Timeline for What happens to Neron-Ogg-Shfarevich when characteristic of the residue field equals the prime at which Tate module is considered?

4 events

when toggle format	what		by	license	comment
Jan 21, 2018 at 21:38	comment	added	nfdc23		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 $\{0,1\}$.
Jan 21, 2018 at 14:27	comment	added	nfdc23		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).
Jan 21, 2018 at 10:35	comment	added	Daniel Loughran		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
Jan 21, 2018 at 10:14	history	asked	Johnny T.	CC BY-SA 3.0