Is Galois representation induced by semistable elliptic curve semistable? $\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\Aut{Aut}$Let $E$ be a semi stable elliptic curve. Let $\overline{\rho_\ell}: \Gal(\overline{\Bbb Q}/\Bbb Q)\to \Aut (E[l]) $  be mod $\ell$ representation of ${\rho_\ell}: \Gal(\overline{\Bbb Q}/\Bbb Q)\to \Aut (T_\ell E) $.
Is it necessary that $\overline{\rho_\ell}| _{G_\ell}$  semistable?
Any reference is well appreciated.
 A: A Galois representation $\rho_\ell:\operatorname{Gal}(\bar{\mathbb Q}_{\ell}/\mathbb Q_{\ell})\longrightarrow\operatorname{GL}_2(\mathbb Q_{\ell})$ can be semistable (technically $B_{st}$-admissible in the sense of Fontaine). A Galois representation $\bar{\rho}_\ell:\operatorname{Gal}(\bar{\mathbb Q}_{\ell}/\mathbb Q_{\ell})\longrightarrow\operatorname{GL}_2(\mathbb F_{\ell})$ cannot be semistable in that sense, as a matter of definition.
Suppose now that $E/\mathbb Q_\ell$ is an elliptic curve with semistable reduction. Then indeed $\rho_{E}:\operatorname{Gal}(\bar{\mathbb Q}_{\ell}/\mathbb Q_{\ell})\longrightarrow\operatorname{GL}_2(\mathbb Q_{\ell})$ is a semistable $\operatorname{Gal}(\bar{\mathbb Q}_{\ell}/\mathbb Q_{\ell})$-representation, but that is not obvious at all: it requires an understanding of the theory of the Tate curve and some understanding of the constructions of the rings $B_{cris}^+$ and $B_{st}$. In fact, from a historical point of view, understanding why $\rho_{E}$ is a semistable $\operatorname{Gal}(\bar{\mathbb Q}_{\ell}/\mathbb Q_{\ell})$-representation amounts exactly to understanding the construction of $B_{cris}$ and $B_{st}$ as Fontaine originally defined $B_{st}$ as the smallest ring $B$ which contains $B_{cris}$ and for which the Tate curve is $B$-admissible. This is all explained in Laurent Berger's notes linked by Chris Wuthrich.
On the other hand, the $\operatorname{Gal}(\bar{\mathbb Q}_{\ell}/\mathbb Q_{\ell})$-representation $\bar{\rho}_{E}$ given by the Galois action on $\ell$-torsion points cannot be semistable, as a matter of definition. Note that it can be very simple indeed: in the situation you describe, it can very well be unramified.
