Ground field $\Bbb{C}$. Algebraic category.
Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve.
Suppose $E$ is an elliptic curve and consider the ruled surface $$ S=\frac{E\times\Bbb{P}^1}{G} $$ where $G$ is a group of translations of $E$, acting on $\Bbb{P}^1$.
Thus $F=E/G$ is an elliptic curve (and $\Bbb{P}^1/G=\Bbb{P}^1$)
Then $S$ is an elliptic surface, for the projection on the second factor induces a morphism $S\rightarrow\Bbb{P}^1$ whose fibers are elliptic curves ($F$, in fact).
An exercise on Beauville's book (chap. IX) says that if $S$ is a ruled surface over an elliptic curve $E$, and $S$ is an elliptic surface, then $S$ is isomorphic to the above example. Any hint for attacking this ? Thank you.
Edit: This question has been answered here.