We give a characterization---and in some cases, a new construction---of integral canonical models of Shimura varieties that uses the notion of an aperture appearing in work of the first author with Gardner on some conjectures of Drinfeld. This applies to Shimura varieties of pre-abelian type at primes of hyperspecial level, recovering and extending previous work of Kisin, Kim--Madapusi and Imai--Kato--Youcis, but also to exceptional Shimura varieties for large enough primes. The characterization in the exceptional case is \emph{a priori} different from the one recently shown by Bakker--Shankar--Tsimerman, and recovers many of their results, such as the existence of prime-to-$p$ Hecke operators, the non-emptiness of the $\mu$-ordinary stratum and the theory of the canonical lift. In fact, we give a uniform proof of the non-emptiness of \emph{all} possible Newton strata, and---in the pre-abelian case---of the non-emptiness of Ekedahl--Oort strata and central leaves as well. An important ingredient in the proofs is a generalization of Tate's full faithfulness theorem for $p$-divisible groups to the context of apertures. This leads to a mapping property for the integral canonical model that characterizes maps into it from all normal, flat and excellent schemes over $\mathbb{Z}_{(p)}$.

We show an equivalence of categories, over general $p$-adic bases, between finite locally $p^n$-torsion commutative group schemes and $\mathbb{Z}/p^n\mathbb{Z}$-modules in perfect $F$-gauges of Tor amplitude $[-1,0]$ with Hodge-Tate weights $0,1$. By relating fppf cohomology of group schemes and syntomic cohomology of $F$-gauges, we deduce some consequences: These include the representability of relative fppf cohomology of finite flat group schemes under proper smooth maps of $p$-adic formal schemes, as well as a reproof of a purity result of Česnavičius-Scholze. We also give a general criterion for a classification in terms of objects closely related to Zink's windows over frames and Lau's divided Dieudonné crystals, and we use this to recover several known classifications, and also give some new ones.

We use the newly developed stacky prismatic technology of Drinfeld and Bhatt-Lurie to give a uniform, group-theoretic construction of smooth stacks $\mathrm{BT}^{G,\mu}_{n}$ attached to a smooth affine group scheme $G$ over $\mathbb{Z}_p$ and $1$-bounded cocharacter $\mu$, verifying a recent conjecture of Drinfeld. This can be viewed as a refinement of results of B\"ultel-Pappas, who gave a related construction using $(G,\mu)$-displays defined via rings of Witt vectors. We show that, when $G = \mathrm{GL}_h$ and $\mu$ is a minuscule cocharacter, these stacks are isomorphic to the stack of truncated $p$-divisible groups of height $h$ and dimension $d$ (the latter depending on $\mu$). This gives a generalization of results of Ansch\"utz-Le Bras, yielding a linear algebraic classification of $p$-divisible groups over very general $p$-adic bases, and verifying another conjecture of Drinfeld. The proofs use deformation techniques from derived algebraic geometry, combined with an animated variant of Lau's theory of higher frames and displays, and---with a view towards applications to the study of local and global Shimura varieties---actually prove representability results for a wide range of stacks whose tangent complexes are $1$-bounded in a suitable sense. As an immediate application, we prove algebraicity for the stack of perfect $F$-gauges of Hodge-Tate weights $0,1$ and level $n$.

I use methods of Chai-Hida and ordinary -Hecke correspondences to study the set of irreducible components of special fibers of special cycles of sufficiently low codimension in integral models of GSpin Shimura varieties, and apply this to prove irreducibility results for the special fibers of the moduli of polarized K3 surfaces. These results are also applied in joint work with Howard on the modularity of generating series of higher codimension cycles on GSpin Shimura varieties.