Mordell-Weil rank in elliptic surfaces

Suppose that an elliptic smooth K3 surface $X$ defined over a number field $k$ has arithmetic Picard rank $r$ and assume that it is equipped with a $k$ fibration over $\mathbb{P}^1$ that has a section over $k$, i.e. $f:X\to \mathbb{P}^1$ is a dominant map with the generic fiber $f^{-1}(\eta)$ being an elliptic curve. Is there a relation between the Mordell-Weil rank of the generic fiber and $r$ ?

• Just to be clear, do you know the answer in case $k$ is instead something algebraically closed? – user64687 Apr 30 '15 at 5:25
• No, not really:( – Captain Darling Apr 30 '15 at 12:41
• OK, let me try to write something that applies in that case, and then maybe you can see how to adapt it to your setting... – user64687 May 1 '15 at 16:49
• Maybe just an explicit example would be ok ? – Captain Darling May 1 '15 at 19:13

Note: let $k$ be a field and let $C/k$ be a smooth projective curve defined over the field $k$ and has genus $g$. The function field of $C/k$ will be denoted by $K=k(C)$. An elliptic surface $\mathcal{E}$ over the curve $C$ is, by definition, a two-dimensional projective variety together with (i) a morphism $\pi: \mathcal{E} \to C$ such that for all but finitely many points $t\in C(\overline{k})$, the fiber $\mathcal{E}_t=\pi^{-1}(t)$ is a non-singular curve of genus $1$, and (ii) a section to $\pi$ (the zero section) $\sigma_0: C \to \mathcal{E}$. With this definition, the generic fibre of $\pi$, $E/K$, may be regarded as an elliptic curve over the field $K$.