RATIONAL CURVES ON ELLIPTIC SURFACES

We prove that a very general elliptic surface epsilon -> P-1 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 congruent to C(u), then the Mordell-Weil group E(L) = 0.