one such effectivity criterion (condition $\ast(\epsilon)$ in [@LR87 Satz.5.21]), but fail to relate it to the more natural one of non-vanishing of $O_{\gamma}(f^p)\cdot TO_{\delta}(\phi_p)$. In this work, we prove this effectivity criterion (Thm. \[thm:LR-Satz5.25\]) and thereby (with a little more work allowing general Hecke correspondences) complete the arguments of Langlands and Rapoport of deriving the Kottwitz formula from their conjecture.[^2] We emphasize that our proof of the effectivity criterion (even though it could be formulated without the language of Galois gerbs) uses the full force of the theory of Galois gerbs and admissible morphisms from [@LR87].

On the other hand, recently Kisin [@Kisin17] obtained a description of ${\mathscr{S}}_{{\mathbf{K}}_p}(G,X)({\bar{\mathbb{F}}_{p}})$ quite similar to (\[eqn:LRconj-ver1\]). But, in his version of the description (\[eqn:LRconj-ver1\]) (i.e. Theorem 0.3 of *loc. cit.*), the action of the group $I_{\phi}({\mathbb Q})$ on $X_p(\phi)\times X^p(\phi)$ (which is somewhat artificial) is not the *natural* one specified in the original conjecture. An unfortunate consequence of this is that the deduction argument above *per se* does not work for such imprecise description.[^3] In fact, in our proof of Kottwitz formula, we do *not* use this weaker form (of Langlands-Rapoport conjecture) itself. Instead, we will just emulate the Langlands-Rapoport arguments above. Fortunately, this 