We are concerned with slight weakening of a result of the Serre-Tate paper "Good reduction of abelian varieties" google. I think the title of the question conveys what we are in here for. So, I'll move on directly to formulate the question.
Let $K$ be a field, $v$ a discrete valuation of K. Denote by $k$ the residue field of $K$ at $v$. Let $K_s$ be a separable closure of $K$ and $\bar{v}$ an extension of $v$ to $K_s$. Denote $I(\bar{v})$ the inertia subgroup of $\mathrm{Gal}(K_s/K)$. Let $A/K$ be an abelian vareity and for any prime number $\ell$ denote $T_{\ell}(A)$ the Tate-module. Let $$ \rho_{\ell}: \mathrm{Gal}(K_s/K)\rightarrow \mathrm{Aut}(T_{\ell}(A)) $$ be the associated $\ell$-adic representation.
Then corollary 1 to Theorem 2 of the paper says the following :
If $k$ is finite of characteristic $p$ and for some prime number $ \ell \neq p $ , the image $\rho_{\ell}(\mathrm{Gal}(K_s/K))$ in $\mathrm{Aut}(T_{\ell}(A))$ is abelian. Then $A$ has potential good reduction at $v$.
My question is the following:
Suppose we have the weaker condition that the image of the inertia subgroup $\rho_{\ell}(I(\bar{v}))$ in $\mathrm{Aut}(T_{\ell}(A))$ is abelian. Can we still conclude that $A/K$ has potential good reduction at $v$ ?
I was thinking that maybe the same proof as that of the above result might work. If this is the case, then can anyone explain me any arguments of the local Class field theory involved here in this case or suggest appropriate references ?
