Sections \[sec:lAdicFailure\] and \[sec:AdelicFailure\] we study the $\ell$-adic and adelic failures respectively. In Section \[sec:CMCounterexample\] we show that one cannot hope to naïvely generalise some of the results in section \[sec:lAdicFailure\] to CM curves. Finally, in Section \[sec:UniformBounds\] we prove Theorem \[thm:UniformIntroduction\] by establishing several auxiliary results about the Galois cohomology of the torsion modules $E[M]$ that might have an independent interest.

Acknowledgements
----------------

It is a pleasure to thank Antonella Perucca for suggesting the problem that led to this paper, for her constant support, and for her useful comments. We are grateful to Peter Bruin for many interesting discussions, and to Peter Stevenhagen and Francesco Campagna for useful correspondence about the results of section \[sec:IntersectionTorsionFields\].

Preliminaries {#sec:Preliminaries}
=============

Notation and definitions
------------------------

The letter $K$ will always denote a number field, $E$ an elliptic curve defined over $K$, and $\alpha$ a point of infinite order in $E(K)$.

For $n$ a positive integer, we denote by $\zeta_n$ a primitive root of unity of order $n$.

Given a prime $\ell$, we denote by $v_\ell$ the usual $\ell$-adic valuation on ${\mathbb{Q}}$ and on ${\mathbb{Q}}_\ell$. If $X$ is a vector in ${\mathbb{Z}}_\ell^n$ or a matrix in $\operatorname{Mat}_{m \times n}({\mathbb{Z}}_\ell)$, we call *valuation* of 