TITLE: Building elliptic curves into a family
QUESTION [11 upvotes]: Suppose $E/ \mathbb{Q}$ is an elliptic curve whose Mordell-Weil group $E(\mathbb{Q})$  has rank r.  When can we realize E as a fiber of an elliptic surface $S\to C$ fibered over some curve, with everything defined over $\mathbb{Q}$, such that the group of $\mathbb{Q}$-rational sections of $S$ has rank at least r?
Edit: Let me also demand that the resulting family is not isotrivial, i.e. the j-invariants of the fibers are not all equal.

REPLY [4 votes]: Evidence that it is not known to always be possible over $\mathbb{QP}^1$: If you look at the records of the elliptic curves with high rank, they are broken down into three categories 

Elliptic curves over $\mathbb{Q}$.
Nonisotrivial curves over $E$, where $E$ is an elliptic curve over $\mathbb{Q}$ with $|E(\mathbb{Q})|$ infinite.
Nonisotrivial curves over $\mathbb{QP}^1$.

If we could always deform, these records would be the same. In fact, according to the tables here and here, the highest known rank of type 1 is 28, of type 2 is 19 and of type 3 is 18.
I do not know whether there are examples where it is known that such a deformation is impossible.