A bound for the Mordell-Weil rank of an elliptic surface after a cyclic base extension

Let epsilon --> P-1 be an elliptic surface defined over a number field K, o r equivalently an elliptic curve defined over K(T). In this note we prove, assuming Tate's conjecture, that the rank of epsilon(K(T-1/n)) is bounded b y F*(epsilon)d(K)(n), where F*(epsilon) is an explicit constant independent of n and d(K)(n) is an explicit elementary function. In particular, if K b oolean AND Q(zeta(d)) = Q for all d\n, then d(K)(n) = d(n) is just the numb er of divisors of n.