arises from a ${\mathbb F}_{p^n}$-valued point of ${\mathscr{S}}$ as (\[eq:K-triple\_for\_isogeny\_adm.pair\]) if and only if $\mathrm{O}_{\gamma}(f^p)\cdot\mathrm{TO}_{\delta}(\phi_p)$ is non-zero.

\[Thm. \[thm:non-emptiness\_of\_NS\]\] \[thm:3rd\_Main\_thm\] Keep the assumptions of Theorem \[thm\_intro:1st\_Main\_thm\], and further assume that $G_{{{\mathbb Q}_p}}$ splits over a *cyclic* tame extension of ${{\mathbb Q}_p}$. Let ${\mathbf{K}}_p$ be a (not necessarily special) parahoric subgroup of $G({{\mathbb Q}_p})$ and put ${\mathbf{K}}={\mathbf{K}}_p{\mathbf{K}}^p$ for a compact open subgroup ${\mathbf{K}}^p$ of $G({\mathbb A}_f^p)$.

\(1) Then, for any $[b]\in B(G_{{{\mathbb Q}_p}},\{\mu_X\}$) (), there exists a special Shimura sub-datum $(T,h\in{\mathrm{Hom}}({\mathbb{S}},T_{{\mathbb R}})\cap X)$ such that for any $g_f\in{\mathbf{K}}^p$, the reduction in ${\mathscr{S}}_{{\mathbf{K}}}\otimes{\bar{\mathbb{F}}_{p}}$ of the special point $[h,g_f\cdot{\mathbf{K}}]\in {\mathrm{Sh}}_{{\mathbf{K}}}(G,X)({\overline{\mathbb Q}})$ has the $F$-isocrystal represented by $[b]$.

\(2) The reduction ${\mathscr{S}}_{{\mathbf{K}}}(G,X)\otimes{\bar{\mathbb{F}}_{p}}$ has non-empty ordinary locus if and only if $\wp$ has absolute height one (i.e. $E(G,X)_{\wp}={\mathbb Q}_p$).

This theorem generalizes Theorem 4.3.1 and Corollary 4.3.2 of [@Lee16] in the hyperspecial cases.

Next, we give some comments on possible generalizations of Thm. \[thm\_intro:Kottwitz\_conj\] and Thm. \[thm\_intro:Kottwitz\_formula\] for bad reductions. In view of the recent works [@KisinPappas15 Thm. 4.7.11] and [@Zhou17], it seems very likely that our methods and results allow us to establish also the conjecture on the semisimple zeta function ([@Rapoport05], [@Haines14]) in special parahoric level case, when $G_{{{\mathbb Q}_p}}$ is unramified and 