Specializations of Elliptic Surfaces, and Divisibility in the Mordell-weil Group

Let E → C be an elliptic surface, defined over a number field k, let P : C → E be a section, and let l be a rational prime. We bound the number of points of low algebraic degree in the l-division hull of P at the fibre Et. Specifically, for t ∈ C(k) with [k(t) : k] ≤ B1 such that Et is non-singular, we obtain a bound on the number of Q ∈ Et(k) such that [k(Q) : k] ≤ B2, and such that lQ = Pt, for some n ≥ 1. This bound depends on E, P , l, B1, and B2, but is independent of t.