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Neron-Ogg-Shafarevich criterion states that an elliptic curve $E$ over a local field $K$ has a good reduction if and only if the Tate module $T_{\ell}(E)$ is unramified for some prime $\ell$ which does not equal the characteristic of $k$, the residue field of $K$.

Just from curiosity I was wondering is there anything one can say when the characteritic of $k$ is equal to $\ell$? Any comments are appreciated. Thank you.

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    $\begingroup$ The correct condition in this case is that the Tate module is "crystalline" (provided $K$ has characteristic $0$). If you search this on google you really find many papers about this. See e.g. arxiv.org/pdf/math/0605326.pdf $\endgroup$
    – Daniel Loughran
    Commented Jan 21, 2018 at 10:35
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    $\begingroup$ SGA7, Exp. IX, Thm. 5.10 (the hypothesis ${\rm{char}}(K)=0$ there can be dropped since deJong later proved the equicharacteristic-$\ell$ case of Tate's result on extending homomorphisms between generic fibers of $\ell$-divisible groups over a discrete valuation ring). $\endgroup$
    – nfdc23
    Commented Jan 21, 2018 at 14:27
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    $\begingroup$ Just to clarify @DanielLoughran's comment, there are two quite distinct equivalences involved: an abelian variety over an $\ell$-adic field has good reduction if and only if its $\ell$-adic Tate module arises from an $\ell$-divisible group over the valuation ring (this is the result in SGA7 mentioned above, and there is no way around it), and an $\ell$-adic representation of the Galois group of such a field (with perfect residue field) arises from an $\ell$-divisible group over the valuation ring if and only if it is crystalline with Hodge-Tate weights in $\{0,1\}$. $\endgroup$
    – nfdc23
    Commented Jan 21, 2018 at 21:38

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