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We define, for each quasi-syntomic ring $R$ (in the sense of Bhatt-Morrow-Scholze), a category $mathrm{DF}(R)$ of textit{filtered prismatic Dieudonne crystals over $R$} and a natural functor from $p$-divisible groups over $R$ to $mathrm{DF}(R)$. We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze.
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We establish an arithmetic intersection theory in the framework of Arakelov geometry over adelic curves. To each projective scheme over an adelic curve, we associate a multi-homogenous form on the group of adelic Cartier divisors, which can be written as an integral of local intersection numbers along the adelic curve. The integrability of the local intersection number is justified by using the theory of resultants.
Let $(A, I)$ be a bounded prism, and $X$ be a smooth $p$-adic formal scheme over $Spf(A/I)$. We consider the notion of crystals on Bhatt--Scholzes prismatic site $(X/A)_{prism}$ of $X$ relative to $A$. We prove that if $X$ is proper over $Spf(A/I)$ of relative dimension $n$, then the cohomology of a prismatic crystal is a perfect complex of $A$-modules with tor-amplitude in degrees $[0,2n]$. We also establish a Poincare duality for the reduced prismatic crystals, i.e. the crystals over the reduced structural sheaf of $(X/A)_{prism}$. The key ingredient is an explicit local description of reduced prismatic crystals in terms of Higgs modules.
This is a survey based on the construction of Siegel modular forms of degree 2 and 3 using invariant theory in joint work with Fabien Clery and Carel Faber.
A qualgebra $G$ is a set having two binary operations that satisfy compatibility conditions which are modeled upon a group under conjugation and multiplication. We develop a homology theory for qualgebras and describe a classifying space for it. This space is constructed from $G$-colored prisms (products of simplices) and simultaneously generalizes (and includes) simplicial classifying spaces for groups and cubical classifying spaces for quandles. Degenerate cells of several types are added to the regular prismatic cells; by duality, these correspond to non-rigid Reidemeister moves and their higher dimensional analogues. Coupled with $G$-coloring techniques, our homology theory yields invariants of knotted trivalent graphs in $mathbb{R}^3$ and knotted foams in $mathbb{R}^4$. We re-interpret these invariants as homotopy classes of maps from $S^2$ or $S^3$ to the classifying space of $G$.
The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field is the least positive integer m such that D[p^m] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of p-divisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of p-divisible groups, we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown to be optimal in the isoclinic case. In particular, the latter disproves a conjecture of Traverso. As an application, we answer a question of Zink on the liftability of an endomorphism of D[p^m] to D.
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