Let $\mathbb{E}$ be an elliptic curve defined over a number field $k$,
and let $a\in\mathbb{E}(\mathbb{C})$ be a complex point. Among the
possible choices of sequences of division points of $a$, $(a_n)_n$
such that $a_1 = a$ and $na_{nm} = a_m$, we can pick out those which
converge in the complex topology to the identity. We show that the
algebraic content of this effect of the complex topology is very
small, in the sense that any set of division sequences which shares
certain obvious algebraic properties with the set of those which
converge to the identity is conjugated to it by a field automorphism
of $\mathbb{C}$ over $k$.
As stated, this is a result of algebra and number theory. However, in
proving it we are led ineluctably to use model theoretic techniques -
specifically the concept of "excellence" introduced by Shelah for the
analysis of $L_{\omega_1,\omega}$ categoricity, which reduces the
question to that of proving certain unusual versions of the theorems
of Mordell-Weil and Kummer-Bashmakov. I will discuss this and other
aspects of the proof, without assuming any model- or number-theoretic
knowledge on the part of my audience.