# Notion of good supersingular reduction for proper smooth variety over a $p$-adic field

Let $X$ be a proper smooth variety over a $p$-adic field $K$. Let $\mathcal{O}_K$ be the ring of integers of $K$ and $k$, its residue field. We say that $X$ has good ordinary reduction if there is a smooth proper model $\mathfrak{X}$ of $X$ over $\mathcal{O}_K$ such with special fibre $Y$ satisfying $H^r (Y, d\Omega_Y^s) = 0$ for all $r$ and $s$, where $d\Omega_Y^s$ is the sheaf of exact differentials on $Y$. This definition is due to Kato, following his paper with S. Bloch [$p$-adic etale cohomologies]. When $X$ is a $d$-dimensional abelian variety, the above definition coincides with the classical definition of ''ordinary" for abelian varieties: that the group of $\bar{k}$-points killed by $p$ on $Y$ has order $p^d$.

Now, for an elliptic curve $E$, we say it has good supersingular reduction if the group of $\bar{k}$-points of the reduction of $E$ killed by $p$ is trivial. (I don't know if this definition is the same for abelian varieties; I would be grateful if somebody will confirm or correct me on this).

Question: Is there a general notion of good supersingular reduction for a proper smooth variety $X$ over a $p$-adic field $K$ as above?

-
The problem is that there are intermediate cases between ordinary and supersingular even for abelian varieties. You can go by $p$-rank (which is a number between $0$ and the dimension of the variety or, a more refined stratification by the $p$-adic Newton polygon of the characteristic polynomial of Frobenius on $H^1$. The latter generalizes for more general varieties with good reduction and higher dim'l cohomology. Ordinary is the general case and supersingular the most degenerate case, but usually there are many intermediate cases. What is it that you really want? – Felipe Voloch Jul 26 '13 at 14:37
People usually call a smooth proper variety $X/k$ of dimension $n$ in characteristic $p>0$ supersingular if the Artin-Mazur formal group that prorepresents $A\mapsto ker (H^n_{et}(X\otimes A, \mathbb{G}_m)\to H^n_{et}(X, \mathbb{G}_m))$ has infinite height. For a K3 surface, this is the formal Brauer group (the notion does not match up exactly with elliptic curves or abelian varieties, but seems to be an analogous replacement in terms of properties). – Matt Jul 26 '13 at 14:42
Thanks Felipe and Matt for your comments. As for Felipe's question, I am really interested in the Galois representation $V = H^i_{et}(X_{\bar{K}}, \mathbb{Q}_p)$. When $X$ has good ordinary reduction, we know how the inertia subgroup of the absolute Galois group $G_K$ acts on $V$ (cf. article of B. Perrin-Riou and its appendix by Illusie in Asterisque 223). I was wondering whether whether there is a known description about the action of $G_K$ on $V$ when $X$ has good supersingular" reduction. But... – Octobris Jul 26 '13 at 16:18
...I thought it was better to know first what it means to have good supersingular" reduction before asking questions about the action of Galois on $V$. – Octobris Jul 26 '13 at 16:19
It sounds like you want a dichotomy between ordinary and supersingular. If you use the height of the Artin-Mazur formal group, then height 1 would correspond roughly to ordinary (it corresponds exactly at least in the case of a K3 surface) and infinite height is supersingular. But the finite height greater than 1 case probably exhibit better Galois representation properties. For example, look at the Nygaard-Ogus paper on proving the Tate conjecture for finite height K3 surfaces. They also tie this to the relation with Newton polygons of Frobenius. – Matt Jul 26 '13 at 18:43