Let $K$ be a number field and let $K(a,b)$ be the field of rational functions with two indeterminates over $K$. Consider the elliptic curve $E$ over $K(a,b)$ defined by the Weierstrass equation
\begin{equation\*}
E : y^2=x^3+ax+b.
\end{equation\*}
What is the torsion subgroup and the rank of the Mordell-Weil group of $E$ over $K(a,b)$?


In the case $K=\mathbf{Q}$, this Mordell-Weil group is trivial because $E$ admits specializations $a,b \in \mathbf{Q}$ such that $E\_{a,b}$ has trivial Mordell-Weil group. In the general case, I would like to show similarly the existence of $a,b \in K$ such that $E\_{a,b}$ has trivial Mordell-Weil group over $K$, but I don't know how to proceed.


Since $K(a,b)$ is a finitely generated field, the Lang-Néron theorem tells us that the Mordell-Weil group of $E$ is finitely generated, but I'm not familiar enough to tell whether the proof actually gives us a way to compute this Mordell-Weil group.



Determining the Mordell-Weil group of a universal elliptic curve
Specialize $a,b$ to functions giving the universal elliptic curve over the modular curve $X\_0(N)$. These are known to have rank zero over the function field of the modular curve with coefficients over $\mathbb{C}$ even. They can have torsion but, by varying $N$, you can show that the torsion is trivial too. 


T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan, 24, (1972) 20-59.


