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Let $E$ be an elliptic curve having invariant $j$ and defined over $\mathbf C(j)$. Let $\sigma$ be an automorphism of $F_{\mathbf C}$, the field of all modular functions of all levels. Let $p$ be a prime and $$L=\mathbf C(j,j^\sigma, E[p],E^\sigma[p]).$$ Lang claims in his Elliptic functions, 7, §3, p. 84 that the Galois group of $F_\mathbf C$ over $L$ contains an open subgroup of the form $$W=\prod_{\ell\in S}W_\ell\times\prod_{\ell \in S}\operatorname{SL}_2(\mathbf Z_\ell)$$ where $S$ is a finite set of primes and $W_\ell$ is a small neighbourhood of $1$ in $\operatorname{SL}_2(\mathbf Z_\ell)$ such that $W_\ell$ is an $\ell$-group and without torsion.

I do not understand how Lang arrived at this conclusion. Lang's purpose is to prove the Shimura's exact sequence $$1\longrightarrow\mathbf Q ^\times\longrightarrow\operatorname{GL}_2^+(\mathbf Q)\prod_p\operatorname{GL}_2(\mathbf Z_p)\longrightarrow\operatorname{Aut}(F)\longrightarrow 1.$$ Here $F$ is the field of modular functions of all levels and with cyclotomic Fourier coefficients.

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    $\begingroup$ By definition of $F_\mathbf{C}=\bigcup_N \mathbf{C}(X(N))$ since $L$ is finitely generated it is contained in some $\mathbf{C}(X(N))$ so that $\{ A\in SL_2(\hat{\mathbf{Z}}), A\equiv I \bmod N\} \subset Aut(F_\mathbf{C}/ L)$. Also $SL_2(\hat{\mathbf{Z}})$ is the set of sequences $(A_m)$ in $M_2(\mathbf{Z})$ that converge modulo $n$ to an element of $SL_2(\mathbf{Z}/n\mathbf{Z})$ for all $n$, the automorphism is $f\mapsto \lim_{m \to \infty} f\circ A_m$. $\endgroup$
    – reuns
    Commented Oct 10, 2019 at 23:38

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