Elliptic curve and Galois representation

Question

For an elliptic curve $E$ over ${\Bbb{Q}}$, let us consider Serre's mod $l$ representation by

$\rho_{E,l} \colon {\mathrm{Gal}}({\overline{\Bbb{Q}}}/{\Bbb{Q}}) \to {\mathrm{Aut}}(\phantom{}_lE) = {\mathrm{GL}}_2({\Bbb{F}}_l)$,

where $\phantom{}_lE$ is the group of $l$-torsion points on $E$.

We say $\rho_{E,l}$ is ``finite" at prime $p$, which is equivalent to that $p$ is unramified if $p \not= l$. For $p = l$, it is equivalent to that there is a ${\Bbb F}_l$-vector space scheme $H$ over ${\Bbb Z}_p$ such that $H(\overline{\Bbb{Q}}_p)$ gives the representation of ${\mathrm{Gal}}(\overline{\Bbb{Q}}_p/\Bbb{Q}_p)$ when $\rho_{E,l}$ is restricted to ${\mathrm{Gal}}(\overline{\Bbb{Q}}_p/{\Bbb{Q}}_p) \subset {\mathrm{Gal}}(\overline{\Bbb{Q}}/\Bbb{Q})$.

Q: Are these conditions equivalent to that $E$ has good reduction at $p$?

Pierre

Answer 1

None of these conditions implies that $E$ has good reduction at $p$. Consider, for instance, the elliptic curve $E = X_0(11)$, for which $E[5] \cong \mathbb{Z}/5\mathbb{Z} \oplus \mu_5$. Then for $l = 5$, the representation $E[l]$ is unramified at every prime $p \neq l$ and is finite flat at $p = l$, but $E$ has bad reduction at $p = 11$.

Your question is purely local (no need to assume that $E$ is over $\mathbb{Q}$). It is instructive to work out for oneself the case of split multiplicative reduction ("Tate curve") to get a clear picture why the answer is "no".