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Let $E$ be an elliptic surface over the curve $C$, defined over a numberfield $k$, let $P$ be a section of $E$, and let $\ell$ be a rational prime. Forany non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of(algebraic) degree at most $D$ over $k$, such that $\ell^n Q=P_t$, for some$n\geq 1$. The bound obtained depends only on $\ell$, the surface and sectionin question, $D$, and the degree $[k(t):k]$; that is, it is uniform across allfibres of bounded degree. In special cases, we obtain more specific, in someinstances sharp, bounds.

Specializations of elliptic surfaces, and divisibility in the Mordell–Weil group