the local zeta function of ${\mathscr{S}}_{{\mathbf{K}}}(G,X)_{\kappa(\wp)}$.

Such deduction arguments were provided by Kottwitz [@Kottwitz84b] and Langlands-Rapoport [@LR87]. Here, we sketch its basic idea in the trivial correspondence case (cf. [@Milne92]). For simplicity, we assume (only here) that the derived group $G^{{\mathrm{der}}}$ of $G$ is simply connected. By an elementary argument [@Kottwitz84b $\S$1.4], one readily sees that for each admissible morphism $\phi$, the corresponding fixed-point set $S(\phi)^{\Phi^m=\mathrm{Id}}$ breaks up further as a disjoint union of subsets $S(\phi,\epsilon)$ indexed by (the equivalence class of) a pair $(\phi,\epsilon)$, where $\epsilon$ is an automorphism of $\phi$ which exists as an element of $G({\overline{\mathbb Q}})$ and is to be regarded as a Frobenius descent datum of $\phi$: $${\mathscr{S}}_{{\mathbf{K}}}(G,X)({\mathbb F}_{q^m}){\stackrel{\sim}{\rightarrow}}\bigsqcup_{[\phi]}S(\phi)^{\Phi^m=\mathrm{id}} = \bigsqcup_{[\phi,\epsilon]} S(\phi,\epsilon)$$ Among such pairs $(\phi,\epsilon)$, we are interested only in the pairs, called *admissible*, satisfying some natural conditions that are necessary (but, not sufficient in general!) for the corresponding set $S(\phi,\epsilon)$ to be non-empty (Def. \[defn:admissible\_pair\]). Then, with any admissible pair $(\phi,\epsilon)$, one can associate a triple of group elements (*Kottwitz triple*) $$(\gamma_0;\gamma=(\gamma_l)_{l\neq p},\delta)\in G({\mathbb Q})\times G({\mathbb A}_f^p)\times G(L_n)$$ satisfying certain compatibilities among themselves, where $L_n$ is the unramified extension of ${{\mathbb Q}_p}$ of degree $n=m[\kappa(\wp):{\mathbb{F}_{p}}]$, and express the cardinality of the set $S(\phi,\epsilon)$ 