elliptic surface

Definition 1.

Let k be a field and let C/k be a smooth projective curve defined over the field k and has genus g. The function fieldMathworldPlanetmath of C/k will be denoted by K=k(C). An elliptic surface E over the curve C is, by definition, a two-dimensional projective variety together with:

  1. 1.

    A morphism π:C such that for all but finitely many points tC(k¯), the fiber t=π-1(t) is a non-singular curve of genus 1,

  2. 2.

    A section to π (the zero section) σ0:C.

With this definition, E/K may be regarded as an elliptic curveMathworldPlanetmath over the field K.

Example 1.

The surface y2=x3+t is an elliptic surface over the curve 1(). It may be regarded as an elliptic curve over the function field (t).


  • 1 R. Miranda, The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica, Dipartimento di Mathematica dell’ Università di Pisa, ETS Editrice Pisa, 1989.
  • 2 J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 151, Springer-Verlag, New York.
Title elliptic surface
Canonical name EllipticSurface
Date of creation 2013-03-22 15:34:16
Last modified on 2013-03-22 15:34:16
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Definition
Classification msc 14J27
Related topic EllipticCurve