# 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 field  of $C/k$ will be denoted by $K=k(C)$. An elliptic surface $\mathcal{E}$ over the curve $C$ is, by definition, a two-dimensional projective variety together with:

1. 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 non-singular curve of genus $1$,

2. 2.

A section to $\pi$ (the zero section) $\sigma_{0}:C\to\mathcal{E}$.

With this definition, $\mathcal{E}/K$ 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’ UniversitÃÂ  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 EllipticSurface 2013-03-22 15:34:16 2013-03-22 15:34:16 alozano (2414) alozano (2414) 5 alozano (2414) Definition msc 14J27 EllipticCurve