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.

If you want to do this directly, you could "partially specialize" to, say $y^2 = x^3 + Ax + T$ with $A\in\mathbb C$. Then I don't think it's very hard to show, via a standard descent, that as an elliptic curve over $\mathbb C(T)$, the rank is $0$, and that for most $A$, the torsion is trivial, too. Actually, for the rank $0$ part, maybe it's easier to show that $y^2=x(x-A)(x-T)$ has rank 0 over $\mathbb C(T)$, since you can more easily do a 2-descent. And you also get that there is at most 2-torsion for most $A$. But it's pretty clear that your original curve over $\mathbb C(a,b)$ has no 2-torsion.