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.



Building elliptic curves into a family
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.


