group-theoretic description of the set of ${\bar{\mathbb{F}}_{p}}$-points of the mod-$p$ reduction of a Shimura variety, as provided with Hecke operators and Frobenius automorphism.

Let $(G,X)$ be a (general) Shimura datum and ${\mathbf{K}}^p\subset G({\mathbb A}_f^p)$, ${\mathbf{K}}_p\subset G({{\mathbb Q}_p})$ compact open subgroups, and set ${\mathbf{K}}:={\mathbf{K}}_p\times {\mathbf{K}}^p\subset G({\mathbb A}_f)$. The original conjecture mainly concerned good reduction cases, where ${\mathbf{K}}_p$ is *hyperspecial*, i.e. ${\mathbf{K}}_p=G_{{{\mathbb Z}_p}}({{\mathbb Z}_p})$ for a reductive ${{\mathbb Z}_p}$-group scheme with generic fiber $G_{{{\mathbb Q}_p}}$. We also choose a place $\wp$ of the reflex field $E(G,X)$ dividing $p$, and let ${\mathcal{O}}_{\wp}$, $\kappa(\wp)$ denote respectively the integer ring of the local field $E(G,X)_{\wp}$ and its residue field. Then, Langlands and Rapoport conjectured that there exists an integral model ${\mathscr{S}}_{{\mathbf{K}}_p}(G,X)$ of ${\mathrm{Sh}}_K(G,X)$ over ${\mathcal{O}}_{\wp}$, for which there is a bijection $$\label{eqn:LRconj-ver1}
{\mathscr{S}}_{{\mathbf{K}}_p}(G,X)({\bar{\mathbb{F}}_{p}}){\stackrel{\sim}{\rightarrow}}\bigsqcup_{[\phi]}S(\phi)$$ where $$S(\phi)=\varprojlim_{{\mathbf{K}}^p} I_{\phi}({\mathbb Q})\backslash X_p(\phi)\times X^p(\phi)/{\mathbf{K}}^p.$$ To give an idea of what these objects look like, suppose that our Shimura variety ${\mathrm{Sh}}_{{\mathbf{K}}_p}(G,X)$ is a moduli space of abelian varieties endowed with a certain prescribed set of additional structures defined by $G$ (called $G$-structure, for short), and that there exists an integral model whose reduction affords a similar moduli description (at least over ${\bar{\mathbb{F}}_{p}}$). Then, roughly speaking, each $\phi$ is supposed to correspond 