Skip to main content
SpringerLink
Log in

The number of genus 2 covers of an elliptic curve

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

The main aim of this paper is to determine the number c N,D of genus 2 covers of an elliptic curve E of fixed degree N ≥ 1 and fixed discriminant divisor D ∈Div (E). In the case that D is reduced, this formula is due to Dijkgraaf.

The basic technique here for determining c N,D is to exploit the geometry of a certain compactification C =C E,N of the universal genus 2 curve over the Hurwitz space H E,N which classifies (normalized) genus 2 covers of degree N of E. Thus, a secondary aim of this paper is to study the geometry of C. For example, the structure of its degenerate fibres is determined, and this yields formulae for the numerical invariants of C which are also of independent interest.

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. Arakelov S. Ju. (1971). Families of algebraic curves with fixed degeneracies. Izv. Akad. Nauk SSSR 35 (1971) = Math. USSR Izv. 5, 1277–1302

  2. Artin M.(1986). Lipman’s proof of resolution of singularities of surfaces. In: Cornell G., Silverman J. (eds) Arithmetic Geometry. Springer, New York, pp. 267–287

    Google Scholar 

  3. Bosch S., Lütkebohmert W., Raynaud M.(1990). Néron Models. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  4. Barth W., Peters C., Van de Ven A.(1984). Compact Complex Surfaces. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  5. Deligne P., Mumford D.(1969). The irreducibility of the space of curves of given genus. Publ. Math. I.H.E.S. 36, 75–109

    MathSciNet  MATH  Google Scholar 

  6. Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. In: Modular functions of one variable, II Lecture Notes in Math, vol. 349, Springer, Berlin Heidelberg New York, pp. 143–316 (1973)

  7. Dijkgraaf R. (1995). Mirror symmetry and elliptic curves. In: Dijkgraaf R., Faber C., van der Geer G. (eds) The Moduli of Curves. Birkhäuser, Basel, pp. 149–163

    Google Scholar 

  8. Grothendieck, A., Dieudonné, J.: Eléments de Géométrie Algébrique I-IV. Ch. I: Springer Berlin Heidelberg New York 1971; ch. II-IV: Publ. Math. I.H.E.S. 8, 11, 17, 20, 24, 28, 32 (1961–1968)

  9. Faltings G.(1983). Arakelov’s theorem for abelian varieties. Invent. math. 73, 337–348

    Article  MathSciNet  MATH  Google Scholar 

  10. Frey, G., Kani, E.: Curves of genus 2 covering elliptic curves and an arithmetical application. In: Arithmetic Algebraic Geometry (van der Geer, G., Oort, F., Steenbrink, J. (eds.)) pp. 153–176. Progress In Math. vol. 89, Birkhäuser, Boston (1991)

  11. Fulton W.(1969). Hurwitz schemes and irreducibility of moduli of algebraic curves. Ann. Math. (2)90: 542–575

    Article  MathSciNet  Google Scholar 

  12. Fulton W.(1984). Intersection Theory. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  13. Grothendieck, A.: Techniques de construction et théorèmes d’existence en géométrie algébrique IV: Les schémas de Hilbert. Séminaire Bourbaki No. 221 (1960/61)

  14. Grothendieck A.(1972). Modeles de Néron et Monodromie. SGA 7-I, Exp. IX, Springer Lect. Notes 288, 313–523

    MathSciNet  Google Scholar 

  15. Hartshorne R.(1977). Algebraic Geometry. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  16. Iitaka S. (1982). Algebraic Geometry. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  17. Kaneko M., Zagier D.(1995). A generalized Jacobi theta function and quasi-modular forms. In: Dijkgraaf R., Faber C., van der Geer G. (eds), The Moduli of Curves. Birkhäuser, Basel, pp. 165–172

    Google Scholar 

  18. Kani E.(1997). The number of curves of genus 2 with elliptic differentials. J. reine angew. Math. 485, 93–121

    MathSciNet  MATH  Google Scholar 

  19. Kani E.(1997). The existence of curves of genus 2 with elliptic differentials. J. Number Theory 64, 130–161

    Article  MathSciNet  MATH  Google Scholar 

  20. Kani E.(2003). Hurwitz spaces of genus 2 covers of an elliptic curve. Collect. Math. 54, 1–51

    MathSciNet  MATH  Google Scholar 

  21. Kleiman S.(1980). Relative duality for quasi-coherent sheaves. Comp. Math. 41, 39–60

    MathSciNet  MATH  Google Scholar 

  22. Kunz E.(1986). Kähler Differentials. Vieweg, Wiesbaden

    MATH  Google Scholar 

  23. Lang S.(1976). Introduction to Modular Forms. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  24. Lichtenbaum S.(1968).Minimal models over discrete valuation rings. Am. J. Math. 90, 380–405

    Article  MathSciNet  MATH  Google Scholar 

  25. Lønsted K., Kleiman S.(1979). Basics on families of hyperelliptic curves. Comp. Math. 38, 83–111

    MATH  Google Scholar 

  26. Mazur, B.: Letter to the author, 9 December (1989)

  27. Mumford D.(1983). Towards an enumerative geometry of the moduli space of curves. In: Artin M., Tate J. (eds) Arithmetic and Geometry - Papers dedicated to I.R. Shafarevich. Birkhäuser, Boston, pp. 271–328

    Google Scholar 

  28. Ogg A.(1966). On pencils of curves of genus 2. Topology 5, 355–362

    Article  MathSciNet  MATH  Google Scholar 

  29. Parshin, A.: Algebraic curves over function fields I. Izv. Akad. Nauk SSSR 32 (1968) = Math. USSR Izv. 2, 1145–1170 (1968)

  30. Parshin, A.: Minimal models of curves of genus 2. Izv. Akad. Nauk SSSR 36 (1972) = Math. USSR Izv. 6, 65–108 (1972)

  31. Saito T.(1989). The discriminants of curves of genus 2. Comp. Math. 69, 229-240

    MATH  Google Scholar 

  32. Serre J.-P.(1959). Groupes algébriques et corps de classes.Hermann, Paris

    MATH  Google Scholar 

  33. Szpiro, L.: Proprietés numériques du faiseaux dualisant relatif. In: Szpiro, L. (ed.) Séminaire sur les pinceaux des courbes des genre aux moins deux. Astérisque 86, 44–78 (1981)

  34. Tate, J.: Algorithm for determining the type of a singular fibre of an elliptic pencil. In: Modular Functions of One Variable IV. Springer Lect. Notes 476, 33–52 (1975)

  35. Ueno K. (1988). Discriminants of curves of genus 2 and arithmetic surfaces. In: Hijikata et al,H. (eds) Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata. Academic Press, Tokyo, pp. 749–770

    Google Scholar 

  36. Viehweg, E., Zuo, K.: A characterization of certain Shimura curves in the moduli stack of abelian varieties. Preprint (2002)

  37. Weil, A.: Zum Beweis des Torellischen Satzes. Göttinger Nachr. 1957, No. 2, 33–53 = Œuvres Sci./Collected Papers II pp. 307–327. Springer, Berlin Heidelberg New York (1979)

  38. Wevers S.: Deformation of tame admissible covers of curves. In: Völklein H. et al. (eds) Aspects of Galois Theory. LMS Lect. Note Ser No. 256. pp. 239–282. Cambridge U Press (1999)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Queen’s University, Kingston, ON, K7L 3N6, Canada

    Ernst Kani

Authors
  1. Ernst Kani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ernst Kani.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kani, E. The number of genus 2 covers of an elliptic curve. manuscripta math. 121, 51–80 (2006). https://doi.org/10.1007/s00229-006-0012-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-006-0012-z

Keywords

Search

Navigation