may not appear as a summation index “effectively” in the sum (\[eqn:formulra\_for\_number\_of\_pts\]) (i.e. the corresponding summand could be zero): one could also have defined $\iota(\gamma_0;\gamma;\delta)$ to be zero if is not effective, but then it is not the definition $\iota(\gamma_0;\gamma,\delta):=|\ker[\ker^1({\mathbb Q},G_{\gamma_0})\rightarrow \ker^1({\mathbb Q},G)]|$ used by Kottwitz in his formula (which is always non-zero). We note that to reconcile the formula (\[eqn:formulra\_for\_number\_of\_pts\]) with the Kottwitz’s formula [@Kottwitz90 (3.1)], it suffices to establish the following *effectivity criterion of Kottwitz triple*: a Kottwitz triple with trivial Kottwitz invariant is effective (in the sense just defined) if the corresponding summand $O_{\gamma}(f^p)\cdot TO_{\delta}(\phi_p)$ is non-zero (one also needs the fact that $\iota(\gamma_0;\gamma,\delta)=|\ker[\ker^1({\mathbb Q},G_{\gamma_0})\rightarrow \ker^1({\mathbb Q},G)]|$ which is however proved in [@LR87 Satz 5.25]).

### The works of Langlands and Rapoport [@LR87] and of Kisin [@Kisin17]

A substantial part of the work [@LR87] of Langlands and Rapoport is devoted to constructing all these objects and establishing their basic properties, especially those facts that are needed to carry out the deduction just sketched of the formula (\[eqn:formulra\_for\_number\_of\_pts\]) from the (conjectural) description (\[eqn:LRconj-ver1\]). The only missing ingredient in completing this deduction arguments was the effectivity criterion of Kottwitz triple stated above. In fact, Langlands and Rapoport also suggest 