= CM Tate module stabilizer
{c}
{title2=$\{a\in K_\ell:aT_\ell(E)\subseteq T_\ell(E)\}=R\otimes\mathbb Z_\ell$}
Let $R=K\cap\operatorname{End}_k(E)$. If $\alpha\in R$ maps the <Tate module> into $\ell^nT_\ell(E)$, it kills $E[\ell^n]$ and factors through the quotient isogeny $[\ell^n]$. The resulting <endomorphism> is $\alpha/\ell^n$, belongs to $K$ and is defined over $k$, hence belongs to $R$. Approximation in $R\otimes\mathbb Z_\ell$ now identifies the displayed stabilizer. This does not assume that the integral <Tate module> is free over a possibly nonmaximal CM order.
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