$X$, denoted by $v_\ell(X)$, the minimum of the $\ell$-adic valuations of its coefficients.

We shall often use divisibility conditions involving the symbols $\ell^\infty$ (where $\ell$ is a prime) and $\infty$. Our convention is that every power of $\ell$ divides $\ell^\infty$, every positive integer divides $\infty$, and $\ell^\infty$ divides $\infty$.

Recall from the Introduction that we denote by $K_M$ the field $K(E[M])$ generated by the coordinates of the $M$-torsion points of $E$, and by $K_{M,N}$ (for $N \mid M$) the field $K(E[M], N^{-1} \alpha)$. We extend this notation by setting $K_{\ell^\infty} = \bigcup_n K_{\ell^n}$, $K_\infty = \bigcup_M K_M$, and more generally, for $M, N \in \mathbb{N}_{>0} \cup \{\ell^\infty, \infty\}$ with $N \mid M$, $$K_M = \bigcup_{d \mid M} K_d, \quad K_{M,N} = \bigcup_{d \mid M} \bigcup_{\substack{e \mid d \\ e \mid N}} K_{d,e}$$

If $H$ is a subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$, we denote by ${\mathbb{Z}}_\ell[H]$ the sub-${\mathbb{Z}}_\ell$-algebra of $\operatorname{Mat}_2({\mathbb{Z}}_\ell)$ generated by the elements of $H$.

Let $G$ be a (profinite) group. We write $G'$ for its derived subgroup, namely, the subgroup of $G$ (topologically) generated by commutators, and $G^{\operatorname{ab}}=G/G'$ for its abelianisation, namely, its largest abelian (profinite) quotient. We say that a finite simple group $S$ *occurs* in a profinite group $G$ 