On the Lau group scheme
Vladimir Drinfeld
TL;DR
This work gives an explicit description of the Lau group scheme $\mathrm{Lau}_n^{d,d'}$, which bands the Lau gerbe $\phi_n: \overline{\mathscr{BT}}_n^{d,d'}\to \mathrm{Disp}_n^{d,d'}$, by applying the Zink functor to a carefully chosen $n$-truncated semidisplay on $\mathrm{Disp}_n^{d,d'}$. It introduces and exploits the notions of $n$-smoothness and $n$-cosmoothness to study the inertia and Frobenius kernels, proving that $\mathrm{Lau}_n^{d,d'}$ is $n$-smooth of rank $d'(d-d')$ and that its pullback to inertia matches the Frobenius kernel $(\mathcal I_n^{d,d'})^{(F^n)}$ up to closed-immersion considerations. The paper further develops the theory of $n$-truncated semidisplays, the Zink functor, and the higher-display framework to obtain an explicit, Shimura-inspired description of Lau's group and its Cartier dual, and to set up conjectural descriptions of the corresponding stacks $\mathrm{BT}_n^{G,\mu}$ for general $(G,\mu)$, with detailed treatment for the $G=GL_d$ case. The results provide powerful structural control over the moduli of BT$_n$-objects and their displays, yielding a robust bridge between crystalline Dieudonné theory, higher displays, and the geometry of Shimura-type stacks in positive characteristic. The constructions have potential implications for explicit presentations of these stacks, as well as for Cartière duality and tensor structures in the truncated display setting. Overall, the work advances a precise, computable description of Lau’s gerbe and its generalizations, enabling new explicit and combinatorial approaches to questions about BT$_n$-groups and their displays in characteristic $p$.
Abstract
In a 2013 article, Eike Lau constructed a canonical morphism from the stack of $n$-truncated Barsotti-Tate groups over $F_p$ to the stack of $n$-truncated displays. He also proved that this morphism is a gerbe banded by a commutative group scheme. In this paper we describe the group scheme explicitly. The stack of $n$-truncated Barsotti-Tate groups over $F_p$ has a generalization related to any pair $(G,μ)$, where $G$ is a smooth group scheme over $Z/p^n$ and $μ$ is a 1-bounded cocharacter of $G$. The same is true for the stack of $n$-truncated displays. We conjecture that in this more general situation the first stack is a gerbe over the second one banded by a commutative group scheme, and we give a conjectural description of this group scheme. We also give a conjectural description of the stack of $n$-truncated Barsotti-Tate groups over the formal spectrum of $Z_p$ and of its $(G,μ)$-generalization.