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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.

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up vote 10 down vote accepted

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.

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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

  1. Elliptic curves over $\mathbb{Q}$.
  2. Nonisotrivial curves over $E$, where $E$ is an elliptic curve over $\mathbb{Q}$ with $|E(\mathbb{Q})|$ infinite.
  3. 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.

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