Rapoport–Zink space

Definition

Fix a perfect field $k$ of characteristic $p$ and a $p$ -divisible group $\mathbb X$ over $k$ . Here a $p$ -divisible group is a compatible system of finite flat group schemes $X[p^n]$ of order $p^{nh}$ , with $X[p^n]=X[p^{n+1}][p^n]$ . In its basic form, the Rapoport–Zink functor assigns to a $p$ -adically nilpotent test formal scheme $S$ the pairs $(X,\rho)$ , where $X$ is a $p$ -divisible group over $S$ and

\rho:\mathbb X_{\bar S}\dashrightarrow X_{\bar S}

is a quasi-isogeny over the special fiber. Under the standard hypotheses this functor is represented by a formal scheme, a Rapoport–Zink space.

EL and PEL versions impose endomorphisms, polarizations, and determinant conditions. More general group-theoretic versions are described by local Shimura data.

Generic fiber and level tower

The adic generic fiber is a rigid-analytic space. Adding level structure on the rational Tate module produces a tower with actions by a $p$ -adic reductive group and by the group of self-quasi-isogenies of $\mathbb X$ . At infinite level many such towers become perfectoid.

Examples

The Lubin–Tate deformation space of a one-dimensional formal group is the basic EL example. Drinfeld's formal half-space gives a closely related tower. These spaces are primary examples of local Shimura varieties.

References

Michael Rapoport and Thomas Zink, Period Spaces for $p$ -divisible Groups, Annals of Mathematics Studies 141, Princeton University Press, 1996.
Peter Scholze and Jared Weinstein, “Moduli of $p$ -divisible groups,” Cambridge Journal of Mathematics 1 (2013), 145–237. arXiv.