related questions are which (${\mathbb R}$-elliptic) stably conjugacy classes in $G({\mathbb Q})$ and which $\sigma$-conjugacy classes in $G(L_n)$ are “effective” (i.e. can be the classes of the elements $\gamma_0$ and $\delta$ attached to some ${\mathbb F}_{p^n}$-point, respectively). This question for (the stable conjugacy class of) an ${\mathbb R}$-elliptic rational element $\gamma_0\in G({\mathbb Q})$ can be regarded as Honda-Tate theorem in the context of Shimura varieties, while the question for $\delta$ is known as non-emptiness problem of Newton strata.

For hyperspecial level, the proof of Theorem \[thm\_intro:Kottwitz\_formula\] answers the first question affirmatively: the natural necessary condition is also sufficient. We also answer the second question too. For that, we consider Shimura varieties of Hodge type and fix an integral model ${\mathscr{S}}_{{\mathbf{K}}}$ over ${\mathcal{O}}_{\wp}$ of the canonical model ${\mathrm{Sh}}_{{\mathbf{K}}}(G,X)_{E_{\wp}}$ with the extension property that every $F$-point of ${\mathrm{Sh}}_{{\mathbf{K}}}(G,X)$ for a finite extension $F$ of $E_{\wp}$ extends uniquely to ${\mathscr{S}}_{{\mathbf{K}}}$ over its local ring (for example, integral model constructed by a suitable normalization); see [@KisinPappas15] for a construction of such integral model in general parhoric levels.

\[Cor. \[cor:geom\_effectivity\_of\_K-triple\]\] Keep the assumptions of Theorem \[thm\_intro:1st\_Main\_thm\]. a Kottwitz triple $(\gamma_0;\gamma,\delta)$ of level $n$ with trivial Kottwitz invariant is *geometrically effective* in the sense that it 