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

Abstract

Let $\mathcal{ℰ}\to C$ be an elliptic surface defined over a number field $k$, let $P:C\to \mathcal{ℰ}$ be a section, and let $\ell$ be a rational prime. We bound the number of points of low algebraic degree in the $\ell$-division hull of $P$ at the fibre ${\mathcal{ℰ}}_t$. Specifically, for $t\in C\left(\bar{k}\right)$ with $\left[k\left(t\right):k\right]\le B_1$ such that ${\mathcal{ℰ}}_t$ is nonsingular, we obtain a bound on the number of $Q\in {\mathcal{ℰ}}_t\left(\bar{k}\right)$ such that $\left[k\left(Q\right):k\right]\le B_2$, and such that ${\ell }^{n}Q=P_t$ for some $n\ge 1$. This bound depends on $\mathcal{ℰ}$, $P$, $\ell$, $B_1$, and $B_2$, but is independent of $t$.