Rational curves on elliptic surfaces

We prove that a very general elliptic surface E → P1 over the complex numbers with a section and with geometric genus pg ≥ 2 contains no rational curves other than the section and components of singular fibers. Equivalently, if E/C(t) is a very general elliptic curve of height d ≥ 3 and if L is a finite extension of C(t) with L ∼= C(u), then the Mordell-Weil group E(L) = 0.