# Does p-adic etale cohomology know the variety has ordinary reduction or not?

For a smooth proper variety $$X$$ over discrete valuation ring $$\mathcal{O}$$ of mixed characteristic $$(0,p)$$, let $$X_K$$ be the generic fibre over a generic point $$\textbf{Spec} K$$ and let $$X_k$$ be the special fibre over a special point $$\textbf{Spec} k$$. We say that the variety $$X_k$$ has ordinary reduction iff the cohomology group $$H^i(X_k,d\Omega^j)=0$$ for any $$i,j$$.

My question is that p-adic etale cohomology $$H^i_{et}(X,\mathbb{Z}_p)$$ of $$X_K$$ knows $$X_k$$ has ordinary reduction or not.

• What is the meaning of $d$ in that cohomology group $H^i(X_k, d\Omega^j)$? Apr 2 at 0:41
• @WillSawin exact differentials of degree $j+1$, often denoted by $B^{j+1}$. Apr 2 at 12:22
• @PiotrAchinger Is $B^{j+1}$ a Zariski sheaf?
– Z. M
Apr 2 at 16:27
• Good point, of course what is meant is the sheaf theoretic image of d, so locally exact differentials. Apr 2 at 16:36

I think you meant the etale cohomology of the geometric generic fiber, equipped with the action of the Galois group of $$K$$. Moreover, in $$p$$-adic Hodge theory it is usually necessary to assume that the residue field $$k$$ is perfect.
In this case, the answer is yes under the additional (customary) assumption that the crystalline and Hodge cohomology groups of $$X$$ are torsion free (this assumption is needed for the equivalence of two "standard" definitions of ordinarity: the one you gave and the one in terms of Hodge and Newton polygons). This essentially follows from the results of Bernadette Perrin-Riou "Representations p-adiques ordinaires" in Asterisque no. 223, 1994 "Periodes p-adiques" MR 1293973. See Proposition 6.7 in https://arxiv.org/abs/2005.02246 (I claim no originality here, but the statement gives what you want).