Canonical liftings of Calabi--Yau hypersurfaces: Dwork hypersurfaces

We explicitly compute canonical liftings modulo $p^2$ in a sense of Achinger–Zdanowicz of Dwork hypersurfaces. The computation involves studying a compatibility between Hodge filtrations and a crystalline Frobenius. In particular, remarkably, we explicitly compute a partial data of the crystalline Frobenius modulo $p^2$.