trivial Kottwitz invariant (Def. \[defn:Kottwitz\_triple\]), if $X(\{\mu_X\},\delta)_{{\mathbf{K}}_p}\neq\emptyset$, there exists an admissible pair $(\phi,\epsilon)$ giving rise to it. There exists an explicit cohomological expression for the number of non-equivalent pairs $(\phi,\epsilon)$ producing a given triple $(\gamma_0;(\gamma_l)_{l\neq p},\delta)$.

This is one of the key ingredients in our proof of the Kottwitz formula as well as in the Langlands-Rapoport’s arguments of deriving it from Langlands-Rapoport conjecture. There is also a similar, second version of effectivity criterion of Kottwitz triple (Thm. \[thm:LR-Satz5.25b2\]) which will be needed in our (unconditional) proof of Thm. \[thm\_intro:Kottwitz\_formula\].

### Some further results and comments

According to the discussion in , Langlands-Rapoport conjecture implies that to any ${\mathbb F}_{p^n}$-point ($n$ being a multiple of $[\kappa(\wp):{\mathbb F}_p]$), one should be able to attach a Kottwitz triple $(\gamma_0;(\gamma_l)_{l\neq p},\delta)$ of level $n$ (i.e. with $\delta\in G(L_n)$). For Hodge-type Shimura varieties with hyperspecial level, this was done by Kisin [@Kisin17 Cor. 2.3.1].[^5] Then, the formula (\[eqn:formulra\_for\_number\_of\_pts\]) implies that only those triples whose corresponding summation term is non-zero should be *geometrically effective*, i.e. arises from an ${\mathbb F}_{p^n}$-point (where $n$ is the level of the triple). Thus, a natural question arises whether such necessary condition for geometric effectivity is also a sufficient condition. Closely 