is *$n$-indivisible over $K$* if there is no $\beta \in E(K)$ such that $n \beta=\alpha$; otherwise we say that $\alpha$ is *$n$-divisible* or *divisible by $n$ over $K$*. Let $\ell$ be a prime number. We say that $\alpha$ is *strongly $\ell$-indivisible over $K$* if the point $\alpha+T$ is $\ell$-indivisible over $K$ for every torsion point $T\in E(K)$ of $\ell$-power order. Finally, we say that $\alpha$ is *strongly indivisible over $K$* if its image in the free abelian group $E(K)/E(K)_{\operatorname{tors}}$ is not divisible by any $n>1$, or equivalently if $\alpha$ is strongly $\ell$-indivisible over $K$ for every prime $\ell$.

Our aim is to give an analogue of the following result, which bounds the index of the image of the Kummer representation, in those cases when the torsion representation is not surjective.

\[thm:JonesRouseSurjectivity\] Assume that the $\ell$-adic torsion representation $\tau_{\ell^\infty}:\operatorname{Gal}(K_{\ell^\infty}\mid K)\to \operatorname{GL}_2({\mathbb{Z}}_\ell)$ is surjective. Assume that $\alpha$ is $\ell$-indivisible in $E(K)$ and, if $\ell=2$, that $K_{2,2}\not\subseteq K_4$. Then the $\ell$-adic Kummer representation $\kappa_{\ell^\infty}:\operatorname{Gal}(K_{\ell^\infty,\ell^\infty}\mid K_{\ell^\infty})\to {\mathbb{Z}}_\ell^2$ is surjective.

An exact sequence
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We shall need to understand the divisibility properties of $\alpha$ not only over the base field $K$, but also over the division fields of $E$. Thus we turn to studying how 