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In this section we introduce three representations of the absolute Galois group of $K$ that will be our main tool for studying the extensions $K_{M,N}$. For further information about these representations see for example [@JonesRouse Section 3], [@2018arXiv180208527B], and [@2016arXiv161202847L].

### The torsion representation

Let $N$ be a positive integer. The group $E[N]$ of $N$-torsion points of $E$ is a free ${\mathbb{Z}}/N{\mathbb{Z}}$-module of rank $2$. Since the multiplication-by-$N$ map is defined over $K$, the absolute Galois group of $K$ acts ${\mathbb{Z}}/N{\mathbb{Z}}$-linearly on $E[N]$, and we get a homomorphism $$\begin{aligned}
\tau_N:\operatorname{Gal}({\overline{K}}\mid K)\to \operatorname{Aut}(E[N]).\end{aligned}$$

The field fixed by the kernel of $\tau_N$ is exactly the $N$-th torsion field $K_N$. Thus, after fixing a ${\mathbb{Z}}/N{\mathbb{Z}}$-basis of $E[N]$, the Galois group $\operatorname{Gal}(K_N\mid K)$ is identified with a subgroup of $\operatorname{GL}_2({\mathbb{Z}}/N{\mathbb{Z}})$ which we denote by $H_N$.

As $N$ varies, and provided that we have made compatible choices of bases, these representations form a compatible projective system, so we can pass to the limit over powers of a fixed prime $\ell$ to obtain the *$\ell$-adic torsion representation* $\tau_{\ell^\infty} : \operatorname{Gal}(\overline{K} \mid K) \to \operatorname{GL}_2({\mathbb{Z}}_\ell)$. We can also take the limit over all integers $N$ (ordered by divisibility) to obtain the *adelic torsion representation* $\tau_\infty : 