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In https://arxiv.org/abs/1505.04769 in the proof of Theorem 5 it is asserted that since $\rho_{E, l}:G_\mathbb{Q}\to\mathrm{GL}_2(\mathbb{Z}_l)$ is surjective then $E_{\mathbb{Q}_\infty}[l^\infty]=0$. Here $\mathbb{Q}_\infty$ means a $\mathbb{Z}_p$-extension of $\mathbb{Q}$ for some prime $p$ (possibly equal to $l$).

I want to fill in the details.

I guess that if $l\neq p$ then $\rho_{E, l}(G_{\mathbb{Q}_\infty})=\mathrm{GL}_2(\mathbb{Z}_l)$ and if $l=p$ then $\rho_{E, l}(G_{\mathbb{Q}_\infty})$ may not be $\mathrm{GL}_2(\mathbb{Z}_l)$ but it contains $\mathrm{SL}_2(\mathbb{Z}_l)$. So either way there are no non-trivial fixed vectors.

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  • $\begingroup$ I think this is basically right, but extra care must be taken for $\ell=2$ because then the comutator subgroup of $GL_2(\mathbb Z_\ell)$ is smaller than $SL_2(\mathbb Z_\ell)$ (as you can see just by looking mod $2$). $\endgroup$
    – Will Sawin
    Commented Sep 12, 2021 at 15:40
  • $\begingroup$ Forgot to mention there is a separate argument for $l=2$ $\endgroup$
    – LgF
    Commented Sep 13, 2021 at 1:39

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