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One version of the Neron-Ogg-Shafarevich criterion for abelian varieties says that for a local field $K$ with valuation ring $R$ and perfect residue field $k$ and an abelian variety over $A$, $A$ has good reduction at $R$ if and only if the action of the inertia group on the $\ell$-torsion points is trivial for $\ell$ invertible in $k$.

Going through the proof of this in the paper by Serre-Tate, I think we can actually take $R$ to be any discrete valuation ring (we only seem to use anything about $R$ in constructing the Neron model). Is this correct?

If so, I have a question about a later part of the Serre-Tate paper. They prove that if $A$ has potentially good reduction, then the kernel of the map $\rho_l : I \to \operatorname{Aut}(T_l)$ is independent of $l$. $I$ is the inertia group of $K$. The proof goes like this:

We can first assume that $K$ has no unramified extensions since we are only interested in the inertia group. Then, if $J\subset I$ is the kernel, $J$ has finite index (since we have potentially good reduction) and so the fixed field $L$ of $J$ is a finite extension of $K$ and $A$ has good reduction over $L$. Since this characterizes $L$ (and hence $J$) and is independent of $l$, we are done.

I don't see why it is important that $J$ has finite index in $I$ assuming the first part of this question is accurate. So is it true that the kernel of the representation on the inertia group is independent of $l$ even without potentially good reduction or am I missing something?

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When Serre and Tate published their paper, the notion of semistable reduction of abelian varieties was not familiar and they consider only the case of potentially good reduction.

You can find the general case in Astérisque 86, exposé of Mireille Deschamps: Réduction semi-stable.

The basic facts are as follows:

Let $R$ be a strictly Henselian DVR with fraction field $K$.

Let $A$ be a $K$-abelian variety.

There exists a minimal finite extension $K'$ of $K$ such that if $R'$ denotes the integral closure of $R$ in $K'$ then $A \times K'$ has a semistable reduction on $R'$.

Moreover, $K'/K$ is unramified and Galois. Let $G$ be its Galois group.

For $\ell$ prime to $p$ = the residue characteristic of $R$ and $\ell ≥ 3$, $G$ acts faithfully on a suitable canonical subgroup $E(\ell)$ of $A(\ell)=$ $($points of order $\ell$ of $A)$. In this case, the reduction of $A$ is potentially good $E(\ell) = A(\ell)$.

Compare with Theorem 5.15 and Theorem 5.18.

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I realized that I was missing something very basic. The problem is that the integral closure of a DVR in an infinite extension of the fraction field need not itself be a DVR. Simple example: $\mathbb Q_p \in \overline{\mathbb Q_p}$...

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