in terms of it as $$|S(\phi,\epsilon)|=\mathrm{vol}(G_{\gamma_0}({\mathbb Q})\backslash G_{\gamma_0}({\mathbb A}_f))\cdot O_{\gamma}(f^p)\cdot TO_{\delta}(\phi_p),$$ where $G_{\gamma_0}$ is the centralizer subgroup of $\gamma_0$ in $G$, $O_{\gamma}(f^p)$ and $TO_{\delta}(\phi_p)$ are some orbital integral and twisted orbital integral, respectively. Thus, the final formula for $|{\mathscr{S}}_{{\mathbf{K}}}(G,X)({\mathbb F}_{q^m})|$ (for any sufficiently small $K^p$) takes the form of a sum, indexed by (the equivalence classes of) Kottwitz triples, of a product of quantities that can be defined purely group theoretically: $$\label{eqn:formulra_for_number_of_pts}
|{\mathscr{S}}_{{\mathbf{K}}}(G,X)({\mathbb F}_{q^m})|=\sum_{(\gamma_0;\gamma,\delta)}\iota(\gamma_0;\gamma,\delta)\cdot \mathrm{vol}(G_{\gamma_0}({\mathbb Q})\backslash G_{\gamma_0}({\mathbb A}_f))\cdot O_{\gamma}(f^p)\cdot TO_{\delta}(\phi_p),$$ where $\iota(\gamma_0;\gamma,\delta)$ is by definition the number of equivalence classes of admissible pairs giving rise to a fixed Kottwitz triple $(\gamma_0;\gamma,\delta)$.

With an explicit cohomological description for $\iota(\gamma_0;\gamma,\delta)$, this is the formula conjectured (and proved in certain PEL-type cases [@Kottwitz92 (19.6)]) by Kottwitz [@Kottwitz90 (3.1)], except that here the sum is only over the *effective* Kottwitz triples, namely those Kottwitz triples attached to some admissible pair $(\phi,\epsilon)$, while in the original formula, one takes *all* Kottwitz triples with trivial Kottwitz invariant (it is known that every effective Kottwitz triple has trivial Kottwitz invariant). We bring the reader’s attention to this usuage of the terminology *effectivity* and the possible confusion that a Kottwitz triple $(\gamma_0;\gamma,\delta)$ which is effective in this sense 