Remark 2.1. We will define bounds on local G-shtukas as (equivalence classes of) certain ind-subschemes of ${\widehat {{\mathcal {F}}\ell }}_{G,R}:={\mathcal {F}}\ell _G\widehat {\displaystyle \times }_{{\mathbb {F}}_q}\operatorname {\mathrm {Spf}} R$ where R is a finite extension of ${\mathbb {F}}_q\mathrm {[}\kern-0.15em\mathrm {[}\zeta \mathrm {]}\kern-0.15em\mathrm {]}$ . In order to define and consider also the generic fibre of the associated moduli spaces, one needs to bound the singularities with respect to $z-\zeta $ of the local G-shtukas. In particular, our definition is more restrictive than the one in [Reference Arasteh Rad and Hartl4, Definitions 4.5 and 4.8]. To encode this condition in our notion of bounds, we have to compare ${\widehat {{\mathcal {F}}\ell }}_{G,R}$ to the following closed ind-subschemes associated with a representation of G.

If $A={\mathbb {F}}_q\mathrm {[}\kern-0.15em\mathrm {[} z\mathrm {]}\kern-0.15em\mathrm {]}$ or $A={\mathbb {F}}_q(\kern-0.15em( z)\kern-0.15em)$ , we let $\operatorname {\mathrm {Rep}}_{A}G$ denote the category of (algebraic) representations of G on finite free A-modules. Here, we consider representations $\rho \colon G\to \operatorname {\mathrm {SL}}_r$ over ${\mathbb {F}}_q\mathrm {[}\kern-0.15em\mathrm {[} z\mathrm {]}\kern-0.15em\mathrm {]}$ and the induced functor ${\mathcal {G}}\mapsto \rho _*{\mathcal {G}}$ from $L^+G$ -torsors to $L^+\operatorname {\mathrm {SL}}_r$ -torsors, which in turn yields a morphism $\rho _*\colon {\widehat {{\mathcal {F}}\ell }}_G\to {\widehat {{\mathcal {F}}\ell }}_{\operatorname {\mathrm {SL}}_r}$ . Here, ${\widehat {{\mathcal {F}}\ell }}_{G}:={\mathcal {F}}\ell _{G,{\mathbb {F}}_q\mathrm {[}\kern-0.15em\mathrm {[}\zeta \mathrm {]}\kern-0.15em\mathrm {]}}$ . The category of $L^+\operatorname {\mathrm {SL}}_r$ -torsors on S is equivalent to the category of pairs $(M,\alpha )$ , where M is a finite locally free ${\mathcal {O}}_S\mathrm {[}\kern-0.15em\mathrm {[} z\mathrm {]}\kern-0.15em\mathrm {]}$ -module of rank r on S and is an isomorphism of ${\mathcal {O}}_S\mathrm {[}\kern-0.15em\mathrm {[} z\mathrm {]}\kern-0.15em\mathrm {]}$ -modules, with isomorphisms as morphisms. We denote the ${\mathcal {O}}_S\mathrm {[}\kern-0.15em\mathrm {[} z\mathrm {]}\kern-0.15em\mathrm {]}$ -module associated with an $L^+\operatorname {\mathrm {SL}}_r$ -torsor ${\mathcal {S}}$ by $M({\mathcal {S}})$ . For example, $M\big ((L^+\operatorname {\mathrm {SL}}_r)_S\big )={\mathcal {O}}_S\mathrm {[}\kern-0.15em\mathrm {[} z\mathrm {]}\kern-0.15em\mathrm {]}^{\oplus r}$ . For a positive integer n we consider the closed ind-subscheme of ${\widehat {{\mathcal {F}}\ell }}_{\operatorname {\mathrm {SL}}_r}$ given by

(2.1)

It is a $\zeta $ -adic formal scheme over $\operatorname {\mathrm {Spf}}{\mathbb {F}}_q\mathrm {[}\kern-0.15em\mathrm {[}\zeta \mathrm {]}\kern-0.15em\mathrm {]}$ by [Reference Hartl and Viehmann56, Proposition 5.5]; see Example 2.8 for more explanations. Note that the compatibility with the isomorphism is equivalent to the assertion that the inclusion of the exterior powers in (2.1) is an equality for $j=r$ , because $\big ({\mathcal {S}},\,\delta \big )\in {\widehat {{\mathcal {F}}\ell }}_{\operatorname {\mathrm {SL}}_r}(S)$ implies $\wedge ^r M(\delta )=\alpha $ . We then require the bounds to factor through some ${\widehat {{\mathcal {F}}\ell }}^{(n)}_{\operatorname {\mathrm {SL}}_r}$ ; cf. Condition (b)(iv).

A different way to formulate such a condition would be to use the isomorphism

(2.2) $$ \begin{align} \lim_{\longrightarrow_n} {\widehat{{\mathcal{F}}\ell}}_G \times_{{\widehat{{\mathcal{F}}\ell}}_{\operatorname{\mathrm{SL}}_r}} {\widehat{{\mathcal{F}}\ell}}^{(n)}_{\operatorname{\mathrm{SL}}_r}\cong \textrm{Gr}(G_X,X) \times_X \operatorname{\mathrm{Spec}} R \end{align} $$

where the right-hand side is the BD-Grassmannian associated with G of [Reference Richarz84, Definition 3.3]. Here we use that G extends by [Reference Richarz84, Lemma 3.1] to a smooth affine group scheme $G_X$ on a smooth connected curve X over ${\mathbb {F}}_q$ on which ${\mathbb {F}}_q\mathrm {[}\kern-0.15em\mathrm {[} z\mathrm {]}\kern-0.15em\mathrm {]}$ is identified with the completion of the local ring at a point $x\in X$ . The map $\operatorname {\mathrm {Spec}} R\to X$ comes from the inclusion ${\mathbb {F}}_q\mathrm {[}\kern-0.15em\mathrm {[} z\mathrm {]}\kern-0.15em\mathrm {]}\hookrightarrow R, z\mapsto \zeta $ . The isomorphism (2.2) also induces a comparison between specific bounds with global Schubert varieties of [Reference Richarz84]; compare Example 2.7.

Remark 2.3. (a) The definition of a bound in Definition 2.2(b) is more restrictive than the one in [Reference Arasteh Rad and Hartl4, Definition 4.8] where only conditions (b)(i) and (b)(ii) were required. The reason is that in [Reference Arasteh Rad and Hartl4] the content of Proposition 2.6 was not needed and the $R[\tfrac {1}{\zeta }]$ -analytic spaces $(\hat {Z}_R)^{\textrm {an}}$ were not considered.

In this article we will mainly consider local G-shtukas that are bounded by $\hat {Z}^{-1}$ . This definition coincides with the notion of boundedness from [Reference Arasteh Rad and Hartl4, Definition 4.8(b)] in the following way. If $\hat {Z}$ is a bound in the sense of [Reference Arasteh Rad and Hartl4, Definition 4.8], like, for example, our bound $\hat Z$ , then by Lemma 2.12 there is a bound $\hat {Z}^{-1}$ in the sense of [Reference Arasteh Rad and Hartl4, Definition 4.8] and $\tau _{\mathcal {G}}^{-1}$ is bounded by $\hat {Z}$ if and only if $\tau _{\mathcal {G}}$ is bounded by $\hat {Z}^{-1}$ .

(b) The reflex ring in Definition 2.2(c) is always the ring of integers of a finite extension of ${\mathbb {F}}_q(\kern-0.15em(\zeta )\kern-0.15em)$ . For a detailed explanation of the definition of the reflex ring and a comparison with the number field case, see [Reference Arasteh Rad and Hartl4, Remark 4.7]. We do not know whether in general $\hat Z$ has a representative over the reflex ring. In contrast, the equivalence class of the $Z_R:=\hat {Z}_R\widehat {\displaystyle \times }_{\operatorname {\mathrm {Spf}} R}\operatorname {\mathrm {Spec}} \kappa _R$ always has a representative $Z\subset {\mathcal {F}}\ell _G\widehat {\displaystyle \times }_{{\mathbb {F}}_q}\operatorname {\mathrm {Spec}}\kappa $ over the residue field $\kappa $ of the reflex ring $R_{\hat Z}$ , because the Galois descent for closed ind-subschemes of ${\mathcal {F}}\ell _G$ is effective. We call Z the special fibre of $\hat Z$ . It is a projective scheme over $\kappa $ by [Reference Hartl and Viehmann56, Lemma 5.4] because ${\mathcal {F}}\ell _G$ is ind-projective.

(c) The condition of Definition 2.2(d) is satisfied for all trivialisations $\alpha $ and $\alpha ^{\prime }$ and for all such finite extensions R of ${\mathbb {F}}_q\mathrm {[}\kern-0.15em\mathrm {[}\zeta \mathrm {]}\kern-0.15em\mathrm {]}$ if and only if it is satisfied for one trivialisation and for one such finite extension. Indeed, by the $L^+G$ -invariance of $\hat Z$ , the definition is independent of the trivialisations. That one finite extension suffices follows from [Reference Arasteh Rad and Hartl4, Remark 4.6].

(d) At first glance one might think that conditions (b)(i) and (b)(v) of Definition 2.2 are related. However, in Example 2.9 we show that we really need to impose both of them.

Remark 2.5. Without the assumption that f is adic, the conclusion of the lemma is false, as the following example shows. Let $Y={\displaystyle \lim _{\longrightarrow }} Y_n$ with $Y_n=\operatorname {\mathrm {Spec}}{\mathbb {F}}_q\mathrm {[}\kern-0.15em\mathrm {[}\zeta \mathrm {]}\kern-0.15em\mathrm {]}[x]/(\zeta ^n)$ and $X={\displaystyle \lim _{\longrightarrow }} X_n$ with $X_n=\operatorname {\mathrm {Spec}}{\mathbb {F}}_q\mathrm {[}\kern-0.15em\mathrm {[}\zeta \mathrm {]}\kern-0.15em\mathrm {]}[x]/(\zeta ,x)^n$ . Then $X=\operatorname {\mathrm {Spf}}{\mathbb {F}}_q\mathrm {[}\kern-0.15em\mathrm {[}\zeta ,x\mathrm {]}\kern-0.15em\mathrm {]}\to Y=\operatorname {\mathrm {Spf}}{\mathbb {F}}_q\mathrm {[}\kern-0.15em\mathrm {[}\zeta \mathrm {]}\kern-0.15em\mathrm {]}\langle x\rangle $ , with the notation of (6.3), is an ind-closed immersion of ind-schemes but not a closed immersion of formal schemes.

Remark. The proof of statements (c) and (d) uses the ind-projectivity of the affine flag variety ${\mathcal {F}}\ell _G$ . If G is not parahoric, it is not clear to us whether $\rho _*\colon \hat {Z}_R\hookrightarrow {\widehat {{\mathcal {F}}\ell }}_{\operatorname {\mathrm {SL}}_{r},R}^{(n)}$ is a locally closed immersion of $\zeta $ -adic formal schemes.

Remark 2.11. We discuss some possible additional assumptions on bounds.

1. (a) The set $\pi _0({\widehat {{\mathcal {F}}\ell }}_{G,R})$ coincides with the set of $\Gamma $ -orbits in $\pi _1(G)_I$ by [Reference Neupert76, Lemma 2.2.6]. Every bound is a disjoint union of its intersections with the various connected components of ${\widehat {{\mathcal {F}}\ell }}_{G,R}$ , and thus we also obtain similar decompositions of base schemes for bounded local G-shtukas and later of the corresponding moduli spaces. If one wants to consider only one of these disjoint parts at a time, one has to assume that there is a $\Gamma $ -orbit of elements $\xi \in \pi _1(G)_I$ such that the $\hat Z_R$ are contained in the corresponding connected component of ${\widehat {{\mathcal {F}}\ell }}_{G,R}$ .

   Also compare the nonemptiness condition of Theorem 4.20, which is in terms of $\pi _1(G)_{\Gamma }$ instead of $\pi _1(G)_{I}/\Gamma $ . The natural projection map $\pi _1(G)_{I}/\Gamma \rightarrow \pi _1(G)_{\Gamma }$ is surjective but in general not injective. Thus, there may be several connected components of a given bound that lead to nonempty parts of the period domain.

2. (b) If one wants to compare properties of the generic and the special fibre of a moduli space of local G-shtukas bounded by $\hat Z^{-1}$ (as defined in Section 3), it might be useful to consider bounds $\hat Z$ satisfying certain flatness or extension properties.

   Assumptions (b)(i) and (b)(v) of Definition 2.2 imply that the closed points of the special fibre Z from Remark 2.3(b), respectively of the analytic space $\hat Z^{\textrm {an}}$ of $\hat Z$ from Proposition 2.6(d), consist of the points of a finite set $N_0$ of $L^+G$ -cosets, respectively a finite set $N_{\textrm {an}}$ of $G\big ({\,{\scriptscriptstyle \bullet }\,}\mathrm {[}\kern-0.15em\mathrm {[} z-\zeta \mathrm {]}\kern-0.15em\mathrm {]}\big )$ -cosets. The former cosets are parametrised by some quotient $\bar W$ of the extended affine Weyl group $\widetilde W$ of G with respect to a chosen maximal torus T of $G_{{\mathbb {F}}_q(\kern-0.15em( z)\kern-0.15em)}$ . Let $L_0\supset {\mathbb {F}}_q(\kern-0.15em(\zeta )\kern-0.15em)$ be a finite separable field extension over which $T_{L_0}:=T\times _{{\mathbb {F}}_q(\kern-0.15em( z)\kern-0.15em),z\mapsto \zeta }\operatorname {\mathrm {Spec}} L_0$ (and therefore also $G_{L_0}$ ) splits. Because $G\big (L_0\mathrm {[}\kern-0.15em\mathrm {[} z-\zeta \mathrm {]}\kern-0.15em\mathrm {]}\big )\subset G\big (L_0(\kern-0.15em( z-\zeta )\kern-0.15em)\big )$ is a hyperspecial maximal bounded open subgroup, every $G\big (L_0\mathrm {[}\kern-0.15em\mathrm {[} z-\zeta \mathrm {]}\kern-0.15em\mathrm {]}\big )$ -coset is of the form $G\big (L_0\mathrm {[}\kern-0.15em\mathrm {[} z-\zeta \mathrm {]}\kern-0.15em\mathrm {]}\big )\mu (z-\zeta )G\big (L_0\mathrm {[}\kern-0.15em\mathrm {[} z-\zeta \mathrm {]}\kern-0.15em\mathrm {]}\big )\big / G\big (L_0\mathrm {[}\kern-0.15em\mathrm {[} z-\zeta \mathrm {]}\kern-0.15em\mathrm {]}\big )$ for a uniquely determined determined $\mu \in X_*(T_{L_0})_{\textrm {dom}}= X_*(T)_{\textrm {dom}}$ where dominance is with respect to some chosen Borel subgroup. It would thus be interesting to see whether such flatness conditions can be formulated in terms of the associated sets $N_0\subset \bar W$ and $N_{\textrm {an}}\subset X_*(T)_{\textrm {dom}}$ . However, for the present article we do not need any such condition.

3. (c) Given the discussion above, one might ask whether bounds should just be defined by the corresponding sets of double cosets of their geometric points. Even under assumptions as discussed above, our definition gives some more freedom in the sense that the nilpotent structure of the bound may still vary.

Remark 3.2. Because ${\underline {{\mathbb {G}}}}_0\cong \big ((L^+G)_{\mathbb {F}},b\sigma ^\ast \big )$ , we can identify $\operatorname {\mathrm {QIsog}}_{\mathbb {F}}({\underline {{\mathbb {G}}}}_0)\cong J_b\big ({\mathbb {F}}_q(\kern-0.15em( z)\kern-0.15em)\big )$ where $J_b$ is the connected algebraic group over ${\mathbb {F}}_q(\kern-0.15em( z)\kern-0.15em)$ that is defined by its functor of points that assigns to an ${\mathbb {F}}_q(\kern-0.15em( z)\kern-0.15em)$ -algebra A the group

(3.1) $$ \begin{align} J_b(A):=J_b^G(A):=\big\{g \in G(A\otimes_{{\mathbb{F}}_q(\kern-0.15em( z )\kern-0.15em)} {{\mathbb{F}}(\kern-0.15em( z )\kern-0.15em)})\colon g^{-1}\,b\,\sigma^\ast(g)=b\big\}\,; \end{align} $$

see [Reference Arasteh Rad and Hartl4, Remark 4.16].

Remark 3.3. As in the arithmetic case (compare [Reference Rapoport and Zink80, 3.48]), we have a Weil descent datum $\alpha $ on the functor ${\breve {\mathcal {M}}}_{{\underline {{\mathbb {G}}}}_0}^{\hat {Z}^{-1}}$ . To define it, let $S\in {{\mathcal {N}}\!\mathit {ilp}}_{\breve R_{\hat Z}}$ and let $f\colon S\rightarrow \operatorname {\mathrm {Spf}}\breve R_{\hat Z}$ and $\bar f\colon \bar S\rightarrow \operatorname {\mathrm {Spec}}\breve R_{\hat Z}/(\zeta )$ be the structure morphisms. Let $\varphi $ be the Frobenius of $\breve R_{\hat Z}={\mathbb {F}}\mathrm {[}\kern-0.15em\mathrm {[}\xi \mathrm {]}\kern-0.15em\mathrm {]}$ over $R_{\hat Z}=\kappa \mathrm {[}\kern-0.15em\mathrm {[}\xi \mathrm {]}\kern-0.15em\mathrm {]}$ and let $S_\varphi $ be the scheme S together with the structure morphism $\varphi \circ f\colon S\rightarrow \operatorname {\mathrm {Spf}}\breve R_{\hat Z}$ . If $\#\kappa =q^e$ , the inclusion ${\mathbb {F}}\hookrightarrow \breve R_{\hat Z}$ is equivariant for the action of $\varphi $ on $\breve R_{\hat Z}$ and the action of $\sigma ^e$ on ${\mathbb {F}}$ . To define , let $({\underline {{\mathcal {G}}}},\bar \delta )$ be a point in the first set. To make the definition more clear, we write ${\underline {{\mathbb {G}}}}_{0,\bar {S}}=\bar f^*({\underline {{\mathbb {G}}}}_0)$ . Then we set $\alpha ({\underline {{\mathcal {G}}}},\bar \delta )=({\underline {{\mathcal {G}}}},\bar f^*(\tau _{{\mathbb {G}}_0}^e)^{-1}\circ \bar \delta )$ ; that is, we replace the quasi-isogeny $\bar \delta $ by the composite ${\underline {{\mathcal {G}}}}_{\bar {S}}\to \bar f^*({\underline {{\mathbb {G}}}}_0)\to \bar f^*\sigma ^{e*}({\underline {{\mathbb {G}}}}_0)=(\bar \varphi \bar f)^*({\underline {{\mathbb {G}}}}_0)$ . Although this Weil descent datum is in general not effective, ${\breve {\mathcal {M}}}_{{\underline {{\mathbb {G}}}}_0}^{\hat {Z}^{-1}}$ always descends to a finite unramified extension of $\breve R_{\hat Z}$ ; see Remark 3.6(b).

Remark 3.4. By its definition, ${\breve {\mathcal {M}}}_{{\underline {{\mathbb {G}}}}_0}^{\hat {Z}^{-1}}$ only depends on the triple $(G,[b],\hat {Z})$ , where $[b]$ is the $LG({\mathbb {F}})$ - $\sigma $ -conjugacy class of b corresponding to the isogeny class of ${\underline {{\mathbb {G}}}}_0\cong \big ((L^+G)_{\mathbb {F}},b\sigma ^\ast \big )$ . In the arithmetic analogue, the corresponding independence of the Rapoport-Zink spaces on additional choices made in their definition is far from obvious. It was conjectured in [Reference Rapoport and Viehmann79, Conjecture 4.16] and proved in [Reference Scholze and Weinstein90, Corollary 23.4.3 and thereafter].

Remark 3.6. (a) $LG({\mathbb {F}})$ - $\sigma $ -conjugacy classes in $LG({\mathbb {F}})$ are in bijection to isogeny classes of local G-shtukas over ${\mathbb {F}}$ . To classify them, Kottwitz associated with every element $b\in LG({\mathbb {F}})$ a slope homomorphism $\nu _b\colon {\mathbb {D}}_{{\mathbb {F}}(\kern-0.15em( z)\kern-0.15em)}\to G_{{\mathbb {F}}(\kern-0.15em( z)\kern-0.15em)}$ , called Newton point (or Newton polygon) of b; see [Reference Kottwitz70, 4.2]. Here ${\mathbb {D}}$ is the diagonalisable pro-algebraic group over ${\mathbb {F}}(\kern-0.15em( z)\kern-0.15em)$ with character group ${\mathbb {Q}}$ . The slope homomorphism is characterised by assigning the slope filtration of

$$ \begin{align*} \big(V\otimes_{{\mathbb{F}}_q(\kern-0.15em( z )\kern-0.15em)}{{\mathbb{F}}(\kern-0.15em( z )\kern-0.15em)},\,\rho(b)\sigma^\ast\big) \end{align*} $$

to any $(V,\rho )$ in $\operatorname {\mathrm {Rep}}_{{\mathbb {F}}_q(\kern-0.15em( z)\kern-0.15em)}G$ ; see [Reference Kottwitz70, Section 4]. Furthermore, he showed that $\sigma $ -conjugating b amounts to conjugating $\nu _b$ by the corresponding element. One can thus associate with the $LG({\mathbb {F}})$ - $\sigma $ -conjugacy class $[b]$ a well-defined $G\big ({\mathbb {F}}(\kern-0.15em( z)\kern-0.15em)\big )$ -conjugacy class $\{\nu _b\}$ in $\operatorname {\mathrm {Hom}}({\mathbb {D}}_{{\mathbb {F}}(\kern-0.15em( z)\kern-0.15em)},G_{{\mathbb {F}}(\kern-0.15em( z)\kern-0.15em)})$ , which moreover is invariant under $\sigma $ by [Reference Kottwitz70, 4.4].

The second important invariant of $[b]$ is defined as follows. Consider again the Kottwitz homomorphism $\kappa _G\colon LG({\mathbb {F}})\rightarrow \pi _1(G)_{I}$ as explained in [Reference Pappas and Rapoport77, 2.a.2]. We compose $\kappa _G$ with the projection to $\pi _1(G)_{\Gamma }$ . This then yields a well-defined map (again denoted $\kappa _G$ ) from the set of $\sigma $ -conjugacy classes $B(G)$ to $\pi _1(G)_{\Gamma }$ (see [Reference Kottwitz71]). Together, $\{\nu _b\}$ and $\kappa _G([b])$ determine $[b]$ uniquely. We denote by $[b]^{\#}$ the images of $\kappa _G(b)$ in $\pi _1(G)_{\Gamma }$ and in $\pi _1(G)_{\Gamma ,{\mathbb {Q}}}:=\pi _1(G)_{\Gamma }\otimes _{\mathbb {Z}}{\mathbb {Q}}$ . The latter equals the image of the conjugacy class $\{\nu _b\}$ in $\pi _1(G)_{\Gamma ,{\mathbb {Q}}}$ ; see [Reference Rapoport and Richartz78, Theorem 1.15(iii)].

(b) Because ${\mathbb {F}}$ is algebraically closed and the generic fibre of G is connected reductive, we may replace b by $h^{-1}b\sigma ^\ast (h)$ and assume that $b\in LG({\mathbb {F}})$ satisfies a decency equation for a positive integer s; that is, $s\nu _b:{\mathbb {D}}_{{\mathbb {F}}(\kern-0.15em( z)\kern-0.15em)}\to G_{{\mathbb {F}}(\kern-0.15em( z)\kern-0.15em)}$ factors through ${\mathbb {G}}_m$ , and

(3.3) $$ \begin{align} (b\sigma)^s\;=\;s\nu_b(z)\,\sigma^s\qquad\text{in}\quad LG({\mathbb{F}})\rtimes \langle\sigma\rangle; \end{align} $$

see [Reference Kottwitz70, Section 4]. Let ${\mathbb {F}}_{q^s}\subset {\mathbb {F}}$ be the finite field extension of ${\mathbb {F}}_q$ of degree s. Then $b\in LG({\mathbb {F}}_{q^s})$ , $\nu _b$ is defined over ${\mathbb {F}}_{q^s}(\kern-0.15em( z)\kern-0.15em)$ ([Reference Rapoport and Zink80, Corollary 1.9]) and $J_b\times _{{\mathbb {F}}_q(\kern-0.15em( z)\kern-0.15em)}{\mathbb {F}}_{q^s}(\kern-0.15em( z)\kern-0.15em)$ is an inner form of the centraliser of the 1-parameter subgroup $s\nu _b$ of G, a Levi subgroup of $G_{{\mathbb {F}}_{q^s}(\kern-0.15em( z)\kern-0.15em)}$ ; see [Reference Rapoport and Zink80, Corollary 1.14]. In particular, $J_b\big ({\mathbb {F}}_q(\kern-0.15em( z)\kern-0.15em)\big )\subset G\big ({\mathbb {F}}_{q^s}(\kern-0.15em( z)\kern-0.15em)\big )= LG({\mathbb {F}}_{q^s})$ . Moreover, in this case ${\breve {\mathcal {M}}}_{{\underline {{\mathbb {G}}}}_0}^{\hat {Z}^{-1}}$ descends by [Reference Arasteh Rad and Hartl4, Theorem 4.18] to a formal scheme locally formally of finite type over $({\mathbb {F}}_{q^s}\cdot \kappa )\mathrm {[}\kern-0.15em\mathrm {[}\xi \mathrm {]}\kern-0.15em\mathrm {]}$ where ${\mathbb {F}}_{q^s}\cdot \kappa $ is the compositum inside ${\mathbb {F}}$ .

(c) If more generally we start with a local G-shtuka ${\underline {{\mathbb {G}}}}_0$ over any field k in ${{\mathcal {N}}\!\mathit {ilp}}_{ R_{\hat Z}}$ , we can define the Rapoport-Zink functor ${{\mathcal {M}}}_{{\underline {{\mathbb {G}}}}_0}^{\hat {Z}^{-1}}$ as in Definition 3.1 on the category ${{\mathcal {N}}\!\mathit {ilp}}_{k\mathrm {[}\kern-0.15em\mathrm {[}\xi \mathrm {]}\kern-0.15em\mathrm {]}}$ . Then ${\underline {{\mathbb {G}}}}_0\otimes _k k^{\textrm {alg}}$ is trivial and decent and hence ${{\mathcal {M}}}_{{\underline {{\mathbb {G}}}}_0}^{\hat {Z}^{-1}}\otimes _k k^{\textrm {alg}}$ is ind-representable by a formal scheme locally formally of finite type over $k^{\textrm {alg}}\mathrm {[}\kern-0.15em\mathrm {[}\xi \mathrm {]}\kern-0.15em\mathrm {]}$ . By an unpublished result of Eike Lau on Galois descent of formal schemes locally formally of finite type, already ${{\mathcal {M}}}_{{\underline {{\mathbb {G}}}}_0}^{\hat {Z}^{-1}}$ is ind-representable by a formal scheme locally formally of finite type over $k\mathrm {[}\kern-0.15em\mathrm {[}\xi \mathrm {]}\kern-0.15em\mathrm {]}$ . However, we will not use this in the rest of this work.

Remark 3.7. The constructions described above are functorial with respect to the group scheme G. To explain this, let $\varepsilon \colon G\to G^{\prime }$ be a morphism of parahoric group schemes over ${\mathbb {F}}_q\mathrm {[}\kern-0.15em\mathrm {[} z\mathrm {]}\kern-0.15em\mathrm {]}$ . Important examples are closed immersions, epimorphisms or the change of the parahoric model, that is, morphisms that are generically isomorphisms. Then $\varepsilon $ induces a functor

$$ \begin{align*} \varepsilon_*\colon\; \{\,\text{local } G\text{-shtukas}\,\}\;\longrightarrow\;\{\,\text{local } G^{\prime}\text{-shtukas}\,\}\,{,}\quad{\underline{{\mathcal{G}}}}\;\longmapsto\; \varepsilon_*{\underline{{\mathcal{G}}}}\;:=\;{\underline{{\mathcal{G}}}}\overset{G}{\times} G^{\prime}\,. \end{align*} $$

The morphism $\varepsilon $ also induces morphisms $\varepsilon \colon L^+G\to L^+G^{\prime }$ and $\varepsilon \colon LG\to LG^{\prime }$ and $\varepsilon \colon {\mathcal {F}}\ell _G\to {\mathcal {F}}\ell _{G^{\prime }}$ . We say that $\varepsilon $ is compatible with two given bounds $\hat {Z}=[(R,\hat {Z}_R)]$ and $\hat {Z}^{\prime }=[(R,\hat {Z}^{\prime }_R)]$ with $\hat {Z}_R\subset {\widehat {{\mathcal {F}}\ell }}_{G,R}$ and $\hat {Z}^{\prime }_R\subset {\widehat {{\mathcal {F}}\ell }}_{G^{\prime }\!,R}$ if $\varepsilon (\hat {Z}_R)\subset \hat {Z}^{\prime }_R$ for all suitable R. If $\varepsilon $ is compatible with $\hat {Z}$ and $\hat {Z}^{\prime }$ and ${\underline {{\mathcal {G}}}}$ is bounded by $\hat {Z}^{-1}$ , then $\varepsilon _*{\underline {{\mathcal {G}}}}$ is bounded by $(\hat {Z}^{\prime })^{-1}$ .

Let ${\underline {{\mathbb {G}}}}_0\cong \big ((L^+G)_{\mathbb {F}},b\sigma ^\ast \big )$ be a local G-shtuka over ${\mathbb {F}}$ with $b\in LG({\mathbb {F}})$ . Then ${\underline {{\mathbb {G}}}}^{\prime }_0:=\varepsilon _*{\underline {{\mathbb {G}}}}_0\cong \big ((L^+G^{\prime })_{\mathbb {F}},b^{\prime }\sigma ^\ast \big )$ with $b^{\prime }=\varepsilon (b)$ . Kottwitz’s classification of isogeny classes is functorial in the sense that $\nu _{b^{\prime }}=\varepsilon \circ \nu _b$ and $\varepsilon \circ \kappa _G=\kappa _{G^{\prime }}\circ \varepsilon $ for the induced $\Gamma $ -equivariant morphism $\varepsilon \colon \pi _1(G)\to \pi _1(G^{\prime })$ . We also obtain a morphism of Rapoport-Zink spaces

(3.4) $$ \begin{align} \varepsilon_*\colon\;{\breve{\mathcal{M}}}_{{\underline{{\mathbb{G}}}}_0}^{\hat{Z}^{-1}}\;\longrightarrow\; {\breve{\mathcal{M}}}_{{\underline{{\mathbb{G}}}}^{\prime}_0}^{{\hat{Z}^{\prime}{}^{-1}}},\quad({\underline{{\mathcal{G}}}},\bar\delta)\;\longmapsto\;(\varepsilon_*{\underline{{\mathcal{G}}}},\varepsilon_*\bar\delta)\,{,} \end{align} $$

which is equivariant for the group $J_b^G$ that acts on the target via the morphism of algebraic groups over ${\mathbb {F}}_q(\kern-0.15em( z)\kern-0.15em)$ ,

(3.5) $$ \begin{align} J_b^G\;\longrightarrow\;J_{b^{\prime}}^{G^{\prime}}\,{,}\quad g\;\longmapsto\;\varepsilon(g)\,. \end{align} $$

Remark 4.2. By [Reference Schauch87, Proposition 2.2.5], a Hodge-Pink structure ${\mathfrak {q}}$ of rank r over X is Zariski-locally on X free of rank r as ${\mathcal {O}}_X\mathrm {[}\kern-0.15em\mathrm {[} z-\zeta \mathrm {]}\kern-0.15em\mathrm {]}$ -module. In particular, it is of the form ${\mathfrak {q}}=\gamma \cdot {\mathcal {O}}_X\mathrm {[}\kern-0.15em\mathrm {[} z-\zeta \mathrm {]}\kern-0.15em\mathrm {]}^r\subset {\mathcal {O}}_X(\kern-0.15em( z-\zeta )\kern-0.15em)^r$ for a uniquely determined Hodge-Pink $\operatorname {\mathrm {GL}}_r$ -structure $\gamma \in \operatorname {\mathrm {Gr}}_{\operatorname {\mathrm {GL}}_r}^{{\mathbf {B}}_{\textrm {dR}}}(X)$ over X. This yields an equivalence between Hodge-Pink $\operatorname {\mathrm {GL}}_r$ -structures and Hodge-Pink structures of rank r over X.

Remark 4.4. (a) The Hodge-Pink structure ${\mathfrak {q}}_D$ on $(D,\tau _D)$ induces a decreasing Hodge-Pink filtration on $D_L:=D\otimes _{{\mathbb {F}}(\kern-0.15em( z)\kern-0.15em),z\mapsto \zeta }L={\mathfrak {p}}_D/(z-\zeta ){\mathfrak {p}}_D$ given by

$$ \begin{align*} Fil^i D_L\;:=\;\big((z-\zeta)^i{\mathfrak{q}}_D\cap{\mathfrak{p}}_D\big)\big/ \big((z-\zeta)^i{\mathfrak{q}}_D\cap(z-\zeta){\mathfrak{p}}_D\big)\;=\;\operatorname{\mathrm{im}}\big((z-\zeta)^i{\mathfrak{q}}_D\cap{\mathfrak{p}}_D\to D_L\big). \end{align*} $$

If $(D,\tau _D,Fil^\bullet D_L)$ is weakly admissible in the sense of Fontaine (see [Reference Fontaine38, Définition 4.1.4]), then $(D,\tau _D,{\mathfrak {q}}_D)$ is weakly admissible, but the converse is false in general. An example is $D={\mathbb {F}}(\kern-0.15em( z)\kern-0.15em)^2$ , $\tau _D=z^{-1}$ , ${\mathfrak {q}}_D=\binom {z-\zeta }{1}L\mathrm {[}\kern-0.15em\mathrm {[} z-\zeta \mathrm {]}\kern-0.15em\mathrm {]}+(z-\zeta )^2{\mathfrak {p}}_D$ and $D^{\prime }=\binom {0}{1}{\mathbb {F}}(\kern-0.15em( z)\kern-0.15em)$ . This is due to the fact that our Hodge slope $t_H(D,\tau _D,{\mathfrak {q}}_D)$ equals Fontaine’s Hodge slope $t_H(D,\tau _D,Fil^\bullet D_L):= \sum _{i\in {\mathbb {Z}}}i\cdot \dim _L Fil^i D_L/Fil^{i+1}D_L$ , see [Reference Hartl50, p. 1290 before Definition 2.2.4] and that the subspace $Fil^iD^{\prime }_L$ of $D^{\prime }_L$ induced by ${\underline {D\!}\,}^{\prime }$ from (4.2) is (in general strictly) contained in $D^{\prime }_L\cap Fil^iD_L$ .

(b) We obtain an ${\mathbb {F}}_q(\kern-0.15em( z)\kern-0.15em)$ -linear tensor functor ${\underline {D\!}\,}_{b,\gamma }\colon V\mapsto {\underline {D\!}\,}_{b,\gamma }(V)$ . Namely, if $f\colon (V,\rho )\to (V^{\prime }\!,\rho ^{\prime })$ is a morphism in $\operatorname {\mathrm {Rep}}_{{\mathbb {F}}_q(\kern-0.15em( z)\kern-0.15em)}G$ and $(D,\tau _D,{\mathfrak {q}}_D)={\underline {D\!}\,}_{b,\gamma }(V)$ and $(D^{\prime }\!,\tau _{D^{\prime }},{\mathfrak {q}}_{D^{\prime }})={\underline {D\!}\,}_{b,\gamma }(V^{\prime })$ , then $\sigma ^*f=f$ and so $f\circ \tau _D=(f\circ \rho (\sigma ^\ast b))\sigma ^\ast =(\rho ^{\prime }(\sigma ^\ast b)\circ f)\sigma ^\ast =(\rho ^{\prime }(\sigma ^\ast b)\circ \sigma ^\ast f)\sigma ^\ast =\tau _{D^{\prime }}\circ \sigma ^*f$ and $f({\mathfrak {q}}_D)\subset {\mathfrak {q}}_{D^{\prime }}$ . Furthermore, the compatibility with tensor products

$$ \begin{align*} (D,\tau_D,{\mathfrak{q}}_D)\otimes(D^{\prime}\!,\tau_{D^{\prime}},{\mathfrak{q}}_{D^{\prime}})\;:=\;(D\otimes_{{\mathbb{F}}(\kern-0.15em( z )\kern-0.15em)}D^{\prime}\!,\,\tau_D\otimes\tau_{D^{\prime}},\,{\mathfrak{q}}_D\otimes_{L\text{[}\kern-0.15em\text{[} z-\zeta\text{]}\kern-0.15em\text{]}}{\mathfrak{q}}_{D^{\prime}}) \end{align*} $$

is clear.