Local Shimura variety

Definition

A local Shimura datum is, in the standard unramified form, a triple

(G,[b],\{\mu\}),

where $G$ is a connected reductive group over a $p$ -adic field $F$ , $[b]\in B(G)$ is a sigma-conjugacy class, and $\{\mu\}$ is a conjugacy class of cocharacters satisfying the usual acceptability condition $[b]\in B(G,\mu)$ .

The associated local Shimura variety is a tower, indexed by compact open level subgroups $K\subset G(F)$ , of rigid-analytic spaces or diamonds $\mathcal M_{(G,b,\mu),K}$ . In the local-shtuka formulation it parametrizes modifications of the $G$ -bundle determined by $b$ at an untilt, bounded by $\mu$ , together with level structure.

Why it is a tower

Changing $K$ changes the amount of trivialization retained by the moduli problem. The inverse limit at infinite level is often a perfectoid space or a diamond with commuting actions of $G(F)$ , the self-quasi-isogeny group $J_b(F)$ , and the local Weil group $W_F$ .

Examples and scope

Lubin–Tate and Drinfeld towers are basic examples. Classical Rapoport–Zink spaces give many local Shimura varieties of EL or PEL type. The term is also used for the more general local-shtuka spaces constructed in diamond form; not every datum has a classical formal-scheme model.

Their compactly supported cohomology is expected to realize local Langlands and Jacquet–Langlands correspondences.

References

Michael Rapoport and Eva Viehmann, “Towards a theory of local Shimura varieties,” Münster Journal of Mathematics 7 (2014), 273–326. arXiv.
Laurent Fargues and Peter Scholze, “Geometrization of the local Langlands correspondence,” 2021, Chapter IV. arXiv.