Families of elliptic curves with genus 2 covers of degree 2

• Autores: Claus Diem
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 57, Fasc. 1, 2006, págs. 1-25
• Idioma: inglés
• Enlaces
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• Resumen

  • We study genus 2 covers of relative elliptic curves over an arbitrary base in which 2 is in invertible. Particular emphasis lies on the case that the co covering degree is 2. We show that the data in the "basic construction" of genus 2 covers of relative elliptic curves determine the cover in a unique way (up to isomorphism). A classical theorem says that a genus 2 cover of an elliptic curve of de degree 2 over a field of characteristic =/= 2 is birational to a product of two elliptic curves over the projective line. We formulate and pro prove a generalization of this theorem for the relative situation. We also pro prove a Torelli theorem for genus 2 curves over an arbitrary base.

• Referencias bibliográficas
  • S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, Springer-Verlag, Berlin, 1990.
  • P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109.
  • C. Diem and G. Frey, Non-constant genus 2 curves with infinite pro-Galois covers, to appear.
  • G. Frey and E. Kani, Curves of genus 2 covering elliptic curves and an arithmetical application, Progr. Math. 89 (1991), 153–176.
  • A. Grothendieck, Revêtements Étales et Groupe Fondamental (Fasc. I), Séminaire de Géométrie Algébrique, 1960–61, Institut des Hautes Études...
  • A. Grothendieck and J. Dieudonné, Éléments de Géométrie Algébrique (I-V), ch. I; Springer-Verlag, Berlin, 1971; ch. II-IV: Publ. Math. Inst....
  • R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977.
  • E. Kani, The number of genus 2 covers of an elliptic curve, to appear.
  • E. Kani, The number of curves of genus two with elliptic differentials, J. Reine Angew. Math. 485 (1997), 93–121.
  • E. Kani, Hurwitz spaces of genus 2 covers of an elliptic curve, Collect. Math. 54 (2003), 1–51.
  • A. Krazer, Lehrbuch der Thetafunktionen, Chelsea Publishing Co., New York, 1970, Reprint of the 1903 edition.
  • R.M. Kuhn, Curves of genus 2 with split Jacobian, Trans. Amer. Math. Soc. 307 (1988), 41–49.
  • E. Kux, Kähler Differentials, Friedr. Vieweg and Sohn, Braunschweig, 1986.
  • K. Lonsted and S.L. Kleiman, Basics on families of hyperelliptic curves, Compositio. Math. 38 (1979), 83–111.
  • H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986.
  • J.S. Milne, Jacobian varieties, in G. Cornell and J. Silverman, editors, Arithmetic Geometry, 167–212, Springer-Verlag, 1986.
  • D. Mumford, Geometric Invariant Theory, Springer-Verlag, Berlin-New York, 1965.
  • D. Mumford, Abelian Varieties, Tata Institute of Fundamental Research, Oxford University Press, London, 1970.