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.
If you require $C = P^1$ then it's probably not possible except for very small values of $r$. If you don't care about $C$, then here is something that might work.
Suppose $E$ is given by $y^2=x^3+ax+b$ and $P_i=(x_i,y_i),i=1,\ldots,r$ is a basis for the Mordell-Weil group. Let $C$ be the curve given by the system of equations $u_i^2 = (t^i+x_i)^3 + a(t^i+x_i) + b + t, i=1,\ldots,r$ in $t,u_1,\ldots,u_r$ and $S$ be the family $y^2 = x^3 + ax + b+t$ pulled back to $C$. So above $t=0$, $C$ has a point with $u_i=y_i$ and and the fiber of $S$ above this point is $E$. Also $C$ is defined so that there are sections of $S$ with $x$-coordinate $x=t^i+x_i$ and I bet they are independent. Finally the family is non isotrivial if $a \ne 0$. If $a=0$ adjust the construction is an obvious way.
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
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.
