elliptic surface
Definition 1.
Let $k$ be a field and let $C\mathrm{/}k$ be a smooth projective curve defined over the field $k$ and has genus $g$. The function field^{} of $C\mathrm{/}k$ will be denoted by $K\mathrm{=}k\mathit{}\mathrm{(}C\mathrm{)}$. An elliptic surface $\mathrm{E}$ over the curve $C$ is, by definition, a twodimensional projective variety together with:

1.
A morphism $\pi :\mathcal{E}\to C$ such that for all but finitely many points $t\in C(\overline{k})$, the fiber ${\mathcal{E}}_{t}={\pi}^{1}(t)$ is a nonsingular curve of genus $1$,

2.
A section to $\pi $ (the zero section) ${\sigma}_{0}:C\to \mathcal{E}$.
With this definition, $\mathrm{E}\mathrm{/}K$ may be regarded as an elliptic curve^{} over the field $K$.
Example 1.
The surface ${y}^{2}={x}^{3}+t$ is an elliptic surface over the curve ${\mathbb{P}}^{1}(\mathbb{Q})$. It may be regarded as an elliptic curve over the function field $\mathbb{Q}(t)$.
References
 1 R. Miranda, The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica, Dipartimento di Mathematica dell’ UniversitÃÂ di Pisa, ETS Editrice Pisa, 1989.
 2 J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 151, SpringerVerlag, New York.
Title  elliptic surface 

Canonical name  EllipticSurface 
Date of creation  20130322 15:34:16 
Last modified on  20130322 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 