A Short Review on Local Shtukas and Divisible Local Anderson Modules

Abstract

We review the analog of crystalline Dieudonné theory for p-divisible groups in the arithmetic of function fields from [21]. In our theory, p-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We also explain their relation to global objects like Drinfeld modules and A-motives. We review the cohomology realizations of local shtukas and their comparison isomorphisms, and in the last section we explain how this yields the function field analog of Fontaine’s theory of p-adic Galois representations.

1 Introduction

The theory of p-adic Galois representations is concerned with the continuous representations

$$\begin{aligned} \rho :{{\,\mathrm{Gal}\,}}(L^\mathrm{alg}/L)\;\longrightarrow \; {{\,\mathrm{GL}\,}}_r({\mathbb {Q}}_p) \end{aligned}$$

(1.1)

of the absolute Galois group ${{\,\mathrm{Gal}\,}}(L^\mathrm{alg}/L)$ of a finite field extension L of ${\mathbb {Q}}_p$. It started with Tate’s introduction of p-divisible groups in [33]. These are also called Barsotti-Tate groups. The Tate module $T_p X$ of a p-divisible group X of height r over L induces Galois representations $V_pX:=T_pX\otimes _{{\mathbb {Z}}_p}{\mathbb {Q}}_p$ and ${{\,\mathrm{H}\,}}^1_{\acute{\mathrm{e}}t}(X,{\mathbb {Q}}_p):={{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_p}(T_pX,{\mathbb {Q}}_p)$ as in (1.1). If X extends to a p-divisible group over $\mathcal{{O}}_L$, one says that X has good reduction. In this case, the special fiber $X_0:=X\otimes _{\mathcal{{O}}_L}\kappa$ of X over the residue field $\kappa$ of $\mathcal{{O}}_L$ can be described by its crystalline cohomology $H^1_\mathrm{cris}\bigl (X_0/W(\kappa )\bigr )$, where $W(\kappa )$ is the ring of p-typical Witt vectors with coefficients in $\kappa$. The p-divisible group X, which can be viewed as a lift of $X_0$ to $\mathcal{{O}}_L$, is described by the F-crystal $H^1_\mathrm{cris}\bigl (X_0/W(\kappa )\bigr )$ together with its Hodge filtration. All this was proved by Messing [27]. Grothendieck [15] reformulated this as a functor relating the p-adic étale cohomology ${{\,\mathrm{H}\,}}_{\acute{\mathrm{e}}\mathrm{t}}^1(X,{\mathbb {Q}}_p)$ to the crystalline cohomology ${{\,\mathrm{H}\,}}^1_\mathrm{cris}(X_0/L_0)$ with its Hodge filtration, where $L_0:=W(\kappa )[\tfrac{1}{p}]$ and ${{\,\mathrm{H}\,}}^1_\mathrm{cris}(X_0/L_0)$ is a filtered isocrystal; see Remark 6.6 below. Grothendieck then posed the problem to extend this functor, which he called the mysterious functor, to general proper smooth schemes X over L with good reduction. For those X, the problem was solved by Fontaine [10,11,12,13], who defined the notion of crystalline p-adic Galois representations and constructed a functor from crystalline p-adic Galois representations to filtered isocrystals. Fontaine conjectured that ${{\,\mathrm{H}\,}}_{\acute{\mathrm{e}}t}^i(X\times _LL^\mathrm{alg},{\mathbb {Q}}_p)$ is crystalline when X is a proper smooth scheme over $\mathcal{{O}}_L$. After contributions by Grothendieck, Tate, Fontaine, Lafaille, Messing, Hyodo, Kato, and many others, Fontaine’s conjecture was proved independently by Faltings [8], Niziol [28], and Tsuji [34].

Our goal in this survey is to describe the function field analog of the above. In this analog, p-divisible groups are replaced by divisible local Anderson modules which we discuss in Sect. 4. The analog of Messing’s [27] theory of crystalline Dieudonné-modules for p-divisible groups is Theorem 4.2. In it, Messings F-crystals are replaced by local shtukas, which we treat first in Sect. 2. The anti-equivalence between divisible local Anderson modules and local shtukas passes through finite flat group schemes and finite shtukas. We review it in Sect. 3. Analogous to the étale and crystalline cohomology we mentioned for p-divisible groups in the previous paragraph, local shtukas possess cohomology realizations as described in Sect. 5. In the final Sect. 6, we explain how the theory of local shtukas provides the function field analog of Fontaine’s theory of p-adic Galois representations (1.1).

2 Local Shtukas

The theory of local shtukas is the function field analog of Fontaine’s theory of p-adic Galois representations. Let $A_\varepsilon$ be a complete discrete valuation ring with finite residue field ${\mathbb {F}}_\varepsilon$ of characteristic p such that the fraction field $Q_\varepsilon$ of $A_\varepsilon$ also has characteristic p. The rings $A_\varepsilon$ and $Q_\varepsilon$ are the function field analogs of ${\mathbb {Z}}_p$ and ${\mathbb {Q}}_p$. We choose a uniformizing parameter $z\in A_\varepsilon$. Then $A_\varepsilon$ is canonically isomorphic to ${\mathbb {F}}_\varepsilon [\![ z]\!]$. Let $\hat{q}=\#{\mathbb {F}}_\varepsilon$ be the cardinality of ${\mathbb {F}}_\varepsilon$. As base rings R over which our objects are defined, we are interested in this article in two kinds of $A_\varepsilon$-algebras:

• (a) The first kind are $A_\varepsilon$-algebras in which the image $\zeta$ of the uniformizer z of $A_\varepsilon$ is nilpotent. We denote the category of these $A_\varepsilon$-algebras by ${{\,\mathrm{\mathcal{{N}}\!{\textit{ilp}}}\,}}_{A_\varepsilon }$.

• (b) Let K be a field which is complete with respect to a non-trivial, non-Archimedean absolute value $|\,.\,|:K\rightarrow {\mathbb {R}}_{\ge 0}$ and let $\mathcal{{O}}_K=\{x\in K:|x|\le 1\}$ be the valuation ring of K. We make $\mathcal{{O}}_K$ into an $A_\varepsilon$-algebra via an injective ring homomorphism $\gamma :A_\varepsilon \hookrightarrow \mathcal{{O}}_K$ such that $\zeta :=\gamma (z)\ne 0$ lies in the maximal ideal ${\mathfrak {m}}_K\subset \mathcal{{O}}_K$.

The relation between the two kinds of base rings is that $\mathcal{{O}}_K/(\zeta ^n)\in {{\,\mathrm{\mathcal{{N}}\!{\textit{ilp}}}\,}}_{A_\varepsilon }$ for all positive integers n.

Let R be a base ring as in (a) or (b). To define local shtukas over R, we consider modules $\hat{M}$ over the power series ring $R[\![ z]\!]$, which Zariski locally on ${{\,\mathrm{Spec}\,}}R$ are free over $R[\![ z]\!]$. We call such a module a locally free $R[\![ z]\!]$-module of rank r. We set $\hat{M}[\tfrac{1}{z-\zeta }]:=\hat{M}\otimes _{R[\![ z]\!]}R[\![ z]\!][\tfrac{1}{z-\zeta }]$, and $\hat{M}[\tfrac{1}{z}]:=\hat{M}\otimes _{R[\![ z]\!]}R(\!(z)\!)$ where $R(\!(z)\!):=R[\![ z]\!][\tfrac{1}{z}]$, and $\hat{\sigma }^*\hat{M}:=\hat{M}\otimes _{R[\![ z]\!],\,\hat{\sigma }}R[\![ z]\!]$ where $\hat{\sigma }$ is the endomorphism of $R[\![ z]\!]$ with $\hat{\sigma }(z)=z$ and $\hat{\sigma }(b)=b^{\hat{q}}$ for $b\in R$. Note that $R[\![ z]\!][\tfrac{1}{z-\zeta }]=R(\!(z)\!)$ if $R\in {{\,\mathrm{\mathcal{{N}}\!{\textit{ilp}}}\,}}_{A_\varepsilon }$ as in (a), but $R[\![ z]\!][\tfrac{1}{z-\zeta }]\ne R(\!(z)\!)$ if R is a valuation ring as in (b). There is a natural $\hat{\sigma }$-semilinear map $\hat{M}\rightarrow \hat{\sigma }^*\hat{M},\,m\mapsto \hat{\sigma }_{\!\hat{M}}^*m:=m\otimes 1$. For a morphism of $R[\![ z]\!]$-modules $f:\hat{M}\rightarrow \hat{M}'$, we set $\hat{\sigma }^*f:=f\otimes {{\,\mathrm{\,id}\,}}:\hat{\sigma }^*\hat{M}\rightarrow \hat{\sigma }^*\hat{M}'$.

Definition 2.1

A local $\hat{\sigma }$-shtuka (or local shtuka) of rank r over R is a pair ${\underline{\hat{M}\!}\,}=(\hat{M},\tau _{\hat{M}})$ consisting of a locally free $R[\![ z]\!]$-module $\hat{M}$ of rank r, and an isomorphism . If $\tau _{\hat{M}}(\hat{\sigma }^*\hat{M})\subset \hat{M}$ then ${\underline{\hat{M}\!}\,}$ is called effective, and if $\tau _{\hat{M}}(\hat{\sigma }^*\hat{M})=\hat{M}$ then ${\underline{\hat{M}\!}\,}$ is called étale. We say that $\tau _{\hat{M}}$ is topologically nilpotent, if ${\underline{\hat{M}\!}\,}$ is effective and there is an integer n such that ${{\,\mathrm{im}\,}}(\tau _{\hat{M}}^n)\subset z\hat{M}$, where $\tau _{\hat{M}}^n:=\tau _{\hat{M}}\circ \hat{\sigma }^*\tau _{\hat{M}}\circ \ldots \circ \hat{\sigma }^{(n-1)*}\tau _{\hat{M}}:\hat{\sigma }^{n*}\hat{M} \rightarrow \hat{M}$.

A morphism of local shtukas $f:(\hat{M},\tau _{\hat{M}})\rightarrow (\hat{M}',\tau _{\hat{M}'})$ over R is a morphism of $R[\![ z]\!]$-modules $f:\hat{M}\rightarrow \hat{M}'$ which satisfies $\tau _{\hat{M}'}\circ \hat{\sigma }^*f = f\circ \tau _{\hat{M}}$. We denote the set of morphisms from ${\underline{\hat{M}\!}\,}$ to ${\underline{\hat{M}\!}\,}'$ by ${{\,\mathrm{Hom}\,}}_R({\underline{\hat{M}\!}\,},{\underline{\hat{M}\!}\,}')$.

A quasi-morphism between local shtukas $f:(\hat{M},\tau _{\hat{M}})\rightarrow (\hat{M}',\tau _{\hat{M}'})$ over R is a morphism of $R(\!(z)\!)$-modules $f:\hat{M}[\tfrac{1}{z}]\rightarrow \hat{M}'[\tfrac{1}{z}]$ with $\tau _{\hat{M}'}\circ \hat{\sigma }^*f=f\circ \tau _{\hat{M}}$. It is called a quasi-isogeny if it is an isomorphism of $R(\!(z)\!)$-modules. We denote the set of quasi-morphisms from ${\underline{\hat{M}\!}\,}$ to ${\underline{\hat{M}\!}\,}'$ by ${{\,\mathrm{QHom}\,}}_R({\underline{\hat{M}\!}\,},{\underline{\hat{M}\!}\,}')$.

For any local shtuka $(\hat{M},\tau _{\hat{M}})$ over $R\in {{\,\mathrm{\mathcal{{N}}\!{\textit{ilp}}}\,}}_{A_\varepsilon }$, the homomorphism $\hat{M}\rightarrow \hat{M}[\tfrac{1}{z-\zeta }]$ is injective by the flatness of $\hat{M}$ and the following.

Lemma 2.2

([21, Lemma 2.2]) Let R be an $A_\varepsilon$-algebra as in (a) or (b). Then the sequence of $R[\![ z]\!]$-modules

is exact. In particular, $R[\![ z]\!]\subset R[\![ z]\!][\frac{1}{z-\zeta }]$.

Of fundamental importance is the following.

Example 2.3

Let ${\mathbb {F}}_q$ be a finite field with q elements, let C be a smooth projective geometrically irreducible curve over ${\mathbb {F}}_q$, and let $Q:={\mathbb {F}}_q(C)$ be the function field of C. Fix a closed point $\infty$ of C, and let $A:=\Gamma (C\smallsetminus \{\infty \},\mathcal{{O}}_C)$ be the ring of regular functions on C outside $\infty$. The rings A and Q are the function field analogs of ${\mathbb {Z}}$ and ${\mathbb {Q}}$.

Let $\varepsilon \subset A$ be a maximal ideal and let $A_\varepsilon$ be the completion of A at $\varepsilon$. Then ${\mathbb {F}}_\varepsilon$ is a field extension of ${\mathbb {F}}_q$ with $\hat{q}:=\#{\mathbb {F}}_\varepsilon =q^{[{\mathbb {F}}_\varepsilon :{\mathbb {F}}_q]}$. Let R be a base $A_\varepsilon$-algebra as in (a) or (b) and denote its structure morphism by $\gamma :A_\varepsilon \rightarrow R$. Set $A_R:=$ $A\otimes _{{\mathbb {F}}_q}R$ and let $\sigma :={{\,\mathrm{\,id}\,}}_A\otimes {{\,\mathrm{Frob}\,}}_{q,R}$ be the endomorphism of $A_R$ with $\sigma (a\otimes b)=a\otimes b^{q}$ for $a\in A$ and $b\in R$. An effective A-motive of rank r over R is a pair ${\underline{M\!}\,}=(M,\tau _M)$ consisting of a locally free $A_R$-module M of rank r and an injective $A_R$-homomorphism $\tau _M:\sigma ^*M\hookrightarrow M$ whose cokernel is a finite free R-module and is annihilated by a power of the ideal $\mathcal{{J}}:=(a\otimes 1-1\otimes \gamma (a):a\in A)=\ker (\gamma \otimes {{\,\mathrm{\,id}\,}}_R:A_R\twoheadrightarrow R)\subset A_R$.

More generally, an A-motive of rank r over R is a pair ${\underline{M\!}\,}=(M,\tau _M)$ consisting of a locally free $A_R$-module M of rank r and an isomorphism of the associated sheaves outside ${{\,\mathrm{V}\,}}(\mathcal{{J}})\subset {{\,\mathrm{Spec}\,}}A_R$. Note that if $A={\mathbb {F}}_q[t]$, then $\mathcal{{J}}=(t-\gamma (t))$ and ${{\,\mathrm{Spec}\,}}A_R\smallsetminus {{\,\mathrm{V}\,}}(\mathcal{{J}})={{\,\mathrm{Spec}\,}}R[t][\tfrac{1}{t-\gamma (t)}]$.

Let ${\underline{M\!}\,}$ be an (effective) A-motive over R. We consider the $\varepsilon$-adic completions $A_{\varepsilon ,R}=\displaystyle \lim _{\longleftarrow } A_R/\varepsilon ^n A_R$ of $A_R$ and ${\underline{M\!}\,}\otimes _{A_R}A_{\varepsilon ,R}:=(M\otimes _{A_R}A_{\varepsilon ,R}\,,\,\tau _M\otimes {{\,\mathrm{\,id}\,}})$ of ${\underline{M\!}\,}$. If ${\mathbb {F}}_\varepsilon ={\mathbb {F}}_q$, and hence $\hat{q}=q$ and $\hat{\sigma }=\sigma$, we have $A_{\varepsilon ,R}=R[\![ z]\!]$ and $\mathcal{{J}}\cdot A_{\varepsilon ,R}=(z-\zeta )$ because $R\otimes _{A_R}A_{\varepsilon ,R}=R$. So ${\underline{M\!}\,}\otimes _{A_R}A_{\varepsilon ,R}$ is an (effective) local shtuka over R which we denote by ${\underline{\hat{M}\!}\,}_\varepsilon ({\underline{M\!}\,})$ and call the local $\hat{\sigma }$-shtuka at $\varepsilon$ associated with ${\underline{M\!}\,}$. If $f:=[{\mathbb {F}}_\varepsilon :{\mathbb {F}}_q]>1$, the construction is slightly more complicated; compare the discussion in [4, after Proposition 8.4]. Namely, we consider the canonical isomorphism and the ideals ${\mathfrak {a}}_i=(a\otimes 1-1\otimes \gamma (a)^{q^i}:a\in {\mathbb {F}}_\varepsilon )\subset A_{\varepsilon ,R}$ for $i\in {\mathbb {Z}}/f{\mathbb {Z}}$, which satisfy $\prod _{i\in {\mathbb {Z}}/f{\mathbb {Z}}}{\mathfrak {a}}_i=(0)$, because $\prod _{i\in {\mathbb {Z}}/f{\mathbb {Z}}}(X-a^{q^i})\in {\mathbb {F}}_q[X]$ is a multiple of the minimal polynomial of a over ${\mathbb {F}}_q$ and even equal to it when ${\mathbb {F}}_\varepsilon ={\mathbb {F}}_q(a)$. By the Chinese remainder theorem, $A_{\varepsilon ,R}$ decomposes

$$\begin{aligned} A_{\varepsilon ,R}\;=\;\prod _{i\in {\mathbb {Z}}/f{\mathbb {Z}}}A_{\varepsilon ,R}/{\mathfrak {a}}_i\,. \end{aligned}$$

(2.1)

Each factor is canonically isomorphic to $R[\![ z]\!]$. The factors are cyclically permuted by $\sigma$ because $\sigma ({\mathfrak {a}}_i)={\mathfrak {a}}_{i+1}$. In particular, $\sigma ^f$ stabilizes each factor. The ideal $\mathcal{{J}}$ decomposes as follows: $\mathcal{{J}}\!\cdot \! A_{\varepsilon ,R}/{\mathfrak {a}}_0=(z-\zeta )$ and $\mathcal{{J}}\!\cdot \! A_{\varepsilon ,R}/{\mathfrak {a}}_i=(1)$ for $i\ne 0$. We define the local $\hat{\sigma }$-shtuka at $\varepsilon$ associated with ${\underline{M\!}\,}$ as ${\underline{\hat{M}\!}\,}_\varepsilon ({\underline{M\!}\,}):=(\hat{M},\tau _{\hat{M}}):$ $=\bigl (M\otimes _{A_R}A_{\varepsilon ,R}/{\mathfrak {a}}_0\,,\,(\tau _M\otimes 1)^f\bigr )$, where $\tau _M^f:=\tau _M\circ \sigma ^*\tau _M\circ \ldots \circ \sigma ^{(f-1)*}\tau _M$. Of course, if $f=1$ we get back the definition of ${\underline{\hat{M}\!}\,}_\varepsilon ({\underline{M\!}\,})$ given above. Also note if ${\underline{M\!}\,}$ is effective, then $M/\tau _M(\sigma ^*M)=\hat{M}/\tau _{\hat{M}}(\hat{\sigma }^*\hat{M})$.

The local shtuka ${\underline{\hat{M}\!}\,}_\varepsilon ({\underline{M\!}\,})$ allows to recover ${\underline{M\!}\,}\otimes _{A_R}A_{\varepsilon ,R}$ via the isomorphism

because for $i\ne 0$ the equality $\mathcal{{J}}\!\cdot \! A_{\varepsilon ,R}/{\mathfrak {a}}_i=(1)$ implies that $\tau _M\otimes 1$ is an isomorphism modulo ${\mathfrak {a}}_i$; see [4, Propositions 8.8 and 8.5] for more details. Note that ${\underline{M\!}\,}\mapsto {\underline{\hat{M}\!}\,}_\varepsilon ({\underline{M\!}\,})$ is a functor.

We quote the next lemma from [21, Lemma 2.3].

Lemma 2.4

Let $(\hat{M},\tau _{\hat{M}})$ be a local shtuka over R. Then there are $e,e'\in {\mathbb {Z}}$ such that $(z-\zeta )^{e'} \hat{M} \subset \tau _{\hat{M}}(\hat{\sigma }^*{\hat{M}}) \subset (z-\zeta )^{-e} \hat{M}$. For any such e, the map $\tau _{\hat{M}}:$ $\hat{\sigma }^*{\hat{M}}\rightarrow (z-\zeta )^{-e}\hat{M}$ is injective, and the quotient $(z-\zeta )^{-e} \hat{M}/\tau _M(\hat{\sigma }^*\hat{M})$ is a locally free R-module of finite rank.

Example 2.5

We discuss the case of the Carlitz module [5]. We keep the notation from Example 2.3 and set $A={\mathbb {F}}_q[t]$. Let ${\mathbb {F}}_q(\theta )$ be the rational function field in the variable $\theta$ and let $\gamma :A\rightarrow {\mathbb {F}}_q(\theta )$ be given by $\gamma (t)=\theta$. The Carlitz motive over ${\mathbb {F}}_q(\theta )$ is the A-motive ${\underline{M\!}\,}=\bigl ({\mathbb {F}}_q(\theta )[t],t-\theta \bigr )$.

Now let $\varepsilon =(z)\subset A$ be a maximal ideal generated by a monic prime element $z=z(t)\in {\mathbb {F}}_q[t]$. Then ${\mathbb {F}}_\varepsilon =A/(z)$ and $A_\varepsilon$ is canonically isomorphic to ${\mathbb {F}}_\varepsilon [\![ z]\!]$. Let $\mathcal{{O}}_K\supset {\mathbb {F}}_\varepsilon [\![ \zeta ]\!]$ be a valuation ring as in (b) and let $\theta =\gamma (t)\in \mathcal{{O}}_K$. The Carlitz motive has good reduction in the sense that it has a model over $\mathcal{{O}}_K$ given by the A-motive ${\underline{M\!}\,}=(\mathcal{{O}}_K[t],t-\theta )$ over $\mathcal{{O}}_K$.

If $\deg _t z(t)=1$, that is, $z(t)=t-a$ for $a\in {\mathbb {F}}_q$, then ${\mathbb {F}}_\varepsilon ={\mathbb {F}}_q$, $\zeta =\theta -a$, and $z-\zeta =t-\theta$. So ${\underline{\hat{M}\!}\,}_\varepsilon ({\underline{M\!}\,})=(\mathcal{{O}}_K[\![ z]\!],z-\zeta )$.

If $\deg _t z(t)=f>1$, then ${\underline{\hat{M}\!}\,}_\varepsilon ({\underline{M\!}\,})=\bigl (\mathcal{{O}}_K[\![ z]\!],(t-\theta )(t-\theta ^q)\cdots (t-\theta ^{q^{f-1}})\bigr )$. Here, the product $(t-\theta )(t-\theta ^q)\cdots (t-\theta ^{q^{f-1}})=(z-\zeta )u$ for a unit $u\in$ ${\mathbb {F}}_\varepsilon [\![ \zeta ]\!][\![ z]\!]^{\scriptscriptstyle \times }$, because $\tau _M(\sigma ^*M)=(t-\theta )M$ implies that ${\underline{\hat{M}\!}\,}_\varepsilon ({\underline{M\!}\,})$ is effective and $\hat{M}/\tau _{\hat{M}}(\hat{\sigma }^*\hat{M})=M/\tau _M(\sigma ^*M)$ is free over $\mathcal{{O}}_K$ of rank 1. In order to get rid of u, we denote the image of t in ${\mathbb {F}}_\varepsilon$ by $\lambda$. Then ${\mathbb {F}}_\varepsilon ={\mathbb {F}}_q(\lambda )$ and z(t) equals the minimal polynomial $(t-\lambda )\cdots (t-\lambda ^{q^{f-1}})$ of $\lambda$ over ${\mathbb {F}}_q$. Moreover, $t\equiv \lambda \;\mathrm{mod}\;zA_\varepsilon$ and $\theta \equiv \lambda \;\mathrm{mod}\;\zeta {\mathbb {F}}_\varepsilon [\![ \zeta ]\!]$. We compute in ${\mathbb {F}}_\varepsilon [\![ \zeta ]\!][\![ z]\!]/(\zeta )$

$$ z(t)\;=\;(t-\lambda )\cdots (t-\lambda ^{q^{f-1}})\;\equiv \;(t-\theta )\cdots (t-\theta ^{q^{f-1}})\;\equiv \;(z-\zeta )u\;\equiv \;zu\;\;\mathrm{mod}\;\zeta \,. $$

Since z is a non-zero-divisor in ${\mathbb {F}}_\varepsilon [\![ \zeta ]\!][\![ z]\!]/(\zeta )$, it follows that $u\equiv$ $1\;\mathrm{mod}\;\zeta \,{\mathbb {F}}_\varepsilon [\![ \zeta ]\!][\![ z]\!]$. We write $u=1+\zeta u'$ and observe that the product

$$ w\;:=\;\prod _{n=0}^\infty \hat{\sigma }^n(u)\;=\;\prod _{n=0}^\infty \hat{\sigma }^n(1+\zeta u')\;=\;\prod _{n=0}^\infty \bigl (1+\zeta ^{\hat{q}^n}\hat{\sigma }^n(u')\bigr ) $$

converges in ${\mathbb {F}}_\varepsilon [\![ \zeta ]\!][\![ z]\!]^{\scriptscriptstyle \times }$ because ${\mathbb {F}}_\varepsilon [\![ \zeta ]\!][\![ z]\!]$ is $\zeta$-adically complete. It satisfies $w=u\cdot \hat{\sigma }(w)$ and so multiplication with w defines a canonical isomorphism .

We conclude that ${\underline{\hat{M}\!}\,}_\varepsilon ({\underline{M\!}\,})=(\mathcal{{O}}_K[\![ z]\!],z-\zeta )$, regardless of $\deg _t z(t)$.

3 Finite Shtukas

In this section, let R be an arbitrary ${\mathbb {F}}_\varepsilon$-algebra. For an R-module $\hat{M}$ we set $\hat{\sigma }^*\hat{M}:=\hat{M}\otimes _{R,\,{{\,\mathrm{Frob}\,}}_{\hat{q}}}R$ where ${{\,\mathrm{Frob}\,}}_{\hat{q}}$ is the $\hat{q}$-Frobenius endomorphism of R with ${{\,\mathrm{Frob}\,}}_{\hat{q}}(b)=b^{\hat{q}}$ for $b\in R$.

Definition 3.1

A finite ${\mathbb {F}}_\varepsilon$-shtuka over R is a pair ${\underline{\hat{M}\!}\,}=(\hat{M},\tau _{\hat{M}})$ consisting of a locally free R-module $\hat{M}$ of finite rank denoted by ${{\,\mathrm{rk}\,}}{\underline{\hat{M}\!}\,}$, and an R-module homomorphism $\tau _{\hat{M}}:\hat{\sigma }^*\hat{M}\rightarrow \hat{M}$ satisfying $f\circ \tau _{\hat{M}}=\tau _{\hat{M}'}\circ \hat{\sigma }^*f$. That is, the following diagram is commutative

A finite ${\mathbb {F}}_\varepsilon$-shtuka over R is called étale if $\tau _{\hat{M}}$ is an isomorphism. We say that $\tau _{\hat{M}}$ is nilpotent if there is an integer n such that $\tau _{\hat{M}}^n:=\tau _{\hat{M}}\circ \hat{\sigma }^*\tau _{\hat{M}}\circ \ldots \circ \sigma _{\!q^{n-1}}^*\tau _{\hat{M}}=0$.

Finite ${\mathbb {F}}_\varepsilon$-shtukas were studied at various places in the literature. They were called “(finite) $\varphi$-sheaves” by Drinfeld [7, § 2], Taguchi and Wan [31, 32], and “Dieudonné ${\mathbb {F}}_q$-modules” by Laumon [25]. Finite ${\mathbb {F}}_\varepsilon$-shtukas over a field admit a canonical decomposition.

Proposition 3.2

([25, Lemma B.3.10]) If $R=L$ is a field, every finite ${\mathbb {F}}_\varepsilon$-shtuka $\hat{\underline{M\!}\,}=(\hat{M},\tau _{\hat{M}})$ is canonically an extension of finite ${\mathbb {F}}_\varepsilon$-shtukas

$$ 0\longrightarrow (\hat{M}_{\acute{\mathrm{e}}\mathrm{t}},\tau _{\acute{\mathrm{e}}\mathrm{t}}) \longrightarrow (\hat{M},\tau _{\hat{M}})\longrightarrow (\hat{M}_\mathrm{nil},\tau _\mathrm{nil})\longrightarrow 0 $$

where $\tau _{\acute{\mathrm{e}}\mathrm{t}}$ is an isomorphism and $\tau _\mathrm{nil}$ is nilpotent. $\hat{\underline{M\!}\,}_{\acute{\mathrm{e}}\mathrm{t}}=(\hat{M}_{\acute{\mathrm{e}}\mathrm{t}},\tau _{\acute{\mathrm{e}}\mathrm{t}})$ is the largest étale finite ${\mathbb {F}}_q$-sub-shtuka of $\hat{\underline{M\!}\,}$ and equals ${{\,\mathrm{im}\,}}(\tau _{\hat{M}}^{{{\,\mathrm{rk}\,}}\hat{\underline{M\!}\,}})$. If L is perfect, this extension splits canonically.

Example 3.3

Every effective local shtuka $(\hat{M},\tau _{\hat{M}})$ of rank r over R yields for every $n\in {\mathbb {N}}$ a finite ${\mathbb {F}}_\varepsilon$-shtuka $\bigl (\hat{M}/z^n\hat{M},\tau _{\hat{M}}\;\mathrm{mod}\;z^n\bigr )$ of rank rn, and $(\hat{M},\tau _{\hat{M}})$ equals the projective limit of these finite ${\mathbb {F}}_\varepsilon$-shtukas.

Thus, from Proposition 3.2 we obtain the following.

Proposition 3.4

If $R=L$ is a field in ${{\,\mathrm{\mathcal{{N}}\!{\textit{ilp}}}\,}}_{A_\varepsilon }$, that is, $\zeta =0$ in L, then every effective local shtuka $(\hat{M},\tau _{\hat{M}})$ is canonically an extension of effective local shtukas

$$ 0\longrightarrow (\hat{M}_{\acute{\mathrm{e}}\mathrm{t}},\tau _{\acute{\mathrm{e}}\mathrm{t}}) \longrightarrow (\hat{M},\tau _{\hat{M}})\longrightarrow (\hat{M}_\mathrm{nil},\tau _\mathrm{nil})\longrightarrow 0 $$

where $\tau _{\acute{\mathrm{e}}t}$ is an isomorphism and $\tau _\mathrm{nil}$ is topologically nilpotent. $(\hat{M}_{\acute{\mathrm{e}}t},\tau _{\acute{\mathrm{e}}t})$ is the largest étale effective local sub-shtuka of $(\hat{M},\tau _{\hat{M}})$. If L is perfect, this extension splits canonically. $\square$

Finite ${\mathbb {F}}_\varepsilon$-shtukas and local shtukas are related to group schemes in the following way. Let ${\underline{\hat{M}\!}\,}=(\hat{M},\tau _{\hat{M}})$ be a finite ${\mathbb {F}}_\varepsilon$-shtuka over R. Let

$$ E={{\,\mathrm{Spec}\,}}\,\bigoplus _{n\ge 0}\,{{\,\mathrm{Sym}\,}}^n_{R} \hat{M} $$

be the geometric vector bundle corresponding to $\hat{M}$ over ${{\,\mathrm{Spec}\,}}R$, and let $F_{\hat{q},E}:E\rightarrow \hat{\sigma }^*E$ be its relative $\hat{q}$-Frobenius morphism over R. On the other hand, the map $\tau _{\hat{M}}$ induces another R-morphism ${{\,\mathrm{Spec}\,}}({{\,\mathrm{Sym}\,}}^\bullet \tau _{\hat{M}}):E\rightarrow \hat{\sigma }^*E$. Drinfeld defines

$$ {{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,}){:=}\ker \bigl (\!{{\,\mathrm{Spec}\,}}({{\,\mathrm{Sym}\,}}^\bullet \tau _{\hat{M}})-F_{\hat{q},E}:E{\rightarrow }\hat{\sigma }^*E\!\bigr ){=}{{\,\mathrm{Spec}\,}}\;\bigl (\bigoplus _{n\ge 0}{{\,\mathrm{Sym}\,}}^n_{R} \hat{M}\bigr )/I $$

where the ideal I is generated by the elements $m^{\otimes q}-\tau _{\hat{M}}(\hat{\sigma }^*m)$ for all elements m of $\hat{M}$. (Here, $m^{\otimes q}$ lives in ${{\,\mathrm{Sym}\,}}^q_{R} \hat{M}$ and $\tau _{\hat{M}}(\hat{\sigma }^*m)$ in ${{\,\mathrm{Sym}\,}}^1_{R} \hat{M}$.) Note that locally on ${{\,\mathrm{Spec}\,}}R$, we have $\hat{M}=\bigoplus _{i=1}^{d} R\cdot m_i$ and $E\cong {{\,\mathrm{Spec}\,}}R[m_1,\ldots ,m_{d}]={\mathbb {G}}_{a,R}^{d}$. The subgroup scheme ${{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,})$ is finite locally free over R of order $\hat{q}^{{{\,\mathrm{rk}\,}}{\underline{\hat{M}\!}\,}}$, that is, the R-algebra $\mathcal{{O}}_{{{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,})}$ is a finite locally free R-module of rank $\hat{q}^{{{\,\mathrm{rk}\,}}{\underline{\hat{M}\!}\,}}$. It is also an ${\mathbb {F}}_\varepsilon$-module scheme over R via the comultiplication $\Delta :m\mapsto m\otimes 1+1\otimes m$ and the ${\mathbb {F}}_\varepsilon$-action $[a]:m\mapsto am$ which it inherits from E. It is even a strict ${\mathbb {F}}_\varepsilon$-module scheme in the sense of Faltings [9] and Abrashkin [2]. For a proof, see [2, Theorem 2] or [21, § 5]. This means that ${\mathbb {F}}_\varepsilon$ acts on the co-Lie complex of ${{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,})$ over R, see Illusie [26, § VII.3.1], via the scalar multiplication through ${\mathbb {F}}_\varepsilon \subset R$. A detailed explanation of strict ${\mathbb {F}}_\varepsilon$-module schemes is given in [21, § 4].

Conversely, let $G={{\,\mathrm{Spec}\,}}A$ be a finite locally free strict ${\mathbb {F}}_\varepsilon$-module scheme over R. Note that on the additive group scheme ${\mathbb {G}}_{a,R}={{\,\mathrm{Spec}\,}}R[x]$, the elements $b\in R$ act via endomorphisms $\psi _b:{\mathbb {G}}_{a,R}\rightarrow {\mathbb {G}}_{a,R}$ given by $\psi _b^*:R[x]\rightarrow R[x],\,x\mapsto bx$. This makes ${\mathbb {G}}_{a,R}$ into an R-module scheme, and in particular, into an ${\mathbb {F}}_\varepsilon$-module scheme via ${\mathbb {F}}_\varepsilon \subset R$. We associate with G the R-module of ${\mathbb {F}}_\varepsilon$-equivariant homomorphisms on R

$$ \hat{M}_{\hat{q}}(G) :={{\,\mathrm{Hom}\,}}_{R\text {-groups},{\mathbb {F}}_\varepsilon \text {-lin}}(G,{\mathbb {G}}_{a,R}) = \bigl \{ x \in A:\Delta (x) = x \otimes 1 + 1 \otimes x,\ [a](x) = a x,\ \forall a \in {\mathbb {F}}_\varepsilon \bigr \}\,, $$

with its action of R via $R\rightarrow {{\,\mathrm{End}\,}}_{R\text {-groups},{\mathbb {F}}_\varepsilon \text {-lin}}({\mathbb {G}}_{a,R})$. It is a finite locally free R-module by [30, Proposition 3.6 and Remark 5.5]; see also [1, VII$_\mathrm{A}$, 7.4.3] in the reedited version of SGA 3 by P. Gille and P. Polo. The composition on the left with the relative $\hat{q}$-Frobenius endomorphism $F_{\hat{q},{\mathbb {G}}_{a,R}}$ of ${\mathbb {G}}_{a,R}={{\,\mathrm{Spec}\,}}R[x]$ given by $x\mapsto x^{\hat{q}}$ defines a map $\hat{M}_{\hat{q}}(G)\rightarrow \hat{M}_{\hat{q}}(G), m\mapsto F_{\hat{q},{\mathbb {G}}_{a,R}}\circ m$ which is not R-linear, but $\hat{\sigma }$-linear, because $F_{\hat{q},{\mathbb {G}}_{a,R}}\circ \psi _b=\psi _{b^{\hat{q}}}\circ F_{\hat{q},{\mathbb {G}}_{a,R}}$. Therefore, $F_{\hat{q},{\mathbb {G}}_{a,R}}$ induces an R-homomorphism $\tau _{\hat{M}_{\hat{q}}(G)}:\hat{\sigma }^*\hat{M}_{\hat{q}}(G)\rightarrow \hat{M}_{\hat{q}}(G)$. Then ${\underline{\hat{M}\!}\,}_{\hat{q}}(G):=\bigl (\hat{M}_{\hat{q}}(G),\tau _{\hat{M}_{\hat{q}}(G)}\bigr )$ is a finite ${\mathbb {F}}_\varepsilon$-shtuka over R. If $f:G \rightarrow H$ is a morphism of finite locally free strict ${\mathbb {F}}_\varepsilon$-module schemes over R, then ${\underline{\hat{M}\!}\,}_{\hat{q}}(f):{\underline{\hat{M}\!}\,}_{\hat{q}}(H)\rightarrow {\underline{\hat{M}\!}\,}_{\hat{q}}(G),\,m\mapsto m\circ f$. This defines the functor ${\underline{\hat{M}\!}\,}_{\hat{q}}$ from the category of finite locally free strict ${\mathbb {F}}_\varepsilon$-module schemes over R to finite ${\mathbb {F}}_\varepsilon$-shtukas over R. It has the following properties.

Theorem 3.5

([21, Theorem 5.2])

• (a) The contravariant functors ${{\,\mathrm{Dr}\,}}_{\hat{q}}$ and ${\underline{\hat{M}\!}\,}_{\hat{q}}$ are mutually quasi-inverse anti-equivalences between the category of finite ${\mathbb {F}}_\varepsilon$-shtukas over R and the category of finite locally free strict ${\mathbb {F}}_\varepsilon$-module schemes over R.

• (b) Both functors are ${\mathbb {F}}_q$-linear and map short exact sequences to short exact sequences. They preserve étale objects and map the canonical decompositions from Propositions 3.2 and 3.6 below to each other.

Let ${\underline{\hat{M}\!}\,}=(\hat{M},\tau _{\hat{M}})$ be a finite ${\mathbb {F}}_\varepsilon$-shtuka over R and let $G={{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,})$. Then

• (c) The ${\mathbb {F}}_\varepsilon$-module scheme ${{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,})$ is radical over R if and only if $\tau _{\hat{M}}$ is nilpotent.

• (d) The order of the R-group scheme ${{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,})$ is $\hat{q}^{{{\,\mathrm{rk}\,}}{\underline{\hat{M}\!}\,}}$.

• (e) There is a canonical isomorphism between ${{\,\mathrm{coker}\,}}\tau _{\hat{M}}=\hat{M}/\tau _{\hat{M}}(\hat{\sigma }^*\hat{M})$ and the co-Lie module $\omega _{{{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,})}:=e^*\Omega ^1_{{{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,})/R}$ where $e:{{\,\mathrm{Spec}\,}}R\rightarrow {{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,})$ is the zero section.

Proposition 3.6

([21, Proposition 4.2]) If $R=L$ is a field, every ${\mathbb {F}}_\varepsilon$-module scheme G over L is canonically an extension $0\rightarrow G^\circ \rightarrow G\rightarrow G^{\acute{\mathrm{e}}t}\rightarrow 0$ of an étale ${\mathbb {F}}_\varepsilon$-module scheme $G^{\acute{\mathrm{e}}t}$ by a connected ${\mathbb {F}}_\varepsilon$-module scheme $G^\circ$. The ${\mathbb {F}}_\varepsilon$-module scheme $G^{\acute{\mathrm{e}}t}$ is the largest étale quotient of G. If L is perfect, $G^{\acute{\mathrm{e}}t}$ is canonically isomorphic to the reduced closed ${\mathbb {F}}_\varepsilon$-module subscheme $G^\mathrm{red}$ of G and the extension splits canonically, $G=G^\circ \times _L G^\mathrm{red}$.

4 Divisible Local Anderson Modules

Let $R\in {{\,\mathrm{\mathcal{{N}}\!{\textit{ilp}}}\,}}_{A_\varepsilon }$ and let ${\underline{\hat{M}\!}\,}=(\hat{M},\tau _{\hat{M}})$ be an effective local shtuka over R. Set ${\underline{\hat{M}\!}\,}_n:=(\hat{M}_n,\tau _{\hat{M}_n}):=(\hat{M}/z^n\hat{M},\tau _{\hat{M}}\;\mathrm{mod}\;z^n)$ and consider the finite locally free strict ${\mathbb {F}}_\varepsilon$-module scheme ${{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,}_n)$ over R from the previous section. ${{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,}_n)$ inherits from ${\underline{\hat{M}\!}\,}_n$ an action of $A_\varepsilon /(z^n)={\mathbb {F}}_\varepsilon [z]/(z^n)$. The canonical epimorphisms ${\underline{\hat{M}\!}\,}_{n+1}\twoheadrightarrow {\underline{\hat{M}\!}\,}_n$ induce closed immersions $i_n:{{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,}_n)\hookrightarrow {{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,}_{n+1})$. The inductive limit ${{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,}):=\displaystyle \lim _{\longrightarrow } {{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,}_n)$ in the category of sheaves on the big fppf-site of ${{\,\mathrm{Spec}\,}}R$ is a sheaf of $A_\varepsilon$-modules that satisfies the following.

Definition 4.1

A z-divisible local Anderson module over R is a sheaf of $A_\varepsilon$-modules G on the big fppf-site of ${{\,\mathrm{Spec}\,}}R$ such that

• (a) G is z-torsion, that is, $G = \displaystyle \lim _{\longrightarrow } G[z^n]$, where $G[z^n]:=\ker (z^n:G\rightarrow G)$;

• (b) G is z-divisible, that is, $z:G \rightarrow G$ is an epimorphism;

• (c) For every n, the ${\mathbb {F}}_\varepsilon$-module $G[z^n]$ is representable by a finite locally free strict ${\mathbb {F}}_\varepsilon$-module scheme over R in the sense of Faltings [9] and Abrashkin [2];

• (d) Locally on R, there exists an integer $d \in {\mathbb {Z}}_{\ge 0}$, such that $(z-\zeta )^d=0$ on $\omega _G$ where $\omega _G := \displaystyle \lim _{\longleftarrow } \omega _{G[z^n]}$ and $\omega _{G[z^n]}=e^*\Omega ^1_{G[z^n]/R}$ is the pullback under the zero section $e:{{\,\mathrm{Spec}\,}}R\rightarrow G[z^n]$. Here, the action of z on $\omega _G$ comes from the structure of $A_\varepsilon$-module on G, while the action of $\zeta$ on $\omega _G$ comes from the structure of R-module on $\omega _G$.

A morphism of z-divisible local Anderson modules over R is a morphism of fppf-sheaves of ${\mathbb {F}}_\varepsilon [\![ z]\!]$-modules. It is shown in [21, Lemma 8.2 and Theorem 10.8] that $\omega _G$ is a finite locally free R-module, and we define the dimension of G as ${{\,\mathrm{rk}\,}}\omega _G$ . Moreover, it follows from [21, Proposition 7.5] that there is a locally constant function $h:{{\,\mathrm{Spec}\,}}R\rightarrow {\mathbb {N}}_0,s\mapsto h(s)$ such that the order of $G[z^n]$ equals $\hat{q}^{nh}$. We call h the height of the z-divisible local Anderson module G.

The category of z-divisible local Anderson modules over R and the category of local shtukas over R are both $A_\varepsilon$-linear. The construction and the equivalence from Sect. 3 extend to an equivalence between the category of effective local shtukas over R and the category of z-divisible local Anderson modules over R.

The quasi-inverse functor to ${\underline{\hat{M}\!}\,}\mapsto {{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,})$ is given as follows. Let $G = \displaystyle \lim _{\longrightarrow } G[z^n]$ be a z-divisible local Anderson module over R. We set

$$ {\underline{\hat{M}\!}\,}_{\hat{q}}(G)=\bigl (\hat{M}_{\hat{q}}(G),\tau _{\hat{M}_{\hat{q}}(G)}\bigr ):=\displaystyle \lim _{\underset{n}{\longleftarrow }} \,\bigl (\hat{M}_{\hat{q}}(G[z^n]),\tau _{\hat{M}_{\hat{q}}(G[z^n])}\bigr )\,. $$

Multiplication with z on G gives $\hat{M}_{\hat{q}}(G)$, the structure of an $R[\![ z]\!]$-module. The following theorem was proved in [21, Theorem 8.3].

Theorem 4.2

Let $R\in {{\,\mathrm{\mathcal{{N}}\!{\textit{ilp}}}\,}}_{A_\varepsilon }$.

• (a) The two contravariant functors ${{\,\mathrm{Dr}\,}}_{\hat{q}}$ and ${\underline{\hat{M}\!}\,}_{\hat{q}}$ are mutually quasi-inverse anti-equivalences between the category of effective local shtukas over R and the category of z-divisible local Anderson modules over R.

• (b) Both functors are $A_\varepsilon$-linear, map short exact sequences to short exact sequences, and preserve (ind-) étale objects.

Let ${\underline{\hat{M}\!}\,}=(\hat{M},\tau _{\hat{M}})$ be an effective local shtuka over R, and let $G={{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,})$ be its associated z-divisible local Anderson module. Then

• (c) G is a formal $A_\varepsilon$-module, i.e. a formal Lie group equipped with an action of $A_\varepsilon$, if and only if $\tau _{\hat{M}}$ is topologically nilpotent.

• (d) The height and dimension of G are equal to the rank and dimension of ${\underline{\hat{M}\!}\,}$.

• (e) The $R[\![ z]\!]$-modules $\omega _{{{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,})}$ and ${{\,\mathrm{coker}\,}}\tau _{\hat{M}}$ are canonically isomorphic.

Example 4.3

In the notation of Example 2.3, let $R\in {{\,\mathrm{\mathcal{{N}}\!{\textit{ilp}}}\,}}_{A_\varepsilon }$ and let r be a positive integer. A Drinfeld A-module of rank r over R is a pair ${\underline{E\!}\,}=(E,\varphi )$ consisting of a smooth affine group scheme E over ${{\,\mathrm{Spec}\,}}R$ of relative dimension 1 and a ring homomorphism $\varphi :A\rightarrow {{\,\mathrm{End}\,}}_{R\text {-groups}}(E),\,a\mapsto \varphi _a$ satisfying the following conditions:

• (a) Zariski-locally on ${{\,\mathrm{Spec}\,}}R$ there is an isomorphism of ${\mathbb {F}}_q$-module schemes such that

• (b) the coefficients of $\Phi _a:=\alpha \circ \varphi _a\circ \alpha ^{-1}{=}\sum \limits _{i\ge 0}b_i(a)\tau ^i{\in }{{\,\mathrm{End}\,}}_{R\text {-groups},{\mathbb {F}}_q\text {-lin}}({\mathbb {G}}_{a,R})=R\{\tau \}$ satisfy $b_0(a)=\gamma (a)$, $b_{r(a)}(a)\in R^{\scriptscriptstyle \times }$ and $b_i(a)$ is nilpotent for all $i>r(a):=-r\,[{\mathbb {F}}_\infty :{\mathbb {F}}_q]{{\,\mathrm{ord}\,}}_\infty (a)$.

Here, $R\{\tau \}:=\bigl \{\,\textstyle \sum \limits _{i=0}^nb_i\tau ^i:n\in {\mathbb {N}}_0, b_i\in R\,\bigr \}$ is the non-commutative polynomial ring with $\tau b=b^q\tau$, and the isomorphism of rings is given by sending $\tau$ to the relative $\hat{q}$-Frobenius endomorphism $F_{\hat{q},{\mathbb {G}}_{a,R}}$ of ${\mathbb {G}}_{a,R}={{\,\mathrm{Spec}\,}}R[x]$ given by $x\mapsto x^{\hat{q}}$ and $b\in R$ to the endomorphism $\psi _b$ given by $\psi _b^*:x\mapsto bx$.

For a Drinfeld A-module ${\underline{E\!}\,}=(E,\varphi )$, we consider the set $M:=M({\underline{E\!}\,}):={{\,\mathrm{Hom}\,}}_{R\text {-groups},{\mathbb {F}}_q\text {-lin}}(E,{\mathbb {G}}_{a,R})$ of ${\mathbb {F}}_q$-equivariant homomorphisms of R-group schemes. It is a locally free module over $A_R:=A\otimes _{{\mathbb {F}}_q}R$ of rank r under the action given on $m\in M$ by

$$ \begin{array}{rlll} A\ni a:&{} M\longrightarrow M, &{} m\mapsto m\circ \varphi _a &{} =: am\\ R\ni b:&{} M\longrightarrow M, &{} m\mapsto \psi _b\circ m &{} =: bm \end{array} $$

In addition, we consider the map $\tau :m\mapsto F_{q,{\mathbb {G}}_{a,R}}\!\circ \, m$ on $m\in M$, where $F_{q,{\mathbb {G}}_{a,R}}$ is the relative q-Frobenius of ${\mathbb {G}}_{a,R}$ over R. Since $F_{q,{\mathbb {G}}_{a,R}}\circ \psi _b=\psi _{b^q}\circ F_{q,{\mathbb {G}}_{a,R}}$, and hence $\tau (bm)=b^q\tau (m)$, the map $\tau$ is $\sigma$-semilinear and induces an $A_R$-linear map $\tau _M:\sigma ^*M\rightarrow M$, which makes ${\underline{M\!}\,}({\underline{E\!}\,}):=\bigl (M({\underline{E\!}\,}),\tau _M)$ into an effective A-motive over R in the sense of Example 2.3. The functor ${\underline{E\!}\,}\mapsto {\underline{M\!}\,}({\underline{E\!}\,})$ is fully faithful and its essential image is described in [18, Theorems 3.5 and 3.9] generalizing Anderson’s description [3, Theorem 1].

Now let ${\underline{\hat{M}\!}\,}:={\underline{\hat{M}\!}\,}_\varepsilon ({\underline{M\!}\,}({\underline{E\!}\,}))$ be the effective local $\hat{\sigma }$-shtuka at $\varepsilon$ associated with ${\underline{M\!}\,}({\underline{E\!}\,})$; see Example 2.3. Let $n\in {\mathbb {N}}$ and let $\varepsilon ^n=(a_1,\ldots ,a_s)\subset A$. Then

$$ {\underline{E\!}\,}[\varepsilon ^n]\;:=\;\ker \bigl (\varphi _{a_1,\ldots ,a_s}:=(\varphi _{a_1},\ldots ,\varphi _{a_s}):E \longrightarrow E^s\bigr ) $$

is called the $\varepsilon ^n$-torsion submodule of ${\underline{E\!}\,}$. It is an $A/\varepsilon ^n$-module via $A/\varepsilon ^n\rightarrow {{\,\mathrm{End}\,}}_R({\underline{E\!}\,}[\varepsilon ^n]),\,\bar{a}\mapsto \varphi _a$ and independent of the set of generators of $\varepsilon ^n$; see [18, Lemma 6.2]. Moreover, by [18, Theorem 7.6] it is a finite locally free R-group scheme and a strict ${\mathbb {F}}_\varepsilon$-module scheme and there are canonical $A/\varepsilon ^n$-equivariant isomorphisms of finite locally free R-group schemes

of finite ${\mathbb {F}}_\varepsilon$-shtukas. In particular, ${\underline{E\!}\,}[\varepsilon ^\infty ]:=\displaystyle \lim _{\longrightarrow } {\underline{E\!}\,}[\varepsilon ^n]={{\,\mathrm{Dr}\,}}_{\hat{q}}({\underline{\hat{M}\!}\,})$ is a z-divisible local Anderson module over R.

5 Cohomology Realizations of Local Shtukas

In this section, we work over a valuation ring $\mathcal{{O}}_K$ as in (b). With local shtukas over $\mathcal{{O}}_K$, one can associate various cohomology realizations, which are related to each other under period isomorphisms. We describe the $\varepsilon$-adic, the de Rham, and the crystalline realizations. These period isomorphisms are used in [20, 22] to study the periods of A-motives with complex multiplication.

Definition 5.1

Let ${\underline{\hat{M}\!}\,}=(\hat{M},\tau _{\hat{M}})$ be a local shtuka over a valuation ring $\mathcal{{O}}_K$ as in (b). Then $\tau _{\hat{M}}$ induces an isomorphism , because $z-\zeta \in K[\![ z]\!]^{\scriptscriptstyle \times }$. We define the (dual) Tate module

$$ {{\,\mathrm{H}\,}}^1_\varepsilon ({\underline{\hat{M}\!}\,},A_\varepsilon )\;:=\;\check{T}_\varepsilon {\underline{\hat{M}\!}\,}\;:=\;(\hat{M}\otimes _{\mathcal{{O}}_K[\![ z]\!]}K^\mathrm{sep}[\![ z]\!])^{\hat{\tau }}\;:=\;\bigl \{m\in \hat{M}\otimes _{\mathcal{{O}}_K[\![ z]\!]}K^\mathrm{sep}[\![ z]\!]:\tau _{\hat{M}}(\hat{\sigma }_{\!\hat{M}}^*m)=m\bigr \} $$

and the rational (dual) Tate module

$$ {{\,\mathrm{H}\,}}^1_\varepsilon ({\underline{\hat{M}\!}\,},Q_\varepsilon )\;:=\;\check{V}_\varepsilon {\underline{\hat{M}\!}\,}\;:=\;\bigl \{m\in \hat{M}\otimes _{\mathcal{{O}}_K[\![ z]\!]}K^\mathrm{sep}(\!(z)\!):\tau _{\hat{M}}(\hat{\sigma }_{\!\hat{M}}^*m)=m\bigr \}\;=\;\check{T}_\varepsilon {\underline{\hat{M}\!}\,}\otimes _{A_\varepsilon }Q_\varepsilon \,. $$

By [19, Proposition 4.2], the Tate modules are free over $A_\varepsilon$, resp. $Q_\varepsilon$ of rank equal to ${{\,\mathrm{rk}\,}}{\underline{\hat{M}\!}\,}$ and carry a continuous action of ${{\,\mathrm{Gal}\,}}(K^\mathrm{sep}/K)$. They are also called the $\varepsilon$-adic realizations of ${\underline{\hat{M}\!}\,}$.

Theorem 5.2

([19, Theorem 4.20]) Assume that $\mathcal{{O}}_K$ is discretely valued. Then the functor $\check{T}_\varepsilon :{\underline{\hat{M}\!}\,}\mapsto \check{T}_\varepsilon {\underline{\hat{M}\!}\,}$ from the category of local shtukas over $\mathcal{{O}}_K$ to the category ${{\,\mathrm{Rep}\,}}_{A_\varepsilon }{{\,\mathrm{Gal}\,}}(K^\mathrm{sep}/K)$ of continuous representations of ${{\,\mathrm{Gal}\,}}(K^\mathrm{sep}/K)$ on finite free $A_\varepsilon$-modules and the functor $\check{V}_\varepsilon :{\underline{\hat{M}\!}\,}\mapsto \check{V}_\varepsilon {\underline{\hat{M}\!}\,}$ from the category of local shtukas over $\mathcal{{O}}_K$ with quasi-morphisms to the category ${{\,\mathrm{Rep}\,}}_{Q_\varepsilon }{{\,\mathrm{Gal}\,}}(K^\mathrm{sep}/K)$ of continuous representations of ${{\,\mathrm{Gal}\,}}(K^\mathrm{sep}/K)$ on finite-dimensional $Q_\varepsilon$-vector spaces are fully faithful.

Definition 5.3

Let $\mathcal{{O}}_K$ be discretely valued. The full subcategory of ${{\,\mathrm{Rep}\,}}_{Q_\varepsilon }{{\,\mathrm{Gal}\,}}(K^\mathrm{sep}/K)$ which is the essential image of the functor $\check{V}_\varepsilon$ from Theorem 5.2 is called the category of equal characteristic crystalline representations.

We will explain the motivation for this definition in Sect. 6.

Example 5.4

We describe the $\varepsilon$-adic (dual) Tate module $\check{T}_\varepsilon {\underline{M\!}\,}=\check{T}_\varepsilon {\underline{\hat{M}\!}\,}_\varepsilon ({\underline{M\!}\,})$ of the Carlitz motive ${\underline{M\!}\,}=(\mathcal{{O}}_K[t],t-\theta )$ from Example 2.5 by using the local shtuka ${\underline{\hat{M}\!}\,}:={\underline{\hat{M}\!}\,}_\varepsilon ({\underline{M\!}\,})=(\mathcal{{O}}_K[\![ z]\!],z-\zeta )$ computed there. For all $i\in {\mathbb {N}}_0$, let $\ell _i\in K^\mathrm{sep}$ be solutions of the equations $\ell _0^{\hat{q}-1}=-\zeta$ and $\ell _i^{\hat{q}}+\zeta \ell _i=\ell _{i-1}$. This implies $|\ell _i|=|\zeta |^{\hat{q}^{-i}/(\hat{q}-1)}<1$. Define the power series $\ell _{\scriptscriptstyle +}=\sum _{i=0}^\infty \ell _iz^i\in \mathcal{{O}}_{K^\mathrm{sep}}[\![ z]\!]$. It satisfies $\hat{\sigma }(\ell _{\scriptscriptstyle +})=(z-\zeta )\!\cdot \!\ell _{\scriptscriptstyle +}$, but depends on the choice of the $\ell _i$. A different choice yields a different power series $\tilde{\ell }^{\scriptscriptstyle +}$ which satisfies $\tilde{\ell }^{\scriptscriptstyle +}=u\ell _{\scriptscriptstyle +}$ for a unit $u\in (K^\mathrm{sep}[\![ z]\!]^{\scriptscriptstyle \times })^{\hat{\sigma }={{\,\mathrm{\,id}\,}}}=A_\varepsilon ^{^{\scriptscriptstyle \times }}$, because $\hat{\sigma }(u)=\frac{\hat{\sigma }(\tilde{\ell }^{\scriptscriptstyle +})}{\hat{\sigma }(\ell _{\scriptscriptstyle +})}=\frac{\tilde{\ell }^{\scriptscriptstyle +}}{\ell _{\scriptscriptstyle +}}=u$. The field extension ${\mathbb {F}}_\varepsilon (\!(\zeta )\!)(\ell _i:i\in {\mathbb {N}}_0)$ of ${\mathbb {F}}_\varepsilon (\!(\zeta )\!)$ is the function field analog of the cyclotomic tower ${\mathbb {Q}}_p(\root p^i \of {1}:i\in {\mathbb {N}}_0)$; see [16, § 1.3 and § 3.4]. There is an isomorphism of topological groups called the $\varepsilon$-adic cyclotomic character

which satisfies $g(\ell _{\scriptscriptstyle +}):=\sum _{i=0}^\infty g(\ell _i)z^i=\chi _\varepsilon (g)\cdot \ell _{\scriptscriptstyle +}$ in $K^\mathrm{sep}[\![ z]\!]$ for g in the Galois group. It is independent of the choice of the $\ell _i$. The $\varepsilon$-adic (dual) Tate module $\check{T}_\varepsilon {\underline{\hat{M}\!}\,}$ of ${\underline{\hat{M}\!}\,}$ and ${\underline{M\!}\,}$ is generated by $\ell _{\scriptscriptstyle +}^{-1}$ on which the Galois group acts by the inverse of the cyclotomic character.

Definition 5.5

Let ${\underline{\hat{M}\!}\,}$ be a local shtuka over a valuation ring $\mathcal{{O}}_K$ as in (b). We denote by $K[\![ z-\zeta ]\!]$ the power series ring over K in the “variable” $z-\zeta$ and by $K(\!(z-\zeta )\!)$ its fraction field. We consider the ring homomorphism $\mathcal{{O}}_K[\![ z]\!]\hookrightarrow K[\![ z-\zeta ]\!],\,z\mapsto z=\zeta +(z-\zeta )$ and define the de Rham realization of ${\underline{\hat{M}\!}\,}$ as

$$\begin{aligned} {{\,\mathrm{H}\,}}^1_\mathrm{dR}\bigl ({\underline{\hat{M}\!}\,},K[\![ z-\zeta ]\!]\bigr ):= & {} \hat{\sigma }^*\hat{M}\otimes _{\mathcal{{O}}_K[\![ z]\!]}K[\![ z-\zeta ]\!]\,,\\ {{\,\mathrm{H}\,}}^1_\mathrm{dR}\bigl ({\underline{\hat{M}\!}\,},K(\!(z-\zeta )\!)\bigr ):= & {} \hat{\sigma }^*\hat{M}\otimes _{\mathcal{{O}}_K[\![ z]\!]}K(\!(z-\zeta )\!)\qquad \text {and}\\ {{\,\mathrm{H}\,}}^1_\mathrm{dR}({\underline{\hat{M}\!}\,},K):= & {} \hat{\sigma }^*\hat{M}\otimes _{\mathcal{{O}}_K[\![ z]\!],\,z\mapsto \zeta }K \\= & {} {{\,\mathrm{H}\,}}^1_\mathrm{dR}\bigl ({\underline{\hat{M}\!}\,},K[\![ z-\zeta ]\!]\bigr )\otimes _{K[\![ z-\zeta ]\!]}K[\![ z-\zeta ]\!]/(z-\zeta )\,.\nonumber \end{aligned}$$

The de Rham realization ${{\,\mathrm{H}\,}}^1_\mathrm{dR}\bigl ({\underline{\hat{M}\!}\,},K(\!(z-\zeta )\!)\bigr )$ contains a full $K[\![ z-\zeta ]\!]$-lattice

$$\begin{aligned} {\mathfrak {q}}^{\underline{\hat{M}\!}\,}\;:=\;\tau _{\hat{M}}^{-1}(\hat{M}\otimes _{\mathcal{{O}}_K[\![ z]\!]}K[\![ z-\zeta ]\!]), \end{aligned}$$

(5.1)

which is called the Hodge-Pink lattice of ${\underline{\hat{M}\!}\,}$. The de Rham realization ${{\,\mathrm{H}\,}}^1_\mathrm{dR}({\underline{\hat{M}\!}\,},K)$ carries a descending separated and exhausting filtration $F^\bullet$ by K-subspaces called the Hodge-Pink filtration of ${\underline{\hat{M}\!}\,}$. It is defined via ${\mathfrak {p}}:={{\,\mathrm{H}\,}}^1_\mathrm{dR}({\underline{\hat{M}\!}\,},K[\![ z-\zeta ]\!])$ and (for $i\in {\mathbb {Z}}$)

$$\begin{aligned} F^i{{\,\mathrm{H}\,}}^1_\mathrm{dR}({\underline{\hat{M}\!}\,},K)\;:=\;\bigl ({\mathfrak {p}}\cap (z-\zeta )^i{\mathfrak {q}}^{\underline{\hat{M}\!}\,}\bigr )\big /\bigl ((z-\zeta ){\mathfrak {p}}\cap (z-\zeta )^i{\mathfrak {q}}^{\underline{\hat{M}\!}\,}\bigr )\;\subset \;{{\,\mathrm{H}\,}}^1_\mathrm{dR}({\underline{\hat{M}\!}\,},K)\,. \end{aligned}$$

(5.2)

If we equip ${{\,\mathrm{H}\,}}^1_\mathrm{dR}\bigl ({\underline{\hat{M}\!}\,},K(\!(z-\zeta )\!)\bigr )$ with the descending filtration $F^i{{\,\mathrm{H}\,}}^1_\mathrm{dR}\bigl ({\underline{\hat{M}\!}\,},K(\!(z-\zeta )\!)\bigr ):=(z-\zeta )^i{\mathfrak {q}}^{\underline{\hat{M}\!}\,}$ by $K[\![ z-\zeta ]\!]$-submodules, then $F^i{{\,\mathrm{H}\,}}^1_\mathrm{dR}({\underline{\hat{M}\!}\,},K)$ is the image of ${{\,\mathrm{H}\,}}^1_\mathrm{dR}\bigl ({\underline{\hat{M}\!}\,},K[\![ z-\zeta ]\!]\bigr )\,\cap \,F^i{{\,\mathrm{H}\,}}^1_\mathrm{dR}\bigl ({\underline{\hat{M}\!}\,},K(\!(z-\zeta )\!)\bigr )$ in ${{\,\mathrm{H}\,}}^1_\mathrm{dR}({\underline{\hat{M}\!}\,},K)$. Since $z=\zeta +(z-\zeta )$ is invertible in $K[\![ z-\zeta ]\!]$, the de Rham realization with Hodge-Pink lattice and filtration is a functor on the category of local shtukas over $\mathcal{{O}}_K$ with quasi-morphisms.

Note, however, that the Hodge-Pink filtration on ${{\,\mathrm{H}\,}}^1_\mathrm{dR}({\underline{\hat{M}\!}\,},K)$ does not behave well under tensor products, as opposed to the Hodge-Pink lattice; see Remark 6.3 below. Therefore, the more important concept is the Hodge-Pink lattice ${\mathfrak {q}}^{\underline{\hat{M}\!}\,}$.

Theorem 5.6

([19, Theorem 4.15]) Let ${\,\overline{\!K}}$ be the completion of an algebraic closure $K^\mathrm{alg}$ of K. There is a canonical functorial comparison isomorphism

which satisfies $h_{\varepsilon ,\mathrm{dR}}\bigl ({{\,\mathrm{H}\,}}^1_\varepsilon ({\underline{\hat{M}\!}\,},Q_\varepsilon )\otimes _{Q_\varepsilon }{\,\overline{\!K}}[\![ z-\zeta ]\!]\bigr )={\mathfrak {q}}^{\underline{\hat{M}\!}\,}\otimes _{K[\![ z-\zeta ]\!]}{\,\overline{\!K}}[\![ z-\zeta ]\!]$ and which is equivariant for the action of ${{\,\mathrm{Gal}\,}}(K^\mathrm{sep}/K)$, where on the source of $h_{\varepsilon ,\mathrm{dR}}$ this group acts on both factors of the tensor product and on the target of $h_{\varepsilon ,\mathrm{dR}}$ it acts only on ${\,\overline{\!K}}$.

Definition 5.7

Let $k=\mathcal{{O}}_K/{\mathfrak {m}}_K$ be the residue field of $\mathcal{{O}}_K$. A z-isocrystal over k is a pair $(D,\tau _D)$ consisting of a finite-dimensional $k(\!(z)\!)$-vector space together with a $k(\!(z)\!)$-isomorphism . A morphism $(D,\tau _D)\rightarrow (D',\tau _{D'})$ is a $k(\!(z)\!)$-homomorphism $f:D\rightarrow D'$ satisfying $\tau _{D'}\circ \hat{\sigma }^*f=f\circ \tau _D$.

Definition 5.8

Let ${\underline{\hat{M}\!}\,}=(\hat{M},\tau _{\hat{M}})$ be local shtuka over a valuation ring $\mathcal{{O}}_K$ as in (b). Then the crystalline realization of ${\underline{\hat{M}\!}\,}$ is defined as the z-isocrystal over $k=\mathcal{{O}}_K/{\mathfrak {m}}_K$

$$\begin{aligned} {{\,\mathrm{H}\,}}^1_\mathrm{cris}\bigl ({\underline{\hat{M}\!}\,},k(\!(z)\!)\bigr )\;:=\;\hat{\sigma }^*(\hat{M},\tau _{\hat{M}})\otimes _{\mathcal{{O}}_K[\![ z]\!]}k(\!(z)\!)\,. \end{aligned}$$

(5.3)

It only depends on the special fiber ${\underline{\hat{M}\!}\,}\otimes _{\mathcal{{O}}_K}k$ of ${\underline{\hat{M}\!}\,}$ and defines a functor ${\underline{\hat{M}\!}\,}\mapsto {{\,\mathrm{H}\,}}^1_\mathrm{cris}\bigl ({\underline{\hat{M}\!}\,},k(\!(z)\!)\bigr )$ from the category of local shtukas over $\mathcal{{O}}_K$ with quasi-morphism to the category of z-isocrystals. This functor is faithful by [19, Lemma 4.24] if $\bigcap _n\hat{\sigma }^n({\mathfrak {m}}_K) = (0)$.

To formulate the comparison between the de Rham and the crystalline realization, we assume that there exists a fixed section $k\hookrightarrow \mathcal{{O}}_K$. Then there is a ring homomorphism

(5.4)

We always consider $K[\![ z-\zeta ]\!]$ and its fraction field $K(\!(z-\zeta )\!)$ as $k(\!(z)\!)$-vector spaces via (5.4).

Theorem 5.9

([19, Theorem 5.18]) Let ${\underline{\hat{M}\!}\,}$ be a local shtuka over $\mathcal{{O}}_K$. Assume that $\mathcal{{O}}_K$ is discretely valued or that ${\underline{\hat{M}\!}\,}={\underline{\hat{M}\!}\,}_\varepsilon ({\underline{M\!}\,})$ for an A-motive ${\underline{M\!}\,}$ over $\mathcal{{O}}_K$ as in Example 2.3. Then there are canonical functorial comparison isomorphisms between the de Rham and crystalline realizations

To formulate the comparison between the crystalline and the $\varepsilon$-adic realizations, we introduce the $\mathcal{{O}}_K$-algebra

$$\begin{aligned} \mathcal{{O}}_K[\![ z,z^{-1}\}:= & {} \bigl \{\,\sum _{i=-\infty }^\infty b_iz^i:\; b_i\in \mathcal{{O}}_K\,,\,|b_i|\,|\zeta |^{ri}\rightarrow 0\;(i\rightarrow -\infty ) \text { for all }r>0\,\bigr \}\,.\nonumber \\ \end{aligned}$$

(5.5)

It is a subring of $K[\![ z-\zeta ]\!]$ via the expansion $\sum \limits _{i=-\infty }^\infty b_iz^i=$ $\sum \limits _{j=0}^\infty \zeta ^{-j}\Bigl (\sum \limits _{i=-\infty }^\infty {i\atopwithdelims ()j}b_i\zeta ^i\Bigr )(z-\zeta )^j$. The homomorphism (5.4) factors through $\mathcal{{O}}_K[\![ z,z^{-1}\}$. We view the elements of $\mathcal{{O}}_K[\![ z,z^{-1}\}$ as functions that converge on the punctured open unit disk $\{0<|z|<1\}$. An example of such a function is

$$\begin{aligned} \ell _{\scriptscriptstyle -}:=\prod _{i\in {\mathbb {N}}_0}(1-\tfrac{\zeta ^{\hat{q}^i}}{z})\;\in \;{\mathbb {F}}_\varepsilon [\![ \zeta ]\!][\![ z,z^{-1}\}\;\subset \;\mathcal{{O}}_K[\![ z,z^{-1}\}\,, \end{aligned}$$

(5.6)

which satisfies $\ell _{\scriptscriptstyle -}=(1-\tfrac{\zeta }{z})\cdot \hat{\sigma }(\ell _{\scriptscriptstyle -})$. In addition, we let ${\,\overline{\!K}}$ be the completion of an algebraic closure $K^\mathrm{alg}$ of K and recall the element $\ell _{\scriptscriptstyle +}\in \mathcal{{O}}_{\,\overline{\!K}}[\![ z]\!]$ from Example 5.4, which satisfies $\hat{\sigma }(\ell _{\scriptscriptstyle +})=(z-\zeta )\cdot \ell _{\scriptscriptstyle +}$. We set

$$\begin{aligned} \ell \;:=\;\ell _{\scriptscriptstyle +}\ell _{\scriptscriptstyle -}\;\in \;\mathcal{{O}}_{\,\overline{\!K}}[\![ z,z^{-1}\}\,. \end{aligned}$$

(5.7)

Then $\hat{\sigma }(\ell )=z\!\cdot \!\ell$ and $g(\ell )=\chi _\varepsilon (g)\!\cdot \!\ell$ for $g\in {{\,\mathrm{Gal}\,}}(K^\mathrm{sep}/K)$ where $\chi _\varepsilon$ is the cyclotomic character from Example 5.4.

Theorem 5.10

([19, Theorem 5.20]) Let ${\underline{\hat{M}\!}\,}$ be a local shtuka over $\mathcal{{O}}_K$. Assume that $\mathcal{{O}}_K$ is discretely valued or that ${\underline{\hat{M}\!}\,}={\underline{\hat{M}\!}\,}_\varepsilon ({\underline{M\!}\,})$ for an A-motive ${\underline{M\!}\,}$ over $\mathcal{{O}}_K$ as in Example 2.3. Then there is a canonical functorial comparison isomorphism between the $\varepsilon$-adic and crystalline realizations

The isomorphism $h_{\varepsilon ,\mathrm{cris}}$ is ${{\,\mathrm{Gal}\,}}(K^\mathrm{sep}/K)$- and $\hat{\tau }$-equivariant, where on the left module ${{\,\mathrm{Gal}\,}}(K^\mathrm{sep}/K)$ acts on both factors and $\hat{\tau }$ is ${{\,\mathrm{\,id}\,}}\otimes \hat{\sigma }$, and on the right module ${{\,\mathrm{Gal}\,}}(K^\mathrm{sep}/K)$ acts only on $\mathcal{{O}}_{\,\overline{\!K}}[\![ z,z^{-1}\}[\ell ^{-1}]$ and $\hat{\tau }$ is $(\tau _D\circ \hat{\sigma }_{\!D}^*)\otimes \hat{\sigma }$. In other words, $h_{\varepsilon ,\mathrm{cris}}=\tau _D\circ \hat{\sigma }^*h_{\varepsilon ,\mathrm{cris}}$. Moreover, $h_{\varepsilon ,\mathrm{cris}}$ satisfies $h_{\varepsilon ,\mathrm{dR}}=(h_{\mathrm{dR},\mathrm{cris}}^{-1}\otimes {{\,\mathrm{\,id}\,}}_{{\,\overline{\!K}}(\!(z-\zeta )\!)})\circ (h_{\varepsilon ,\mathrm{cris}}\otimes {{\,\mathrm{\,id}\,}}_{{\,\overline{\!K}}(\!(z-\zeta )\!)})$. It allows to recover ${{\,\mathrm{H}\,}}^1_\varepsilon ({\underline{\hat{M}\!}\,},Q_\varepsilon )$ from ${{\,\mathrm{H}\,}}^1_\mathrm{cris}\bigl ({\underline{\hat{M}\!}\,},k(\!(z)\!)\bigr )$ as the intersection inside ${{\,\mathrm{H}\,}}^1_\mathrm{cris}\bigl ({\underline{\hat{M}\!}\,},k(\!(z)\!)\bigr )\otimes _{k(\!(z)\!)}{\,\overline{\!K}}(\!(z-\zeta )\!)$

$$ h_{\varepsilon ,\mathrm{cris}}\bigl ({{\,\mathrm{H}\,}}^1_\varepsilon ({\underline{\hat{M}\!}\,},Q_\varepsilon )\bigr )\;=\;\bigl ({{\,\mathrm{H}\,}}^1_\mathrm{cris}\bigl ({\underline{\hat{M}\!}\,},k(\!(z)\!)\bigr )\otimes _{k(\!(z)\!)}\mathcal{{O}}_{\,\overline{\!K}}[\![ z,z^{-1}\}[\ell ^{-1}]\bigr )^{\hat{\tau }={{\,\mathrm{\,id}\,}}}\,\cap \,{\mathfrak {q}}_D\otimes _{K[\![ z-\zeta ]\!]}{\,\overline{\!K}}[\![ z-\zeta ]\!]\,, $$

where ${\mathfrak {q}}_D\subset {{\,\mathrm{H}\,}}^1_\mathrm{cris}\bigl ({\underline{\hat{M}\!}\,},k(\!(z)\!)\bigr )\otimes _{k(\!(z)\!)}K(\!(z-\zeta )\!)$ is the Hodge-Pink lattice of ${\underline{\hat{M}\!}\,}$.

6 Crystalline Representations over Function Fields

We explain the motivation for Definition 5.3; compare [19, Remarks 5.13 and 6.17].

Let $\mathcal{{O}}_K$ be discretely valued and let ${\underline{\hat{M}\!}\,}$ be a local shtuka over $\mathcal{{O}}_K$. Theorem 5.9 allows to define a Hodge-Pink lattice and a Hodge-Pink filtration on ${{\,\mathrm{H}\,}}^1_\mathrm{cris}\bigl ({\underline{\hat{M}\!}\,},k(\!(z)\!)\bigr )$. More precisely, we equip the finite-dimensional $k(\!(z)\!)$-vector space $D:={{\,\mathrm{H}\,}}^1_\mathrm{cris}\bigl ({\underline{\hat{M}\!}\,},k(\!(z)\!)\bigr )$ with the Hodge-Pink lattice

$$ {\mathfrak {q}}_D\;:=\;(h_{\mathrm{dR},\mathrm{cris}}\otimes {{\,\mathrm{\,id}\,}}_{K(\!(z-\zeta )\!)})({\mathfrak {q}}^{\underline{\hat{M}\!}\,})\;\subset \; D\otimes _{k(\!(z)\!)}K(\!(z-\zeta )\!)\,, $$

where ${\mathfrak {q}}^{\underline{\hat{M}\!}\,}\subset {{\,\mathrm{H}\,}}^1_\mathrm{dR}\bigl ({\underline{\hat{M}\!}\,},K(\!(z-\zeta )\!)\bigr )$ is the Hodge-Pink lattice from (5.1). Together with the Frobenius $\tau _D:=\hat{\sigma }^*\tau _{\hat{M}}\otimes {{\,\mathrm{\,id}\,}}_{k(\!(z)\!)}$ on $D={{\,\mathrm{H}\,}}^1_\mathrm{cris}\bigl ({\underline{\hat{M}\!}\,},k(\!(z)\!)\bigr )$ from (5.3), the triple ${\underline{D\!}\,}({\underline{\hat{M}\!}\,}):={\underline{D\!}\,}=(D,\tau _D,{\mathfrak {q}}_D)$ forms a z-isocrystal with a Hodge-Pink structure as in the following.

Definition 6.1

A z-isocrystal with Hodge-Pink structure over $\mathcal{{O}}_K$ is a triple ${\underline{D\!}\,}=(D,\tau _D,{\mathfrak {q}}_D)$ consisting of a z-isocrystal $(D,\tau _D)$ over k and a $K[\![ z-\zeta ]\!]$-lattice ${\mathfrak {q}}_D$ in $D\otimes _{k(\!(z)\!)}K(\!(z-\zeta )\!)$ of full rank, which is called the Hodge-Pink lattice of ${\underline{D\!}\,}$. The dimension of D is called the rank of ${\underline{D\!}\,}$ and is denoted by ${{\,\mathrm{rk}\,}}{\underline{D\!}\,}$.

A morphism $(D,\tau _D,{\mathfrak {q}}_D)\rightarrow (D',\tau _{D'},{\mathfrak {q}}_{D'})$ is a $k(\!(z)\!)$-homomorphism $f:D\rightarrow D'$ satisfying $\tau _{D'}\circ \hat{\sigma }^*f=f\circ \tau _D$ and $(f\otimes {{\,\mathrm{\,id}\,}})({\mathfrak {q}}_D)\subset {\mathfrak {q}}_{D'}$.

A strict subobject ${\underline{D\!}\,}'\subset {\underline{D\!}\,}$ is a z-isocrystal with Hodge-Pink structure of the form ${\underline{D\!}\,}'=\bigl (D',\,\tau _D|_{\hat{\sigma }^*\!D'},\,{\mathfrak {q}}_D\cap D'\otimes _{k(\!(z)\!)}K(\!(z-\zeta )\!)\bigr )$ where $D'\subset D$ is a $k(\!(z)\!)$-subspace with $\tau _D(\hat{\sigma }^*D')=D'$.

On a z-isocrystal with Hodge-Pink structure ${\underline{D\!}\,}$, there always is the tautological $K[\![ z-\zeta ]\!]$-lattice ${\mathfrak {p}}_D:=D\otimes _{k(\!(z)\!)}K[\![ z-\zeta ]\!]$. Since $K[\![ z-\zeta ]\!]$ is a principal ideal domain, the elementary divisor theorem provides basis vectors $v_i\in {\mathfrak {p}}_D$ such that ${\mathfrak {p}}_D=\bigoplus _{i=1}^rK[\![ z-\zeta ]\!]\cdot v_i$ and ${\mathfrak {q}}_D=\bigoplus _{i=1}^rK[\![ z-\zeta ]\!]\cdot (z-\zeta )^{\mu _i}\cdot v_i$ for integers $\mu _1\ge \ldots \ge \mu _r$. We call $(\mu _1,\ldots ,\mu _r)$ the Hodge-Pink weights of ${\underline{D\!}\,}$. Alternatively, if e is large enough such that ${\mathfrak {q}}_D\subset (z-\zeta )^{-e}{\mathfrak {p}}_D$ or $(z-\zeta )^e{\mathfrak {p}}_D\subset {\mathfrak {q}}_D$, then the Hodge-Pink weights are characterized by

$$\begin{aligned} (z-\zeta )^{-e}{\mathfrak {p}}_D/{\mathfrak {q}}_D\cong & {} \bigoplus _{i=1}^n K[\![ z-\zeta ]\!]/(z-\zeta )^{e+\mu _i}\,,\\ \text {or}\quad {\mathfrak {q}}_D/(z-\zeta )^e{\mathfrak {p}}_D\cong & {} \bigoplus _{i=1}^n K[\![ z-\zeta ]\!]/(z-\zeta )^{e-\mu _i}. \end{aligned}$$

Like in (5.2), the Hodge-Pink lattice ${\mathfrak {q}}_D$ induces a descending filtration of $D_K:=D\otimes _{k(\!(z)\!),\,z\mapsto \zeta }K$ by K-subspaces as follows. Consider the natural projection

$$ {\mathfrak {p}}_D\;\twoheadrightarrow \;{\mathfrak {p}}_D/(z -\zeta ){\mathfrak {p}}_D\;=\;D_K\,. $$

The Hodge-Pink filtration $F^\bullet D_K=(F^i D_K)_{i\in {\mathbb {Z}}}$ is defined by letting $F^i D_K$ be the image in $D_K$ of ${\mathfrak {p}}_D\cap (z-\zeta )^i{\mathfrak {q}}_D$ for all $i\in {\mathbb {Z}}$. This means, $F^i D_K=\bigl ({\mathfrak {p}}_D\cap (z-\zeta )^i{\mathfrak {q}}_D\bigr )\big /\bigl ((z-\zeta ){\mathfrak {p}}_D\cap (z-\zeta )^i{\mathfrak {q}}_D\bigr )$.

Definition 6.2

Let ${\underline{D\!}\,}=(D,\tau _D,{\mathfrak {q}}_D)$ be a z-isocrystal with Hodge-Pink structure over $\mathcal{{O}}_K$ and set $r=\dim _{k(\!(z)\!)}D$.

• (a) Choose a $k(\!(z)\!)$-basis of D and let $\det \tau _D$ be the determinant of the matrix representing $\tau _D$ with respect to this basis. The number $t_N({\underline{D\!}\,}):={{\,\mathrm{ord}\,}}_z(\det \tau _D)$ is independent of this basis and is called the Newton slope of ${\underline{D\!}\,}$.

• (b) The integer $t_H({\underline{D\!}\,}):=-\mu _1-\ldots -\mu _r$, where $\mu _1,\ldots ,\mu _r$ are the Hodge-Pink weights of ${\underline{D\!}\,}$ from Definition 6.1, satisfies $\wedge ^r{\mathfrak {q}}_D=(z-\zeta )^{-t_H({\underline{D\!}\,})}\wedge ^r{\mathfrak {p}}_D$ and is called the Hodge slope of ${\underline{D\!}\,}$.

• (c) ${\underline{D\!}\,}$ is called weakly admissible if

  $$ t_H({\underline{D\!}\,})=t_N({\underline{D\!}\,})\qquad \text {and}\qquad t_H({\underline{D\!}\,}')\le t_N({\underline{D\!}\,}')\quad \text {for every strict subobject }{\underline{D\!}\,}'\subset {\underline{D\!}\,}\text {.} $$

Remark 6.3

One can show that the tensor product

$$ {\underline{D}}\otimes {\underline{D}}' \; = \; \bigl (D\otimes _{k(\!(z)\!)} D', \tau _D\otimes \tau _{D'}, {\mathfrak {q}}_D\otimes _{K[\![ z-\zeta ]\!]}{\mathfrak {q}}_{D'}\bigr ) $$

of two weakly admissible z-isocrystals with Hodge-Pink structures ${\underline{D\!}\,}$ and ${\underline{D\!}\,}'$ over $\mathcal{{O}}_K$ is again weakly admissible. It was Pink’s insight that for this result the Hodge-Pink filtration does not suffice, but one needs the finer information present in the Hodge-Pink lattice. The problem arises if the field extension $K/{\mathbb {F}}_q(\!(\zeta )\!)$ is inseparable; see [29, Example 5.16]. This is Pink’s ingenious discovery.

Proposition 6.4

([19, Corollary 6.11]) Let ${\underline{\hat{M}\!}\,}$ be a local shtuka over $\mathcal{{O}}_K$. Assume that $\mathcal{{O}}_K$ is discretely valued or that ${\underline{\hat{M}\!}\,}={\underline{\hat{M}\!}\,}_\varepsilon ({\underline{M\!}\,})$ for an A-motive ${\underline{M\!}\,}$ over $\mathcal{{O}}_K$ as in Example 2.3. Then the z-isocrystal with Hodge-Pink structure ${\underline{D\!}\,}({\underline{\hat{M}\!}\,})$ constructed at the beginning of this section is weakly admissible. The functor ${\underline{\hat{M}\!}\,}\longmapsto {\underline{D\!}\,}({\underline{\hat{M}\!}\,})$ from the category of local shtukas over $\mathcal{{O}}_K$ with quasi-morphisms to the category of weakly admissible z-isocrystals with Hodge-Pink structure is fully faithful.

There is a converse to this proposition.

Theorem 6.5

([14, Théorème 7.3], [17, Theorem 2.5.3]) If $\mathcal{{O}}_K$ is discretely valued, then every weakly admissible z-isocrystal with Hodge-Pink structure ${\underline{D\!}\,}$ over $\mathcal{{O}}_K$ is of the form ${\underline{D\!}\,}({\underline{\hat{M}\!}\,})$ for a local shtuka ${\underline{\hat{M}\!}\,}$ over $\mathcal{{O}}_K$.

Remark 6.6

The theory presented here has as analog, the theory of p-adic Galois representations. There L is a discretely valued extension of ${\mathbb {Q}}_p$ with perfect residue field $\kappa$ and $L_0:=W(\kappa )[\tfrac{1}{p}]$ is the maximal, absolutely unramified subfield of L. Let $\hat{\sigma }:=W({{\,\mathrm{Frob}\,}}_p)$ be the lift to $L_0$ of the p-Frobenius on $\kappa$ which fixes the uniformizer p of $L_0$. Crystalline p-adic Galois representations are described by filtered isocrystals ${\underline{D\!}\,}=(D,\tau _D,F^\bullet D_L)$ over L, where D is a finite-dimensional $L_0$-vector space, is an $L_0$-isomorphism, and $F^\bullet D_L$ is a descending filtration on $D_L:=D\otimes _{L_0}L$ by L-subspaces. More precisely, the Theorem of Colmez and Fontaine [6, Théorème A] says that a continuous representation of ${{\,\mathrm{Gal}\,}}(L^\mathrm{sep}/L)$ in a finite-dimensional ${\mathbb {Q}}_p$-vector space is crystalline if and only if it isomorphic to $F^0({\underline{D\!}\,}\otimes _{L_0}{\widetilde{{\mathbf {B}}}}_\mathrm{rig})^{\tau \,={{\,\mathrm{\,id}\,}}}$ for a weakly admissible filtered isocrystal ${\underline{D\!}\,}=(D,\tau _D,F^\bullet D_L)$ over L. Here, ${\widetilde{{\mathbf {B}}}}_\mathrm{rig}$ is a certain period ring from Fontaine’s theory of p-adic Galois representations, which carries a filtration and a Frobenius endomorphism ${{\,\mathrm{Frob}\,}}_p$. The function field analog of ${\widetilde{{\mathbf {B}}}}_\mathrm{rig}$ is the $Q_\varepsilon$-algebra $\mathcal{{O}}_{\,\overline{\!K}}[\![ z,z^{-1}\}[\ell ^{-1}]$; see [16, §§ 2.5 and 2.7]. In the function field case, when K is discretely valued, we could therefore define the category of equal characteristic crystalline representations of ${{\,\mathrm{Gal}\,}}(K^\mathrm{sep}/K)$ as the essential image of the functor

$$\begin{aligned} {\underline{D\!}\,}=(D,\tau _D,{\mathfrak {q}}_D)\;\longmapsto \; \bigl ({\underline{D\!}\,}\otimes _{k(\!(z)\!)}\mathcal{{O}}_{\,\overline{\!K}}[\![ z,z^{-1}\}[\ell ^{-1}]\bigr )^{\tau ={{\,\mathrm{\,id}\,}}}\,\cap \,{\mathfrak {q}}_D\otimes _{K[\![ z-\zeta ]\!]}{\,\overline{\!K}}[\![ z-\zeta ]\!]\ \end{aligned}$$

(6.1)

from weakly admissible z-isocrystals with Hodge-Pink structure ${\underline{D\!}\,}$ to continuous representations of ${{\,\mathrm{Gal}\,}}(K^\mathrm{sep}/K)$ in finite-dimensional $Q_\varepsilon$-vector spaces. By Theorems 6.5, 5.10, and 5.2 and Proposition 6.4, this functor is fully faithful and this definition coincides with our Definition 5.3 above.