Moduli stacks of crystals and isocrystals

Given a liftable smooth proper variety over ${\mathbb{F}}_p$, we construct the moduli stacks of crystals and isocrystals on it. We show that the former is a formal algebraic stack over ${\mathbb{Z}}_p$ and the latter is an adic stack – Artin stack in rigid geometry – over ${\mathbb{Q}}_p$. Both stacks come equipped with the Verschiebung endomorphism $V$ corresponding to the Frobenius pullback of (iso)crystals. We study the geometry of the $V$-fixed points over the open substack of irreducible isocrystals, which we use to geometrically count the rank one $F$-isocrystals. Along the way, we carefully develop the theory of adic stacks.