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Abstract: We use the newly developed “stacky” prismatic technology of Drinfeld and Bhatt-Lurie to give a uniform, group-theoretic construction of smooth stacks $\textup{BT}_n^{\mathcal{G},\mu}$ associated to a smooth affine group scheme $\mathcal{G}$ 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 Bultel-Pappas, who gave a related construction using $(\mathcal{G},\mu)$-displays defined via $p$-typical Witt vectors. When $\mathcal{G}=\textup{GL}_h$ and $\mu$ is minuscule, we show that $\textup{BT}_n^{\mathcal{G},\mu}$ is isomorphic to the stack of $n$-truncated $p$-divisible groups of height $h$ and dimension $d$ (with the latter determined by $\mu$). This generalizes results of Anschutz-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. Since we are interested in applications to the study of local and global Shimura varieties, we actually prove representability results for a wide range of stacks whose tangent complex is $1$-bounded in a suitable sense. As an immediate consequence, we prove algebraicity for the stack of perfect $F$-gauges of level $n$ with Hodge-Tate weights contained in $[0,1]$.

Abstract: We develop the basic theory of derived quasi-coherent ideals for stacks relative to a given derived algebraic context. We compare different notions of completeness with respect to derived ideals, define and compare formal spectra and formal completions along closed immersions, and connect the theory of derived ideals to that of derived extended Rees algebras. A first application is the construction of derived scheme-theoretic images in full generality. We further show that the deformation space of any nonconnectively affine morphism of derived stacks is nonconnectively affine over the base. We close with a first exploration of transmutation cohomology and filtrations thereof in this more general context.

Abstract: Let $\mathcal{O}$ be the ring of integers of a $p$-adic local field with chosen uniformizer $\varpi$. Inspired by the theory of $\mathcal{O}$-prisms introduced by K. Ito and S. Marks, we first construct $\mathcal{O}$-typical analogues of prismatization, filtered prismatization, and syntomification. Then, given a smooth affine group scheme $\mathcal{G}$ over $\mathcal{O}$ and a $1$-bounded cocharacter $\mu$, we study $(\mathcal{G},\mu)$-apertures and show that they come equipped with an analogue of Grothendieck-Messing theory. Moreover, we show that the moduli of $(\mathcal{G},\mu)$-apertures satisfies an analogue of Grothendieck’s smoothness theorem for $p$-divisible groups and that $(\mathcal{G},\mu)$-apertures identify with $\varpi$-divisible groups in the general linear case. We then treat several applications to the theory of Rapoport-Zink spaces.

Abstract: Let $\mathscr{M}_{2d,\mathbb{C}}^{\circ}$ denote the moduli stack of degree $2d$ primitively polarized complex K$3$ surfaces. The classical global Torelli theorem states that the map $\widetilde{\mathscr{M}}_{2d,\mathbb{C}}^{\circ}\to\textup{Sh}(\Lambda_d)_{\mathbb{C}}$ is an etale monomorphism, where $\widetilde{\mathscr{M}}_{2d,\mathbb{C}}^{\circ}$ is a certain $2$-fold “orientation” cover of the moduli, $\Lambda_d$ is the associated primitive cohomology lattice, and $\textup{Sh}(\Lambda_d)$ is the corresponding orthogonal Shimura variety over $\mathbb{Q}$. In this paper, we investigate analogues of the classical global Torelli theorem for $p$-adic formal K$3$ surfaces using prismatic and syntomic techniques, building on previous work of K. Madapusi, Kim-Madapusi, and Ito-Ito-Koshikawa. Our techniques make use of the theory of apertures recently introduced in work of one of the authors joint with K. Madapusi, as well as work in preparation of K. Madapusi and A. Youcis on the syntomic characterization of integral canonical models of Shimura varieties.