Weak ordinarity conjecture for smooth proper Calabi–Yau families

Let $S$ be a scheme smooth and dominant over $\operatorname{Spec} \mathbb{Z}$, and let $Y_S\to S$ be a smooth proper morphism whose fibers are Calabi–Yau varieties. Weak ordinarity conjecture. The fiber $Y_{\mathfrak p}$ is weakly ordinary for a Zariski-dense set of closed points $\mathfrak p\in S$. This is an important open problem in arithmetic geometry and in the algebraic geometry of positive characteristic; the paper studies related mixed-characteristic analogues for hypersurfaces.