, 41–49.Google Scholar
[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.2009.027Google Scholar
[19] Milne, J. S., Abelian varieties. In: Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 103–150.Google Scholar
[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/BF02567449Google Scholar
[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] Shaska, T., Genus 2 fields with degree 3 elliptic subfields. Forum Math. 16(2004), no. 2, 263–280. doi:10.1515/form.2004.013Google Scholar
[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] Smith, B., Explicit endomorphisms and correspondences. Ph.D. thesis, University of Sydney, 2005, http://hdl.handle.net/2123/1066.Google Scholar