Much of my current work is focused around Drinfeld’s idea of developing Shimurian generalizations of $p$-divisible groups, for a fixed rational prime $p$. The cornerstone is joint work with Keerthi Madapusi and Akhil Mathew, where we construct and study the moduli stack $\textup{BT}_n^{G,\mu}$ of $n$-truncated prismatic $(G,\mu)$-displays. Here are the basics of the setup.
- $G$ is a smooth affine group scheme over $\mathbb{Z}_p$ (crucially, it doesn’t have to be reductive).
- $\mu: \mathbb{G}_{m,\mathcal{O}}\to G_{\mathcal{O}}$ is a cocharacter defined over the ring of integers $\mathcal{O}$ of a finite unramified extension of $\mathbb{Q}_p$, which is $1$-bounded in the sense that the adjoint action $$G_{\mathcal{O}}\times\mathbb{G}_{m,\mathcal{O}}\to G_{\mathcal{O}},\qquad (g,z)\mapsto\mu(z)g\mu(z)^{-1}$$ of $\mu$ induces a grading on the Lie algebra $\mathfrak{g}$ of $G_{\mathcal{O}}$ with $\mathfrak{g}_i=0$ for $i>1$. If $G$ is reductive then this is the same as $\mu$ being minuscule.
- The quotient of the adjoint action of $\mu$ yields a graded stack $G\{\mu\}\to B\mathbb{G}_{m,\mathcal{O}}$ which is appropriately “group-like.” This can be made more precise by working with the semidirect product $G_{\mathcal{O}}\rtimes_{\mu}\mathbb{G}_{m,\mathcal{O}}$.
- $\textup{BT}_n^{G,\mu}$ is a derived $p$-adic formal stack over $\textup{Spf}(\mathcal{O})$. This means that it is an appropriately “sheafy” functor which takes in an (animated) $p$-nilpotent $\mathcal{O}$-algebra $R$ and outputs an ($\infty$-)groupoid $\textup{BT}_n^{G,\mu}(R)$. We can formally extend this construction to allow $R$ to be an (animated, derived) $p$-complete $\mathcal{O}$-algebra $R$. In fact, we can formally extend even more to make sense of $\textup{BT}_n^{G,\mu}(\mathfrak{X})$ for any (derived) $p$-adic formal scheme $\mathfrak{X}$. (Feel free to ignore the terms in parentheses for a first viewing.)
- More explicitly, $\textup{BT}_n^{G,\mu}(R)$ is the $\infty$-groupoid of $G\{\mu\}$-torsors over $R^{\textup{syn}}\otimes\mathbb{Z}/p^n\to B\mathbb{G}_{m,\mathcal{O}}$ satisfying an appropriate triviality condition. Here, $R^{\textup{syn}}$ is the syntomification of $R$ with grading defined by the Breuil-Kisin twist of the structure sheaf of $R^{\textup{syn}}$.
- There are natural maps $\textup{BT}_{n+1}^{G,\mu}(R)\to\textup{BT}_n^{G,\mu}(R)$ for each $n\geq1$. We define $\textup{BT}_{\infty}^{G,\mu}(R)$ to be the limit of this inverse system . This yields the moduli stack $\textup{BT}_{\infty}^{G,\mu}$ of prismatic $(G,\mu)$-displays.
The advantage of this setup is that we obtain several powerful results. Let’s first describe the general case.
Theorem (G.-Madapusi-Mathew):
- $\textup{BT}_n^{G,\mu}$ is a represented by a zero-dimensional quasicompact smooth Artin formal stack over $\textup{Spf}(\mathcal{O})$ with affine diagonal. Moreover, the natural maps $\textup{BT}_{n+1}^{G,\mu}\to\textup{BT}_n^{G,\mu}$ are smooth and so $\textup{BT}_{\infty}^{G,\mu}$ is pro-smooth.
- Let $\textup{F-Zip}^{G,\mu}$ be the moduli stack of $(G,\mu)$-zips as defined (albeit with slightly different terminology) by Pink-Wedhorn-Ziegler and $k=\mathcal{O}/p$ the residue field of $\mathcal{O}$. There is a natural map $\textup{BT}_1^{G,\mu}\otimes\mathbb{F}_p\to\textup{F-Zip}^{G,\mu}$ that is relatively representable by a smooth zero-dimensional Artin stack over $\textup{Spec}(k)$ with relatively affine diagonal. In fact, it is a gerbe banded by the Lau group scheme, which is a specific finite flat commutative $p$-group scheme of height $1$. In particular, $\textup{BT}_1^{G,\mu}\otimes\mathbb{F}_p$ is a smooth zero-dimensional Artin stack over $\textup{Spec}(k)$ with affine diagonal.
- $\textup{BT}_n^{G,\mu}$ satisfies an analogue of Grothendieck-Messing theory: certain commutative squares associated to nilpotent animated PD-thickenings are Cartesian. The connection to classical Grothendieck-Messing theory comes from the fact that $R^{\textup{syn}}$ is obtained by gluing together the two distinguished copies of the prismatization of $R$ inside of the Nygaard filtered prismatization $R^{\mathcal{N}}$. $R^{\mathcal{N}}$ geometrizes the notion of the Nygaard filtration on prismatic cohomology, which is itself a generalization of the classical notion of the Hodge filtration on crystalline cohomology.
Suppose now that $G=\textup{GL}_h$ ($h$ for height), so that $\mu$ then encodes the dimension $d$. For simplicity, we write $\textup{BT}_n^{h,d}$ in place of $\textup{BT}_n^{\textup{GL}_h,\mu}$. The basic idea is to compare $\textup{BT}_n^{h,d}$ with the moduli stack $\mathcal{BT}_n^{h,d}$ of $n$-truncated $p$-divisible groups of height $h$ and dimension $d$. This can be accomplished by working with syntomoid $\mathbb{Z}_p$-algebras: an animated commutative ring $R$ is syntomoid if it is (derived) $p$-complete and $\Omega_{(\pi_0(R)/p)/\mathbb{F}_p}^1$ is a finitely generated $\pi_0(R)$-module. This condition ensures that $\textup{Spf}(R)$ has a $p$-quasisyntomic cover by $\textup{Spf}(R’)$ for $R’$ a semiperfectoid $\mathbb{Z}_p$-algebra. Examples of such $R$ arise from the following (sufficient but not necessary) conditions:
- $\pi_0(R)$ is a finitely generated $\mathbb{Z}/p^m$-algebra for $m\geq1$.
- $R$ is a complete local Noetherian $\mathbb{Z}_p$-algebra with residue field $\kappa$ such that $[\kappa:\kappa^p]<\infty$.
- $R$ is a semiperfect $\mathbb{F}_p$-algebra.
We say that a (derived) affine $p$-adic formal scheme $\textup{Spf}(R)$ is syntomoid if $R$ is syntomoid. More generally, we say that a (derived) $p$-adic formal scheme $\mathfrak{X}$ is locally syntomoid if it is covered by syntomoid (derived) affine $p$-adic formal schemes. For $\mathfrak{X}$ any (derived) $p$-adic formal scheme, we have a canonical equivalence of categories
$$\textup{Vect}^{[0,1]}(\mathfrak{X}^{\textup{syn}}\otimes\mathbb{Z}/p^n)^{\textup{grpd}}\simeq\bigsqcup_{1\leq d\leq h}\textup{BT}_n^{h,d}(\mathfrak{X}),$$
whence we conclude that $\mathfrak{X}\mapsto\textup{Vect}^{[0,1]}(\mathfrak{X}^{\textup{syn}}\otimes\mathbb{Z}/p^n)^{\textup{grpd}}$ is a smooth $p$-adic Artin formal stack over $\textup{Spf}(\mathbb{Z}_p)$. Here, the superscript $[0,1]$ signals that we look only at vector bundles with Hodge-Tate weights contained in $\{0,1\}$ and the superscript $\textup{grpd}$ indicates that we discard any non-invertible morphisms. We also have a canonical equivalence of categories
$$\mathcal{BT}_n(\mathfrak{X})^{\textup{grpd}}\simeq\bigsqcup_{1\leq d\leq h}\mathcal{BT}_n^{h,d}(\mathfrak{X})$$
for $\mathcal{BT}_n(\mathfrak{X})$ the full category of $n$-truncated $p$-divisible groups over $\mathfrak{X}$. It is then natural to compare $\textup{Vect}^{[0,1]}(\mathfrak{X}^{\textup{syn}}\otimes\mathbb{Z}/p^n)$ and $\mathcal{BT}_n(\mathfrak{X})$.
Theorem (G.-Madapusi-Mathew): Let $\mathfrak{X}$ be a locally syntomoid (derived) $p$-adic formal scheme. Then, there is an equivalence of categories
$$\mathbb{D}_{\textup{Mon}}^{-1}: \textup{Vect}^{[0,1]}(\mathfrak{X}^{\textup{syn}}\otimes\mathbb{Z}/p^n)\xrightarrow{\simeq}\mathcal{BT}_n(\mathfrak{X})$$
given explicitly as follows: for a vector bundle $\mathcal{E}$ and a point $f: \textup{Spf}(C)\to\mathfrak{X}$,
$$\mathbb{D}_{\textup{Mon}}^{-1}(\mathcal{E})(C)=\tau^{\leq0}R\Gamma(C^{\textup{syn}}\otimes\mathbb{Z}/p^n,(f^{\textup{syn}})^*\mathcal{E}).$$
Moreover, $\mathbb{D}_{\textup{Mon}}^{-1}$
- is functorial in $n$ and $\mathfrak{X}$,
- satisfies $p$-quasisyntomic descent,
- is inverse to the Dieudonné functor $\mathbb{D}_{\textup{Mon}}$ of S. Mondal,
- is compatible with Cartier duality on both sides,
- preserves height and dimension.
Taking the limit over $n$ therefore yields a canonical equivalence
$$\mathbb{D}_{\textup{Mon}}^{-1}: \textup{Vect}^{[0,1]}(\mathfrak{X}^{\textup{syn}})\xrightarrow{\simeq}\mathcal{BT}(\mathfrak{X})$$
with the RHS the full category of $p$-divisible groups over $\mathfrak{X}$.
So, then, where to go from here?
- An obvious generalization of this construction is to allow $G$ to be defined over $\mathcal{O}_L$ for $L$ any finite extension of $\mathbb{Q}_p$. In forthcoming work I accomplish this by constructing $L$-typical analogues of prismatization, Nygaard filtered prismatization, and syntomification. Note that such constructions have applications outside of the above that I would like to explore more in the future (c.f. the work of Devalapurkar-Misterka, S. Marks, and Y. Sulyma).
- An equally obvious but seemingly less tractable generalization is to allow $\mu$ to be defined over the ring of integers $\mathcal{O}$ of any (potentially ramified) finite extension of $\mathbb{Q}_p$.
- As noted above, for suitable $\mathfrak{X}$ we have an explicit equivalence between $n$-truncated $p$-divisible groups over $\mathfrak{X}$ and suitable vector bundles over $\mathfrak{X}^{\textup{syn}}\otimes\mathbb{Z}/p^n$. Since both sides admit a Grothendieck-Messing theory, it makes sense to explicitly relate both theories. This will be accomplished in future work.
- The most salient thing to do with prismatic $(G,\mu)$-displays is to give a prismatic construction of Rapoport-Zink spaces that applies even outside of abelian type. “Pure” representability for $\textup{GL}_h$ (inspired by Bartling-Hoff). Notion of (quasi-)isogeny given forthcoming work of Si Ying Li and Keerthi Madapusi.
- Integral models of local Shimura varieties
- Applications to the program of Goldring, Imai, Koskivirta, etc.
- “Topological” applications of prismatic $(G,\mu)$-displays
All of the above is tied to the choice of a particular rational prime $p$. In joint work with Jeroen Hekking and Edith Hübner, the goal is to relax this constraint. The most promising step in this direction comes from recent work of A. Frederick, Y. Sulyma, and A. Yuan on the relationship between Tambara functors and prisms. It is also interesting to understand what can be extracted simply from $\delta$-ring structure, and more generally from $\lambda$-ring structure (or other, fancier structures).
In the past I spent some time thinking about derived integral and local models for Shimura varieties, as well as (highly experimental and speculative) derived arithmetic intersection theory (DAIT Learning Group).