Suppose $X$ is an abelian variety over a $p$-adic field $K$, and it's well known that $X$ has good reduction is equivalent to the étale cohomology of $X$ is crystalline, and $X$ has semistable reduction equivalent to semistable étale cohomology. I wonder is there a good reference (with proof) for these two things?
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1$\begingroup$ Coleman-Iovita. $\endgroup$– Satan's MinionCommented Mar 19 at 5:33
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5$\begingroup$ @Satan'sMinion Could you provide the title of it? $\endgroup$– RichardCommented Mar 19 at 5:36
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1 Answer
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As Satan's Minion says, the good reduction case is
R. Coleman, A. Iovita, The Frobenius and monodromy operators for curves and abelian varieties, Duke Math. J. 97 (1999), 171--215.
For the semistable reduction case, see
C. Breuil, Groupes $p$-divisibles, groupes finis et modules filtrés. Ann. of Math. (2) 152 (2000), no.2, 489–549.
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$\begingroup$ Note there seem to be two versions of the former paper online: one turns up publicly available on Google, but seems to be less complete than the other version. $\endgroup$– Vik78Commented Mar 21 at 12:49