Local Shtukas, Hodge-Pink Structures and Galois Representations

Urs Hartl and Wansu Kim December 21, 2015

Abstract

We review the analog of Fontaine’s theory of crystalline p-adic Galois representations and their classification by weakly admissible filtered isocrystals in the arithmetic of function fields over a finite field. There crystalline Galois representations are replaced by the Tate modules of so-called local shtukas. We prove that the Tate module functor is fully faithful. In addition to this étale realization of a local shtuka we discuss also the de Rham and the crystalline cohomology realizations and construct comparison isomorphisms between these realizations. We explain how local shtukas and these cohomology realizations arise from Drinfeld modules and Anderson’s t-motives. As an application we construct equi-characteristic crystalline deformation rings, establish their rigidanalytic smoothness and compute their dimension. Mathematics Subject Classification (2000): 11G09, (11S20, 14F30, 14F40, 14L05)

Contents

1 Introduction 1 2 Local Shtukas 4 3 Divisible local Anderson modules 7 4 Tate modules 10 5 Hodge-Pink structures 17 6 Admissibility and weak admissibility 27 7 Torsion local shtukas and torsion Galois representations 32 8 Deformation theory of Galois representations 43 References 50

1 Introduction

Let X be a smooth proper scheme over a number field L. With X one associates various cohomology realizations, for example the étale ℓ-adic realization H_{ét}^{i}(X ×_{L} L^{alg}, ℚ_{ℓ}), the de Rham realization H_{dR}^{i}(X/L) and the Betti realization H_{Betti}^{i}(X(ℂ), ℤ). One has comparison isomorphisms between these realizations. This lead Grothendieck to postulate a category of “motives” as a universal cohomology theory through which all cohomology realizations factor.

There exists a rich parallel between the above number field situation and the “Arithmetic of function fields”. In the latter, Anderson [And86] introduced the concept of “t-motives”. We slightly generalize his definition as follows. Let F be a finite field extension of the rational function field 𝔽_{q}(ϑ) in the variable ϑ over a finite field 𝔽_{q} with q elements. Equip F with a morphism γ: 𝔽_{q}[t] → F, t ↦ ϑ where 𝔽_{q}[t] is the polynomial ring. Let σ be the endomorphism of F [t] with σ(t) = t and σ(b) = b^{q} for b ∈ F. An 𝔽_{q}[t]-motive of rank r over F is a pair \underline{M} = (M, τ_{M}) consisting of a free F [t]-module M of rank r and an F [t][\frac{1}{t−ϑ}]-isomorphism τ_{M}: σ^{∗}M[\frac{1}{t−ϑ}] ^{∼} ⟶ M[\frac{1}{t−ϑ}], where σ^{∗}M := M ⊗_{F [t],σ} F [t]. A t-motive in the sense of Anderson [And86] is an 𝔽_{q}[t]-motive \underline{M} such that in addition τ_{M}(σ^{∗}M) ⊂ M and M is finitely generated over the non-commutative polynomial ring F {τ} := { ∑_{i=0}^{n} b_{i}τ^{i}: n ∈ ℕ, b_{i} ∈ F } with τb = b^{q}τ which acts on m ∈ M by τ(m) := τ_{M}(σ_{M}^{∗}m), where σ_{M}^{∗}m := m ⊗ 1 ∈ σ^{∗}M.

Various “cohomology” realizations are attached to an 𝔽_{q}[t]-motive \underline{M}; see [Gos94, § 2]. Namely, for a maximal ideal v ∈ 𝔽_{q}[t] one has a v-adic realization given by the v-adic (dual) Tate module

Ť_{v}\underline{M} := (M ⊗_{F [t]} A_{v,F^{sep}})^{τ} := \{m ∈ M ⊗_{F [t]} A_{v,F^{sep}} : τ_{M}(σ_{M}^{∗}m) = m\},

where F^{sep} is a separable closure of F and A_{v,F^{sep}} is the v-adic completion of F^{sep}[t]. For example if v = (t) then A_{v,F^{sep}} = F^{sep}[[t]]. Ť_{v}\underline{M} is a free module over the v-adic completion A_{v} of 𝔽_{q}[t] of rank equal to the rank of \underline{M} and carries a continuous action of Gal(F^{sep}/F). It is sometimes also denoted H_{v}^{1}(\underline{M}, A_{v}). The de Rham realization is defined as H_{dR}^{1}(\underline{M}, F) = σ^{∗}M/(t − ϑ)σ^{∗}M. If \underline{M} is “uniformizable” there is also a Betti realization with a Hodge structure, which is described for example in [HJ16, § 3.4].

Assume now that \underline{M} has good reduction \overline{\underline{M}} at a maximal ideal 𝔪 of the integral closure O_{F} of 𝔽_{q}[ϑ] in F; see Example 2.2 for the precise definition. Set k := O_{F} /𝔪 and ε := ker(𝔽_{q}[t] → O_{F} /𝔪), and let F_{𝔪} be the completion of F at the place 𝔪. If v ≠ ε, the assumption of good reduction implies that the Galois action on Ť_{v}\underline{M} is unramified at 𝔪 by [Gar02, Theorem 1.1] and so the decomposition group Gal(F_{𝔪}^{sep}/F_{𝔪}) acts through its quotient Gal(k^{sep}/k). This representation of Gal(k^{sep}/k) is canonically isomorphic to the (dual) Tate module Ť_{v}\overline{\underline{M}} := (\overline{M} ⊗_{k[t]} A_{v,k^{sep}})^{τ} of \overline{\underline{M}}.

If v = ε however, the reduction \overline{\underline{M}} does not possess a v-adic Tate module. In the parallel between function fields and number fields the Gal(F_{𝔪}^{sep}/F_{𝔪})-representation Ť_{ε}\underline{M} corresponds to what is called a p-adic Galois representation (see [Ber04b] or [BC09] for an introduction), because F_{𝔪} has the same “residue 𝔽_{q}[t]-characteristic” ε as A_{ε}. One aim of this survey article is to explain the function field analog of Fontaine’s theory [Fon79, Fon82, Fon90, Fon94] of p-adic Galois representations. Namely, Fontaine defined the notion of “crystalline p-adic Galois representation” and conjectured that the Galois representations arising from the p-adic étale cohomology of a smooth proper variety with good reduction over a finite field extension of ℚ_{p} is crystalline. After contributions by Grothendieck, Tate, Fontaine, Lafaille, Messing, Kato and many others, Fontaine’s conjecture was finally proved independently by Faltings [Fal89], Niziol [Niz98] and Tsuji [Tsu99]. Moreover, Fontaine classified crystalline representations by “filtered Frobenius-isocrystals” whose function field analogs we will explain in Section 5 below.

It turns out that the appropriate function field analogs of crystalline p-adic Galois representations are given by local shtukas; see Definition 2.1. To describe these let \underline{M} be an 𝔽_{q}[t]-motive over F_{𝔪} with good reduction, and assume for simplicity that ε := ker(𝔽_{q}[t] → O_{F_{𝔪}}/𝔪) = (t). Set z := t and ζ := ϑ. Then the z-adic completion of the (unique) good model of \underline{M} over O_{F_{𝔪}} is a free O_{F_{𝔪}}[[z]]- module M̂ equipped with an isomorphism τM̂: σ^{∗}M̂[\frac{1}{z−ζ}] ^{∼} ⟶ M̂[\frac{1}{z−ζ}]. The pair \underline{M̂} = (M̂, τM̂) is the local shtuka at ε associated with \underline{M}; see Example 2.2 for more details. Via their Tate modules Ť_{ε}\underline{M̂} := (M̂ ⊗_{O_{F𝔪}[[z]]} F_{𝔪}^{sep}[[z]])^{τ} and V̌_{ε}\underline{M̂} := Ť_{ε}\underline{M̂} ⊗_{A_{ε}} Frac(A_{ε}) local shtukas give rise to actual representations of Gal(F_{𝔪}^{sep}/F_{𝔪}) on finite dimensional vector spaces over the field Q_{ε} := Frac(A_{ε}) which also equals the completion of 𝔽_{q}(t) at the place ε. The category of equal characteristic crystalline representations of Gal(F_{𝔪}^{sep}/F_{𝔪}) is defined to be the essential image of the functor \underline{M̂} ↦ V̌_{ε}\underline{M̂} on local shtukas over O_{F_{𝔪}}. We explain the reason for this in Remarks 5.13 and 6.16.

Let us outline the content of this article. For ε ≠ (t) the definition of local shtukas and of local shtukas at ε associated with 𝔽_{q}[t]-motives is given in Section 2. In Section 3 we discuss their relation with the function field analogs of p-divisible groups and in Section 4 we explain the Tate module functor and show that it is fully faithful. Moreover, we define the de Rham realization of local shtukas and 𝔽_{q}[t]-motives and establish a comparison isomorphism h_{ε,dR} between the ε-adic (dual) Tate module and the de Rham realization. In Section 5 we define the crystalline realization of local shtukas and 𝔽_{q}[t]-motives and we construct the comparison isomorphisms h_{dR,cris} between the de Rham and the crystalline realization and h_{ε,cris} between the ε-adic (dual) Tate module and the crystalline realization. The former provides on the crystalline realization a Hodge structure in the sense of Pink. These are the function field analogs of Fontaine’s filtered Frobenius-isocrystals which classify crystalline p-adic Galois representations. In Section 6 we describe which function field isocrystals with Hodge-Pink structure arise from local shtukas using the analog of Fontaine’s notion of weak admissibility.

The function field analog of Fontaine’s theory has a somewhat surprising application to Galois deformation theory; namely, there exists an equi-characteristic local analog of flat deformation rings (or rather, crystalline deformation rings, see Theorems 8.4 and 8.9), which have been intensively studied in the p-adic setting for applications to modularity lifting theorems and congruences of modular forms; see for example [Kis08, Kis09a, Kis09b, GK14]. The existence of equi-characteristic Galois deformation theory is somewhat surprising because the usual (unrestricted) deformation ring does not exist in the category of complete local noetherian rings since the tangent space cannot be bounded; cf. Remark 8.2. Nonetheless, the equi-characteristic analog of Fontaine’s theory (or rather, its torsion version) provides a means to bound the tangent space of the deformation functor. We give an expository overview of the equi-characteristic local Galois deformation theory in Section 8, which is analogous to, and was inspired by Kisin’s technique to study flat deformation rings [Kis09a]. In order to develop Galois deformation theory, we need some techniques to handle torsion Galois representations – some analog of Raynaud’s theory of finite flat group schemes – which is explained in Section 7.

Acknowledgments. These notes grew out of two lectures given by the authors in the fall of 2009 at the conference “t-motives: Hodge structures, transcendence and other motivic aspects” at the Banff International Research Station BIRS in Banff, Canada and a lecture course given by the first author at the Centre de Recerca Mathemàtica CRM in Barcelona, Spain in the spring of 2010. The authors are grateful to BIRS and CRM for their hospitality. The first author acknowledges support of the DFG (German Research Foundation) in form of SFB 878.

1.1 Notation

Throughout this article we denote by

𝔽_{q} a finite field with q elements, C a smooth projective geometrically irreducible curve over 𝔽_{q}, Q := 𝔽_{q}(C) the function field of C, ∞ a fixed closed point of C, A := Γ(C ∖ {∞}, O_{C}) the ring of regular functions on C outside ∞. This generalizes the rings A = 𝔽_{q}[t] and Q = 𝔽_{q}(t) from the introduction, which correspond to C = ℙ_{𝔽_{q}}^{1}.

K a field which is complete with respect to a non-trivial absolute value | . |: K → ℝ_{≥0}

|K^{×}| = {|x|: x ∈ K^{×}} its value group. We do not assume that |K^{×}| is discrete in ℝ_{>0}. R = {x ∈ K: |x| ≤ 1} the valuation ring of K. 𝔪_{R} = {x ∈ R: |x| < 1} the maximal ideal of R, k = R/𝔪_{R} the residue field of R, γ: A ↪ R an injective ring homomorphism,

ε := γ^{−1}(𝔪_{R})  the kernel of the induced homomorphism A → k. It is called the A- characteristic of k. We assume that ε ⊂ A is a maximal ideal.

𝔽_{ε} = A/ε the finite residue field at ε, q̂ = #𝔽_{ε} the cardinality of 𝔽_{ε}. It satisfies q̂ = q^{[𝔽_{ε}:𝔽_{q}]}. z a fixed uniformizing parameter of A at ε, ζ = γ(z) the image of z in R, A_{ε} the completion of the stalk O_{C,ε} at ε, which is canonically isomorphic to 𝔽_{ε}[[z]], Q_{ε} = Frac(A_{ε}) its fraction field, R[[z]] the power series ring in the variable z over R, σ̂ the endomorphism of R[[z]] with σ̂(z) = z and σ̂(b) = bq̂ for b ∈ R.

2 Local Shtukas

The function field analog of Fontaine’s theory of p-adic Galois representations rests on the concept of local shtukas. We are interested in two kinds of base schemes S in this article. The first kind is S = Spec R where R is a valuation ring as in Notation 1.1. The second kind are schemes S in the category Nilp_{A_{ε}} of A_{ε}-schemes on which a uniformizer z of A_{ε} is locally nilpotent. We denote by ζ the image of z in R, respectively in the structure sheaf O_{S} of S. The relation between the two kinds of base schemes is that Spec R/(ζ^{n}) ∈ Nilp_{A_{ε}} for all positive integers n.

To define local shtukas over S = Spec R we consider finite free modules M̂ over the power series ring R[[z]] and set σ̂^{∗}M̂ := M̂ ⊗_{R[[z]], σ̂}R[[z]], and M̂[\frac{1}{z−ζ}] := M̂ ⊗_{R[[z]]}R[[z]][\frac{1}{z−ζ}], and M̂[\frac{1}{z}] := M̂ ⊗_{R[[z]]}R[[z]][\frac{1}{z}]. For more general S we let O_{S}[[z]] be the sheaf of O_{S}-algebras on S for the fpqc-topology whose ring of sections on an S-scheme Y is the ring of power series O_{S}[[z]](Y) := Γ(Y, O_{Y})[[z]]. This is indeed a sheaf being the countable direct product of O_{S}. Let O_{S}[[z]][\frac{1}{z−ζ}], respectively O_{S}((z)), be the fpqc-sheaf of O_{S}- algebras on S associated with the presheaf Y ↦ Γ(Y, O_{Y})[[z]][\frac{1}{z−ζ}], respectively Y ↦ Γ(Y, O_{Y})[[z]][\frac{1}{z}]. If Y is quasi-compact then Γ(Y, O_{S}[[z]][\frac{1}{z−ζ}]) = Γ(Y, O_{Y})[[z]][\frac{1}{z−ζ}] and Γ(Y, O_{S}((z))) = Γ(Y, O_{Y})[[z]][\frac{1}{z}] by [Har77, Exercise II.1.11]. Also O_{S}[[z]][\frac{1}{z−ζ}] = O_{S}((z)) if S ∈ Nilp_{A_{ε}}. Let σ̂ be the endomorphism of O_{S}[[z]] with σ̂(z) = z and σ̂(b) = bq̂ for sections b of O_{S}. For a sheaf of O_{S}[[z]]-modules M̂ we set σ̂^{∗}M̂ := M̂ ⊗_{O_{S}[[z]], σ̂} O_{S}[[z]], as well as M̂[\frac{1}{z−ζ}] := M̂ ⊗_{O_{S}[[z]]} O_{S}[[z]][\frac{1}{z−ζ}] and M̂[\frac{1}{z}] := M̂ ⊗_{O_{S}[[z]]} O_{S}((z)). There is a natural σ̂-semilinear map M̂ → σ̂^{∗}M̂, m ↦ σ̂∗ ; M̂m := m ⊗ 1. For a morphism of O_{S}[[z]]- modules f: M̂ → M̂′ we set σ̂^{∗}f := f ⊗ id: σ̂^{∗}M̂ → σ̂^{∗}M̂′. Note that by [HV11, Proposition 2.3] a sheaf of O_{S}[[z]]-modules which fpqc-locally on S is isomorphic to O_{S}[[z]]^{⊕r} is already Zariski-locally on S isomorphic to O_{S}[[z]]^{⊕r}. We therefore call such a sheaf simply a locally free O_{S}[[z]]-module of rank r. If S = Spec R for a valuation ring R as above, such a sheaf corresponds to a free R[[z]]-module of rank r. Note that in this case R[[z]][\frac{1}{z−ζ}] ≠ R[[z]][\frac{1}{z}]

Definition 2.1. A local σ̂-shtuka (or local shtuka) of rank r over S is a pair \underline{M̂} = (M̂, τM̂) consisting of a locally free O_{S}[[z]]-module M̂ of rank r, and an isomorphism τM̂: σ̂^{∗}M̂[\frac{1}{z−ζ}] ^{∼} ⟶ M̂[\frac{1}{z−ζ}]. If τM̂(σ̂^{∗}M̂) ⊂ M̂ then \underline{M̂} is called effective, and if τM̂(σ̂^{∗}M̂) = M̂ then \underline{M̂} is called étale.

A morphism of local shtukas f: (M̂, τM̂) → (M̂′, τM̂′) over S is a morphism of O_{S}[[z]]-modules f: M̂ → M̂′ which satisfies τM̂′ ∘ σ̂^{∗}f = f ∘ τM̂. We denote the set of morphisms from \underline{M̂} to \underline{M̂}′ by Hom_{S}(\underline{M̂}, \underline{M̂}′).

A quasi-morphism between local shtukas f: (M̂, τM̂) → (M̂′, τM̂′) over S is a morphism of O_{S}((z))- modules f: M[\frac{1}{z}] ^{∼} ⟶ M′[\frac{1}{z}] with τM̂′ ∘σ̂^{∗}f = f ∘τM̂. It is called a quasi-isogeny if it is an isomorphism of O_{S}((z))-modules. We denote the set of quasi-morphisms from \underline{M̂} to \underline{M̂}′ by QHom_{S}(\underline{M̂}, \underline{M̂}′).

Of fundamental importance is the following

Example 2.2. As in Notation 1.1 let R be a valuation ring and set A_{R} := A ⊗_{𝔽_{q}} R, and let σ := id_{A} ⊗ Frob_{q,R} be the endomorphism of A_{R} with σ(a ⊗ b) = a ⊗ b^{q} for a ∈ A and b ∈ R. By an effective A-motive of rank r over R we mean a pair \underline{M} = (M, τ_{M}) consisting of a locally free A_{R}-module M of rank r and an injective A_{R}-homomorphism τ_{M}: σ^{∗}M ↪ M whose cokernel is a finite free R-module and annihilated by a power of the ideal J := (a⊗1−1⊗γ(a): a ∈ A) = ker(γ ⊗ id_{R}: A_{R} → → R) ⊂ A_{R}.

More generally, an A-motive of rank r over R is a pair \underline{M} = (M, τ_{M}) consisting of a locally free A_{R}- module M of rank r and an isomorphism τ_{M}: σ^{∗}M|_{Spec A_{R}∖V(J)} ^{∼} ⟶ M|_{Spec A_{R}∖V(J)} of the associated sheaves outside V(J) ⊂ Spec A_{R}. Note that if A = 𝔽_{q}[t] then J = (t − γ(t)) and Spec A_{R} ∖ V(J) = Spec R[t][\frac{1}{t−γ(t)}]. An A-motive over R should be viewed as a model over R with good reduction of the A-motive \underline{M} ⊗_{R} K over K of generic A-characteristic (0) = ker(γ: A → K). Its reduction \underline{M} ⊗_{R} k is an A-motive over k of A-characteristic ε = ker(A → k). Note that we discussed the case where A = 𝔽_{q}[t] in the introduction.

Let \underline{M} be an (effective) A-motive. We consider the ε-adic completions A_{ε,R} of A_{R} and \underline{M} ⊗_{A_{R}} A_{ε,R} := (M ⊗_{A_{R}} A_{ε,R} , τ_{M} ⊗ id) of \underline{M}. If 𝔽_{ε} = 𝔽_{q}, and hence q̂ = q and σ̂ = σ, we have A_{ε,R} = R[[z]] and J · A_{ε,R} = (z − ζ) because R ⊗_{A_{R}} A_{ε,R} = R. So \underline{M} ⊗_{A_{R}} A_{ε,R} is an (effective) local shtuka over Spec R which we denote by \underline{M̂}_{ε}(\underline{M}) and call the local σ̂-shtuka at ε associated with \underline{M}. If f := [𝔽_{ε} : 𝔽_{q}] > 1 the construction is slightly more complicated. Namely, by continuity the map γ extends to a ring homomorphism γ: A_{ε} ↪ R. We consider the canonical isomorphism 𝔽_{ε}[[z]] ^{∼} ⟶ A_{ε} and the ideals 𝔞_{i} = (a ⊗ 1 − 1 ⊗ γ(a)^{q^{i}}: a ∈ 𝔽_{ε}) ⊂ A_{ε,R} for i ∈ ℤ/fℤ. By the Chinese remainder theorem A_{ε,R} decomposes

A_{ε,R} = ∏_{i∈ℤ/fℤ} A_{ε,R}/𝔞_{i} .

Each factor is canonically isomorphic to R[[z]]. The factors are cyclically permuted by σ because σ(𝔞_{i}) = 𝔞_{i+1}. In particular σ^{f} stabilizes each factor. The ideal J decomposes as follows J ·A_{ε,R}/𝔞_{0} = (z − ζ) and J · A_{ε,R}/𝔞_{i} = (1) for i ≠ 0. We define the local shtuka at ε associated with \underline{M} as \underline{M̂}_{ε}(\underline{M}) := (M̂, τM̂) := (M ⊗_{A_{R}} A_{ε,R}/𝔞_{0} , (τ_{M} ⊗ 1)^{f}), where τ_{M}^{f} := τ_{M} ∘ σ^{∗}τ_{M} ∘ … ∘ σ^{(f−1)∗}τ_{M}. Of course if f = 1 we get back the definition of \underline{M̂}_{ε}(\underline{M}) given above. Also note if \underline{M} is effective, then M/τ_{M}(σ^{∗}M) = M̂/τM̂(σ̂^{∗}M̂).

The local shtuka \underline{M̂}_{ε}(\underline{M}) allows to recover \underline{M} ⊗_{A_{R}} A_{ε,R} via the isomorphism

⨁_{i=0}^{f−1}(τ_{M} ⊗ 1)^{i} mod 𝔞_{i}: ( ⨁_{i∈ℤ/fℤ} σ^{i∗}(M ⊗_{A_{R}} A_{ε,R}/𝔞_{0}), (τ_{M} ⊗ 1)^{f} ⊕ ⨁_{i≠0} id) ^{∼} ⟶ \underline{M} ⊗_{A_{R}} A_{ε,R} ,

because for i ≠ 0 the equality J ·A_{ε,R}/𝔞_{i} = (1) implies that τ_{M} ⊗ 1 is an isomorphism modulo 𝔞_{i}; see [BH11, Propositions 8.8 and 8.5] for more details.

The philosophy is that \underline{M̂}_{ε}(\underline{M}) encodes all the local information of \underline{M} at ε like the (dual) Tate module Ť_{v}\underline{M} encodes all the local information of \underline{M} at v ≠ ε. The whole example also works over a base scheme S ∈ Nilp_{A_{ε}} instead of S = Spec R; see [Har15, Example 6.2] and [HS15, § 6].

We will need the following lemma whose second part is proved more generally for S ∈ Nilp_{A_{ε}} in [HS15, Lemma 2.2].

Lemma 2.3. Let \underline{M̂} = (M̂, τM̂) be a local shtuka of rank r over a valuation ring R as in Notation 1.1.

(a) There is an integer d ∈ ℤ such that det τM̂ ∈ (z − ζ)^{d}·R[[z]]^{×}.

(b) If \underline{M̂} is effective, the integer d from (a) satisfies d ≥ 0 and M̂/τM̂(σ̂^{∗}M̂) is a free R-module of rank d which is annihilated by (z − ζ)^{d}.

Proof. Compare [Har11, Proposition 2.1.3 and Lemma 2.1.2].

(a) Since R[[z]] is a local ring, we may choose a basis of M̂ and non-negative integers s, t such that the matrices of (z−ζ)^{s}·τM̂ and (z−ζ)^{t}·τ −1 ; M̂ with respect to this basis lie in M_{r}(R[[z]]). Set f = (z−ζ)^{rs} det τM̂ and g = (z − ζ)^{rt} det τ −1 ; M̂. Then f, g ∈ R[[z]] satisfy fg = (z − ζ)^{r(s+t)}. Since R is an integral domain, (z − ζ)R[[z]] is a prime ideal. So f ∈ (z − ζ)^{u} ·R[[z]]^{×} for a non-negative integer u, and so det τM̂ ∈ (z − ζ)^{u−rs}·R[[z]]^{×}.

(b) Let d be the integer from (a). Since \underline{M} is effective we may take s = 0, and hence d ≥ 0. By Cramer’s rule (e.g. [Bou70, III.8.6, Formulas (21) and (22)]) the matrix of τ −1 ; M̂ lies in M_{r}((z − ζ)^{−d}R[[z]]). This implies that M̂/τM̂(σ̂^{∗}M̂) is annihilated by (z − ζ)^{d}. By [Har11, Lemma 2.1.2] it is a finite free R-module. We can compute its rank after reducing modulo 𝔪_{R}. Then we are over the principal ideal domain k[[z]] and the elementary divisor theorem tells us that dim_{k}(M̂/τM̂(σ̂^{∗}M̂)) ⊗_{R} k = ord_{z}(det τM̂ mod 𝔪_{R}) = ord_{z}((z − ζ)^{d} mod 𝔪_{R}) = d.

More precisely [Har11, Proposition 2.1.3] (and the lemma) say that all (effective) local shtukas over Spec R are bounded (by (d, 0, …, 0) for d = rk_{R} M̂/τM̂(σ̂^{∗}M̂)) as in the following

Definition 2.4. Let μ_{1} ≥ … ≥ μ_{r} be a decreasing sequence of integers. A local shtuka (M̂, τM̂) of rank r over S is bounded by (μ_{1}, …, μ_{r}) if

τM̂(∧^{i}σ̂^{∗}M̂) ⊂ (z − ζ)^{μ_{r−i+1}+...+μ_{r}} · ∧^{i}M̂ for 1 ≤ i ≤ r with equality for i = r .

Although over a general base boundedness is not preserved under quasi-isogenies this is true over a valuation ring.

Lemma 2.5. Let \underline{M̂} and \underline{M̂}′ be two isogenous local shtukas over a valuation ring R as in Notation 1.1. If \underline{M̂} is bounded by (d, 0, …, 0), respectively satisfies (z − ζ)^{a}M̂ ⊂ τM̂(σ̂^{∗}M̂) ⊂ (z − ζ)^{b}M̂, then the same is true for \underline{M̂}′.

Proof. Choosing bases of M and M′ we write τ_{M} and τ_{M′} as matrices T, T′ ∈ GL_{r}(R[[z]][\frac{1}{z−ζ}]) and the quasi-isogeny f: \underline{M}[\frac{1}{z}] ^{∼} ⟶ \underline{M}′[\frac{1}{z}] as a matrix F ∈ GL_{r}(R[[z]][\frac{1}{z}]) satisfying T′ = F·T·σ̂^{∗}(F)^{−1}. Depending on the assumption on \underline{M}, the matrix coefficients of T′, (z − ζ)^{−d} det(T′) and (z − ζ)^{d} det(T′)^{−1}, respectively (z − ζ)^{−b}T′ and (z − ζ)^{a}(T′)^{−1} lie in R[[z]][\frac{1}{z}]. Since they also lie in R[[z]][\frac{1}{z−ζ}], we can write them as f/z^{s} = g/(z − ζ)^{t}. The term (z − ζ)^{t}f = z^{s}g lies in the prime ideal (z − ζ) ⊂ R[[z]], but z does not. Therefore g is divisible by (z − ζ)^{t} and all the matrix coefficients g/(z − ζ)^{t} lie in R[[z]] as desired.

Remark 2.6. If we are considering a local shtuka \underline{M̂} over Spec R for a valuation ring R as in Notation 1.1, we obtain for all i ∈ ℕ a local shtuka \underline{M̂}^{(i)} := \underline{M̂} ⊗_{R} R/(ζ^{i}) over Spec R/(ζ^{i}). The \underline{M̂}^{(i)} form a local shtuka over Spf R by which we mean a projective system (\underline{M̂}^{(i)})_{i∈ℕ} of local shtukas \underline{M̂}^{(i)} over Spec R/(ζ^{i}) together with isomorphisms \underline{M̂}^{(i+1)}⊗_{R/(ζ^{i+1})}R/(ζ^{i}) ^{∼} ⟶ \underline{M̂}^{(i)}. By [HV11, Proposition 3.16 and § 4] the functor \underline{M̂} ↦ (\underline{M̂}^{(i)})_{i∈ℕ} is an equivalence between local shtukas bounded by (μ_{1}, …, μ_{r}) over Spec R and over Spf R. In that sense the theory of local shtukas over Spec R is subsumed under the theory of bounded local shtukas over schemes S ∈ Nilp_{A_{ε}}.

Example 2.7. We discuss the case of the Carlitz module [Car35]. We keep the notation from Example 2.2 and set A = 𝔽_{q}[t]. Let 𝔽_{q}(ϑ) be the rational function field in the variable ϑ and let γ: A → 𝔽_{q}(ϑ) be given by γ(t) = ϑ. The Carlitz motive over 𝔽_{q}(ϑ) is the A-motive \underline{M} = (𝔽_{q}(ϑ)[t], t − ϑ).

Now let ε = (z) ⊂ A be a maximal ideal generated by a monic prime element z = z(t) ∈ 𝔽_{q}[t]. Then 𝔽_{ε} = A/(z) and A_{ε} is canonically isomorphic to 𝔽_{ε}[[z]]. Let R ⊃ 𝔽_{ε}[[ζ]] be a valuation ring as in Notation 1.1 and let ϑ = γ(t) ∈ R. The Carlitz motive has a model over R with good reduction given by the A-motive \underline{M} = (R[t], t − ϑ) over R.

If deg_{t} z(t) = 1, that is z(t) = t − a for a ∈ 𝔽_{q}, then 𝔽_{ε} = 𝔽_{q}, ζ = ϑ − a, and z − ζ = t − ϑ. So \underline{M̂}_{ε}(\underline{M}) = (R[[z]], z − ζ).

If deg_{t} z(t) = f > 1, then \underline{M̂}_{ε}(\underline{M}) = (R[[z]], (t − ϑ)(t − ϑ^{q}) · . . . · (t − ϑ^{q^{f−1}})). Here the product (t − ϑ)(t − ϑ^{q}) · . . . · (t − ϑ^{q^{f−1}}) = (z − ζ)u for a unit u ∈ 𝔽_{ε}[[ζ]][[z]]^{×}, because τ_{M}(σ^{∗}M) = (t − ϑ)M implies that \underline{M̂}_{ε}(\underline{M}) is effective and the d from Lemma 2.3 is 1. In order to get rid of u we denote the image of t in 𝔽_{ε} by λ. Then 𝔽_{ε} = 𝔽_{q}(λ) and z(t) equals the minimal polynomial (t − λ) · . . . · (t − λ^{q^{f−1}}) of λ over 𝔽_{q}. Moreover, t ≡ λ mod zA_{ε} and ϑ ≡ λ mod ζ𝔽_{ε}[[ζ]]. We compute in 𝔽_{ε}[[ζ]][[z]]/(ζ)

z(t) = (t − λ) · . . . · (t − λ^{q^{f−1}}) ≡ (t − ϑ) · . . . · (t − ϑ^{q^{f−1}}) ≡ (z − ζ)u ≡ zu mod ζ .

Since z is a non-zero-divisor in 𝔽_{ε}[[ζ]][[z]]/(ζ) it follows that u ≡ 1 mod ζ 𝔽_{ε}[[ζ]][[z]]. We write u = 1+ζu′ and observe that the product

w := ∏_{n=0}^{∞} σ̂^{n}(u) = ∏_{n=0}^{∞} σ̂^{n}(1 + ζu′) = ∏_{n=0}^{∞}(1 + ζ^{q̂^{n}}σ̂^{n}(u′))

converges in 𝔽_{ε}[[ζ]][[z]]^{×} because 𝔽_{ε}[[ζ]][[z]] is ζ-adically complete. It satisfies w = u · σ̂(w) and so multiplication with w defines a canonical isomorphism (R[[z]], z − ζ) ^{∼} ⟶ \underline{M̂}_{ε}(\underline{M}).

We conclude that \underline{M̂}_{ε}(\underline{M}) = (R[[z]], z − ζ), regardless of deg_{t} z(t).

3 Divisible local Anderson modules

Let S ∈ Nilp_{A_{ε}} and let \underline{M̂} = (M̂, τM̂) be an effective local shtuka over S. Set \underline{M̂}_{n} := (M̂_{n}, τM̂_{n}) :=

(M̂/z^{n}M̂, τM̂ mod z^{n}). For m ∈ M̂_{n} set σ̂∗ ; M̂m := m⊗1 ∈ M̂_{n}⊗_{O_{S}[[z]], σ̂}O_{S}[[z]] =: σ̂^{∗}M̂_{n}. Drinfeld [Dri87,

§ 2] associates with \underline{M̂}_{n} a group scheme

Drq̂(\underline{M̂}_{n}) := Spec(Sym_{O_{S}}(M̂_{n})/(m^{⊗q̂} − τM̂(σ̂∗ ; M̂m): m ∈ M̂_{n}))  (3.1)
= Spec O_{S}[m_{1}, …, m_{nr}]/(mq̂ ; i − τM̂(σ̂∗ ; M̂m_{i}): 1 ≤ i ≤ nr),  (3.2)

if M̂_{n} = ⨁_{i=1}^{nr} O_{S} · m_{i} locally on S. It has the following properties

• Drq̂(\underline{M̂}_{n}) ⊂ Spec Sym_{O_{S}}(M̂_{n}) is a finite locally free subgroup scheme over S of order nr, that is, the O_{S}-algebra O_{Drq̂(\underline{M̂}_{n})} is a finite locally free O_{S}-module of rank nr. Note that locally on S

we have M̂_{n} = ⨁_{i=1}^{nr} O_{S} · m_{i} and Spec Sym_{O_{S}}(M̂_{n}) ≅ Spec O_{S}[m_{1}, …, m_{nr}] = 𝔾_{a,S}^{nr}.

• Drq̂(\underline{M̂}_{n}) inherits from \underline{M̂}_{n} an action of A_{ε}/(z^{n}) = 𝔽_{ε}[z]/(z^{n}).

• The Verschiebung map is zero on Drq̂(\underline{M̂}_{n}).

• There is a canonical isomorphism between coker τM̂_{n} = M̂/(z^{n}M̂ + τM̂(σ̂^{∗}M̂)) and the co-Lie module ω_{Drq̂(\underline{M̂}_{n})} := e^{∗}Ω_{Drq̂(\underline{M̂}_{n})/S}^{1} where e: S → Drq̂(\underline{M̂}_{n}) is the zero section. See [HS15, Theorem 5.2(e)] for a proof.

• Drq̂(\underline{M̂}_{n}) is a strict 𝔽_{ε}-scheme in the sense of Faltings [Fal02] and Abrashkin [Abr06]. See [Abr06, Theorem 2] for a proof.

Conversely we recover M̂_{n} as the A_{ε}/(z^{n})-module of 𝔽_{ε}-linear morphisms of S-group schemes Hom_{S-groups,𝔽_{ε}-lin}(Drq̂(\underline{M̂}_{n}) , 𝔾_{a,S}) by [Abr06, Theorem 2] or [HS15, Theorem 5.2]. Moreover the structure as O_{S}-module is given via the action of O_{S} on the additive group scheme 𝔾_{a,S} and τM̂_{n} corresponds to the σ̂-semilinear map m ↦ Frob_{q̂,𝔾_{a,S}} ∘m, where Frob_{q̂,𝔾_{a,S}} is the absolute q̂-Frobenius of 𝔾_{a,S}.

The canonical epimorphisms \underline{M̂}_{n+1} → → \underline{M̂}_{n} induce closed immersions i_{n}: Drq̂(\underline{M̂}_{n}) ↪ Drq̂(\underline{M̂}_{n+1}). The inductive limit Drq̂(\underline{M̂}) := lim_{⟶} Drq̂(\underline{M̂}_{n}) in the category of sheaves on the big fppf-site of S is a sheaf of 𝔽_{ε}[[z]]-modules that satisfies the following

Definition 3.1. A z-divisible local Anderson module over S is a sheaf of 𝔽_{ε}[[z]]-modules G on the big fppf-site of S such that

(a) G is z-torsion, that is G = lim_{⟶} G[z^{n}], where G[z^{n}] := ker(z^{n}: G → G),

(b) G is z-divisible, that is z: G → G is an epimorphism,

(c) For every n the 𝔽_{ε}-module G[z^{n}] is representable by a finite, locally free, strict 𝔽_{ε}-module scheme over S in the sense of Faltings [Fal02] and Abrashkin [Abr06], and

(d) locally on S there exist an integer d ∈ ℤ_{≥0}, such that (z − ζ)^{d} = 0 on ω_{G} where ω_{G} := lim_{⟵} ω_{G[z^{n}]}

and ω_{G[z^{n}]} = e^{∗}Ω_{G[z^{n}]/S}^{1} is the pullback under the zero section e: S → G[z^{n}].

A morphism of z-divisible local Anderson modules over S is a morphism of fppf-sheaves of 𝔽_{ε}[[z]]- modules. The category of divisible local Anderson modules is 𝔽_{ε}[[z]]-linear. It is shown in [HS15, Lemma 8.2] that ω_{G} is a finite locally free O_{S}-module and we define the dimension of G as rk ω_{G}.

Remark 3.2. By [HS15, Theorem 8.3] the functor \underline{M̂} ↦ Drq̂(\underline{M̂}) is an anti-equivalence between the category of effective local σ̂-shtukas over S and the category of z-divisible local Anderson modules over S. Moreover, it is A_{ε}-linear and exact. Various properties are preserved under this anti-equivalence. More precisely let \underline{M̂} be an effective local σ̂-shtuka over S and let G = Drq̂(\underline{M̂}).

Then the O_{S}[[z]]-modules M̂/τM̂(σ̂^{∗}M̂) and ω_{G} are canonically isomorphic. In particular, \underline{M̂} is étale, if and only if ω_{G} = (0), if and only if all G[z^{n}] are étale.

The map τM̂ is topologically nilpotent, in the sense that locally on S there is an integer n such that im(τ n ; M̂: σ̂^{n∗}M̂ → M̂) ⊂ zM̂, if and only if G is a formal Lie group.

If \underline{M̂} is bounded by (d, 0, …, 0) we say that G is bounded by d. In this case (z − ζ)^{d} · ω_{G} = (0) globally on S in axiom (d) and dim G = d as can be seen from the elementary divisor theorem applied to the pullback s^{∗}\underline{M̂} to a closed point s: Spec κ(s) → S, where κ(s) is the residue field at s.

Remark 3.3. If \underline{M̂} is a local shtuka over a valuation ring R as in Notation 1.1 and we view it as a projective system of local shtukas \underline{M̂}^{(i)} := (M̂^{(i)}, τM̂^{(i)}) := \underline{M̂} ⊗_{R} R/(ζ^{i}) over Spec R/(ζ^{i}) as in Remark 2.6, then the following are equivalent.

(a) The map τM̂ is topologically nilpotent in the sense that τ n ; M̂(σ^{n∗}M̂) ⊂ ζM̂ + zM̂ for n ≫ 0.

(b) For all i ∈ ℕ the map τM̂^{(i)} is topologically nilpotent, that is τ n ; M̂^{(i)}(σ̂^{n∗}M̂^{(i)}) ⊂ zM̂^{(i)} for n ≫ 0.

(c) There exists an i ∈ ℕ_{>0} for which the map τM̂^{(i)} is topologically nilpotent.

(d) Drq̂(\underline{M̂}) := lim_{⟵_{i}} Drq̂(\underline{M̂}^{(i)}) is a formal Lie group over Spf R.

Indeed, (a) yields τ jn ; M̂(σ^{jn∗}M̂) ⊂ ζ^{1+q̂^{n}+...+q̂^{(j−1)n}} M̂ + zM̂ for all j ∈ ℕ. This implies (b), from which (c) follows trivially. Conversely if (c) holds for some i then τ n ; M̂(σ^{n∗}M̂) ⊂ ζ^{i}M̂ +zM̂ for n ≫ 0, whence (a). Finally, by Remark 3.2 assertion (b) implies (d), and (d) implies that every Drq̂(\underline{M̂}^{(i)}) is a formal Lie group, whence (b).

Example 3.4. Let S = Spec B ∈ Nilp_{A_{ε}} be affine and let d and r be positive integers. An abelian Anderson A-module of rank r and dimension d over S is a pair \underline{E} = (E, φ) consisting of a smooth affine group scheme E over S of relative dimension d, and a ring homomorphism φ: A → End_{S-groups}(E), a ↦ φ_{a} such that

(a) there is a faithfully flat morphism S′ → S for which E ×_{S} S′ ≅ 𝔾_{a,S′}^{d} as 𝔽_{q}-module schemes,

(b) (φ_{a} − γ(a))^{d} = 0 on ω_{E} := e^{∗}Ω_{E/S}^{1} for all a ∈ A, where e: S → E is the zero section,

(c) the set M := M(\underline{E}) := Hom_{S-groups,𝔽_{q}-lin}(E, 𝔾_{a,S}) of 𝔽_{q}-equivariant homomorphisms of S-group schemes is a locally free module over A_{B} := A ⊗_{𝔽_{q}} B of rank r under the action given on m ∈ M by

\begin{matrix}A ∋ a: & M ⟶ M, & m ↦ m ∘ φ_{a} \\ B ∋ b: & M ⟶ M, & m ↦ b ∘ m \end{matrix}

In addition we consider the map τ: m ↦ Frob_{q,𝔾_{a,S}}∘ m on m ∈ M. Since τ(bm) = b^{q}τ(m) the map τ is σ-semilinear and induces an A_{B}-linear map τ_{M}: σ^{∗}M → M, which makes \underline{M}(\underline{E}) := (M(\underline{E}), τ_{M}) into an effective A-motive over S in the sense of Example 2.2. The functor \underline{E} ↦ \underline{M}(\underline{E}) is fully faithful and its essential image is described in [Har15, Theorem 2.10] generalizing Anderson’s description [And86, Theorem 1].

Now let \underline{E} = (E, φ) be an abelian Anderson A-module over S and let \underline{M̂} := \underline{M̂}_{ε}(\underline{M}(\underline{E})) be its associated effective local σ̂-shtuka at ε; see Example 2.2. Let n ∈ ℕ and let ε^{n} = (a_{1}, …, a_{s}) ⊂ A. Then

\underline{E}[ε^{n}] := ker(φ_{a_{1},...,a_{s}} := (φ_{a_{1}}, …, φ_{a_{s}}): E ⟶ E^{s})

is called the ε^{n}-torsion submodule of \underline{E}. It is an A/ε^{n}-module via A/ε^{n} → End_{S}(\underline{E}[ε^{n}]), ā ↦ φ_{a} and independent of the set of generators of ε^{n}; see [Har15, Lemma 5.2]. Moreover, by [Har15, Theorem 6.6] it is a finite S-group scheme of finite presentation and a strict 𝔽_{ε}-module scheme and there are canonical A/ε^{n}-equivariant isomorphisms of finite locally free S-group schemes

Drq̂(\underline{M̂}/ε^{n}\underline{M̂}) ^{∼} ⟶ \underline{E}[ε^{n}]  and
\underline{M̂}/ε^{n}\underline{M̂} ^{∼} ⟶ Hom_{S-groups,𝔽_{ε}-lin}(\underline{E}[ε^{n}] , 𝔾_{a,S})

of torsion local shtukas in the sense of Definition 7.1 below. In particular, \underline{E}[ε^{∞}] := lim_{⟶} \underline{E}[ε^{n}] =

Drq̂(\underline{M̂}) is a z-divisible local Anderson module over S.

Example 3.5. We continue with Example 2.7. Let A = 𝔽_{q}[t] and let ε = (z) ⊂ A be a maximal ideal generated by a monic prime element z = z(t) ∈ 𝔽_{q}[t]. Let R = 𝔽_{ε}[[ζ]] and let \underline{C} := (𝔾_{a,R}, φ) with φ_{t} = ϑ + Frob_{q,𝔾_{a,R}} be the Carlitz module over R. That is, if 𝔾_{a,R} = Spec R[x] then φ_{t}^{∗}(x) = ϑ x + x^{q}. Then \underline{M}(\underline{C}) = (R[t], t − ϑ) is the Carlitz 𝔽_{q}[t]-motive over R from Example 2.7 with associated local shtuka \underline{M̂} := \underline{M̂}_{ε}(\underline{M}(\underline{C})) ≅ (R[[z]], z − ζ) at ε. The ε^{n}-torsion submodule of \underline{C} is \underline{C}[ε^{n}] := ker(φ_{z^{n}}) = Spec R[x]/(φ_{z^{n}}^{∗}(x)). We can compute \underline{M̂}/z^{n}\underline{M̂} = ⨁_{i=1}^{n} R · m_{i} with m_{i} = z^{i−1} and

τM̂(σ̂∗ ; M̂m_{i}) = (z − ζ)m_{i} = m_{i+1} − ζm_{i} for 1 ≤ i < n and τM̂(σ̂∗ ; M̂m_{n}) = −ζm_{n}. By Example 3.4 this

implies

\underline{C}[ε^{n}] ≅ Drq̂(\underline{M̂}/ε^{n}\underline{M̂}) ≅ Spec R[m_{1}, …, m_{n}]/(mq̂ ; n + ζm_{n} , mq̂ ; i − m_{i+1} + ζm_{i}: 1 ≤ i < n) .

On it z acts via φ_{z}^{∗}(m_{i}) = m_{i+1} = ζm_{i} + mq̂ ; i for 1 ≤ i < n and φ_{z}^{∗}(m_{n}) = 0.

Since τM̂(σ^{∗}M̂) = (z − ζ)M̂ ⊂ ζM̂ + zM̂, Remark 3.3 implies that G := \underline{C}[ε^{∞}] ≅ Drq̂(\underline{M̂}) is a formal Lie group over Spf R. Its dimension is 1 because ω_{G} ≅ M̂/τM̂(σ̂^{∗}M̂) = R. Setting x := m_{1} it follows that G ≅ Spf R[[x]] is the formal additive group scheme with the action of A_{ε} = 𝔽_{ε}[[z]] given by φ_{z}^{∗}(x) = ζ x + xq̂ and φ_{a}^{∗}(x) = a x for a ∈ 𝔽_{ε}. So we can alternatively describe \underline{C}[ε^{n}] as Spec R[x]/(φ_{z^{n}}^{∗}(x)) with the latter expression for φ_{z}^{∗}.

4 Tate modules

With a local shtuka one can associate a Galois representation. More precisely let \underline{M̂} = (M̂, τM̂) be a local shtuka over Spec R for a valuation ring R as in Notation 1.1. Then τM̂ induces an isomorphism τM̂: σ̂^{∗}M̂ ⊗_{R[[z]]} K[[z]] ^{∼} ⟶ M̂ ⊗_{R[[z]]} K[[z]], because z − ζ ∈ K[[z]]^{×}. So one could say that \underline{M̂} ⊗_{R[[z]]} K[[z]] is an “étale local shtuka over K”.

Definition 4.1. With \underline{M̂} as above one associates the (dual) Tate module

Ť_{ε}\underline{M̂} := (M̂ ⊗_{R[[z]]} K^{sep}[[z]])τ̂ := \{m ∈ M̂ ⊗_{R[[z]]} K^{sep}[[z]]: τM̂(σ̂∗ ; M̂m) = m\}

and the rational (dual) Tate module

V̌_{ε}\underline{M̂} := \{m ∈ M̂ ⊗_{R[[z]]} K^{sep}((z)): τM̂(σ̂∗ ; M̂m) = m\} = Ť_{ε}\underline{M̂} ⊗_{A_{ε}} Q_{ε} .

One also sometimes writes H_{ε}^{1}(\underline{M̂}, A_{ε}) = Ť_{ε}\underline{M̂} and H_{ε}^{1}(\underline{M̂}, Q_{ε}) = V̌_{ε}\underline{M̂} and calls this the ε-adic realization of \underline{M̂}.

Proposition 4.2. Ť_{ε}\underline{M̂} is a free A_{ε}-module of rank equal to rk \underline{M̂} and V̌_{ε}\underline{M̂} is a Q_{ε}-vector space of dimension equal to rk \underline{M̂}. Both carry a continuous action of Gal(K^{sep}/K). Moreover the inclusion Ť_{ε}\underline{M̂} ⊂ M̂ ⊗_{R[[z]]} K^{sep}[[z]] defines a canonical isomorphism of K^{sep}[[z]]-modules

Ť_{ε}\underline{M̂} ⊗_{A_{ε}} K^{sep}[[z]] ^{∼} ⟶ \underline{M̂} ⊗_{R[[z]]} K^{sep}[[z]]  (4.1)

which is Gal(K^{sep}/K)- and τ̂-equivariant, where on the left module Gal(K^{sep}/K)-acts on both factors and τ̂ is id ⊗σ̂, and on the right module Gal(K^{sep}/K) acts only on K^{sep}[[z]] and τ̂ is (τM̂ ∘ σ̂∗ ; M̂) ⊗ σ̂.

In particular one can recover \underline{M̂} ⊗_{R[[z]]} K[[z]] = (Ť_{ε}\underline{M̂} ⊗_{A_{ε}} K^{sep}[[z]])^{Gal(K^{sep}/K)} as the Galois invariants.

Remark 4.3. We actually prove the stronger statement that for every s > 1/q̂ the isomorphism (4.1) extends to a Gal(K^{sep}/K)- and τ̂-equivariant isomorphism

h: Ť_{ε}\underline{M̂} ⊗_{A_{ε}} K^{sep}⟨\frac{z}{ζ^{s}} ⟩ ^{∼} ⟶ \underline{M̂} ⊗_{R[[z]]} K^{sep}⟨\frac{z}{ζ^{s}} ⟩,  (4.2)

where for a field extension K of 𝔽_{ε}((ζ)) and an s ∈ ℝ_{>0} we use the notation

K⟨\frac{z}{ζ^{s}} ⟩ := \{ ∑_{i=0}^{∞} b_{i}z^{i}: b_{i} ∈ K, |b_{i}| |ζ|^{si} → 0 (i → +∞) \} .

These are subrings of K[[z]] and the endomorphism σ̂: ∑_{i} b_{i}z^{i} ↦ ∑_{i} bq̂ ; iz^{i} of K[[z]] restricts to a homomorphism σ̂: K⟨\frac{z}{ζ^{s}} ⟩ → K⟨\frac{z}{ζ^{sq̂}}⟩. Note that the τ̂-equivariance of h means h⊗ id_{K⟨ \frac{z}{ζ^{sq̂}} ⟩} = τM̂ ∘σ̂^{∗}h.

Proof of Proposition 4.2 and Remark 4.3. That Ť_{ε}\underline{M̂} is an A_{ε}-module and V̌_{ε}\underline{M̂} is a Q_{ε}-vector space comes from the fact that the subring of σ̂-invariants in K^{sep}[[z]] is 𝔽_{ε}[[z]] = A_{ε}.

We set r = rk \underline{M̂}, choose an R[[z]]-basis of M̂, and write τM̂ with respect to this basis as a matrix T ∈ GL_{r}(R[[z]][\frac{1}{z−ζ}]). Since z − ζ is a unit in K⟨\frac{z}{ζq̂}⟩ with (z − ζ)^{−1} = − ∑_{i=0}^{∞} ζ^{−i−1}z^{i} there is an inclusion R[[z]][\frac{1}{z−ζ}] ↪ K⟨\frac{z}{ζq̂}⟩, and we consider T as a matrix in GL_{r}(K⟨\frac{z}{ζq̂}⟩). We claim that there is a matrix U ∈ GL_{r}(K^{sep}[[z]]) with σ̂(U) = T^{−1}U. We write T^{−1} = ∑_{i=0}^{∞} T_{i}z^{i} and U = ∑_{n=0}^{∞} U_{n}z^{n}. We must solve the equations

σ̂(U_{0}) = T_{0} · U_{0}  and  σ̂(U_{n}) = ∑_{i=0}^{n} T_{n−i} · U_{i} .  (4.3)

By Lang’s theorem [Lan56, Corollary on p. 557] there exists a matrix U_{0} ∈ GL_{r}(K^{alg}) satisfying (4.3). Since the morphism GL_{r}(K^{alg}) → GL_{r}(K^{alg}), U_{0} ↦ U_{0} · σ̂(U_{0})^{−1} = T_{0}^{−1} is étale, we actually have U_{0} ∈ GL_{r}(K^{sep}). Then the second equation takes the form

σ̂(U_{0}^{−1}U_{n}) − U_{0}^{−1}U_{n} = ∑_{i=0}^{n−1} σ̂(U_{0})^{−1}T_{n−i}U_{0} · (U_{0}^{−1}U_{i}) .

This is a system of Artin-Schreier equations for the coefficients of U_{0}^{−1}U_{n} which can be solved in K^{sep} and this establishes our claim.

We show that the matrix U ∈ GL_{r}(K^{sep}[[z]]) obtained in this way lies in GL_{r}(K^{sep}⟨\frac{z}{ζ^{s}} ⟩) for every 1 ≥ s > 1/q̂. Since T^{−1} ∈ GL_{r}(K⟨\frac{z}{ζ^{sq̂}}⟩) ⊂ GL_{r}(K⟨\frac{z}{ζq̂}⟩) there is a constant c ≥ 1 with |T_{i}ζ^{q̂i}| ≤ c for all i, where |T_{i}ζ^{q̂i}| denotes the maximal absolute value of the entries of the matrix T_{i}ζ^{q̂i}. We write (4.3) as

σ̂(U_{n}ζ^{n}) = ∑_{i=0}^{n}(T_{n−i}ζ^{q̂(n−i)})(U_{i}ζ^{i}) ζ^{i(q̂−1)} .

In view of |ζ| < 1 this implies the estimate

|U_{n}ζ^{n}|q̂ ≤ c · max{ |U_{i}ζ^{i}| : 0 ≤ i ≤ n }

from which induction yields |U_{n}ζ^{n}| ≤ c^{1/(q̂−1)} for all n ≥ 0. In particular, if s > 1/q̂, then sq̂ > 1 and U ∈ M_{r}(K^{sep}⟨\frac{z}{ζ^{sq̂}}⟩). But now the equation σ̂(U) = T^{−1}U shows that σ̂(U) ∈ M_{r}(K^{sep}⟨\frac{z}{ζ^{sq̂}}⟩), hence U ∈ M_{r}(K^{sep}⟨\frac{z}{ζ^{s}} ⟩). A similar reasoning with the equation σ̂(U^{−1}) = U^{−1}T shows that also U^{−1} ∈ M_{r}(K^{sep}⟨\frac{z}{ζ^{s}} ⟩) and U ∈ GL_{r}(K^{sep}⟨\frac{z}{ζ^{s}} ⟩) as desired.

Multiplication with U provides an isomorphism (K^{sep}⟨\frac{z}{ζ^{s}} ⟩^{⊕r}, τ̂ = id) ^{∼} ⟶ \underline{M̂} ⊗_{R[[z]]} K^{sep}⟨\frac{z}{ζ^{s}} ⟩. It follows that Ť_{ε}\underline{M̂} = U ·A_{ε}^{⊕r} is free of rank r, and the inclusion Ť_{ε}\underline{M̂} ⊂ U ·K^{sep}⟨\frac{z}{ζ^{s}} ⟩^{⊕r} induces the equivariant isomorphism (4.2). Since K^{sep}⟨\frac{z}{ζ^{s}} ⟩ ⊂ K^{sep}[[z]] this induces the isomorphism (4.1).

The continuity of the Galois representation ρ_{\underline{M̂}}: Gal(K^{sep}/K) → Aut_{A_{ε}}(Ť_{ε}\underline{M̂}) means that for all n ∈ ℕ the subgroup \{ g ∈ Gal(K^{sep}/K): ρ_{\underline{M̂}}(g) ≡ id mod z^{n} \} is open. This is true because ρ_{\underline{M̂}} is given by the Galois action on the coefficients of the matrix U, and so this subgroup contains the open subgroup Gal(K^{sep}/K(U_{0}, …, U_{n−1})).

Remark 4.4. There is a statement similar to Proposition 4.2 for an étale local shtuka \underline{M̂} over a connected scheme S ∈ Nilp_{A_{ε}}. For a geometric base point s̄ ∈ S the (dual) Tate module and the rational (dual) Tate module

Ť_{ε}\underline{M̂} := (\underline{M̂} ⊗_{O_{S}[[z]]} κ(s̄)[[z]])τ̂  and  V̌_{ε}\underline{M̂} := Ť_{ε}\underline{M̂} ⊗_{A_{ε}} Q_{ε}

are free of rank rk \underline{M̂} and carry a continuous action of the étale fundamental group π_{1}^{ét}(S, s̄). Moreover, the functor \underline{M̂} ↦ Ť_{ε}\underline{M̂} is an equivalence between the category of étale local shtukas over S and the category of representations of π_{1}^{ét}(S, s̄) on finite free A_{ε}-modules. Similarly, the functor \underline{M̂} ↦ V̌_{ε}\underline{M̂} is an equivalence between the category of étale local shtukas over S with quasi-morphisms and the category of representations of π_{1}^{ét}(S, s̄) on finite Q_{ε}-vector spaces; see [AH15, Proposition 3.4].

Back in the situation over a valuation ring R as in Notation 1.1 there is the following

Proposition 4.5. If \underline{M̂} = \underline{M̂}_{ε}(\underline{M}) for an A-motive \underline{M} = (M, τ_{M}) over R as in Example 2.2, the εadic (dual) Tate module of \underline{M} defined by Ť_{ε}\underline{M} := \{m ∈ M ⊗_{A_{R}} A_{ε,K^{sep}} : τ_{M}(σ_{M}^{∗}m) = m\} is canonically isomorphic to Ť_{ε}\underline{M̂} as representations of Gal(K^{sep}/K).

Proof. Consider the decomposition M ⊗_{A_{R}} A_{ε,K^{sep}} = ∏_{i∈ℤ/fℤ} M ⊗_{A_{R}} K^{sep}[[z]] discussed in Example 2.2.

An element m = (m_{i})_{i} ∈ ∏_{i∈ℤ/fℤ} M ⊗_{A_{R}} K^{sep}[[z]] satisfies τ_{M}(σ_{M}^{∗}m) = m if and only if m_{i+1} =

τ_{M}(σ_{M}^{∗}m_{i}) for all i and m_{0} = τ_{M}^{f}(σ_{M}^{f∗}m_{0}) = τM̂(σ̂∗ ; M̂m_{0}). So the isomorphism Ť_{ε}\underline{M} ^{∼} ⟶ Ť_{ε}\underline{M̂} is given by (m_{i})_{i} ↦ m_{0}.

Definition 4.6. Let R be a valuation ring as in Notation 1.1. For a z-divisible local Anderson module G over R the Tate module T_{ε}G and the rational Tate module V_{ε}G are defined as

T_{ε}G := Hom_{A_{ε}}(Q_{ε}/A_{ε}, G(K^{sep}))  and  V_{ε}G := T_{ε}G ⊗_{A_{ε}} Q_{ε} .

They are A_{ε}-modules with a Gal(K^{sep}/K)-action. For an abelian Anderson A-module \underline{E} over R as in Example 3.4 the ε-adic Tate module T_{ε}\underline{E} and the rational ε-adic Tate module V_{ε}\underline{E} are defined as

T_{ε}\underline{E} := Hom_{A}(Q_{ε}/A_{ε}, \underline{E}(K^{sep}))  and  V_{ε}\underline{E} := T_{ε}\underline{E} ⊗_{A_{ε}} Q_{ε} .

Since every element of Q_{ε}/A_{ε} is annihilated by a power of ε, we have T_{ε}\underline{E} = T_{ε} \underline{E}[ε^{∞}].

After choosing a uniformizing parameter z of A_{ε} there is an isomorphism

T_{ε}G ^{∼} ⟶ lim_{⟵_{n}}(G[z^{n}](K^{sep}), z) := {(P_{n})_{n} ∈ ∏_{n∈ℕ_{0}} G[z^{n}](K^{sep}): z(P_{n+1}) = P_{n}}

which sends f ∈ Hom_{A_{ε}}(Q_{ε}/A_{ε}, G(K^{sep})) to the tuple P_{n} := f(z^{−n}). This is indeed an isomorphism, because from (P_{n})_{n} we can reconstruct f as follows. Every x ∈ Qε^{×} is of the form x = uz^{−n} for an integer n and a unit u ∈ Aε^{×}. Then f(x) = u(P_{n}).

To describe the relation between T_{ε}G and Ť_{ε}\underline{M̂} consider the A_{ε}-module Hom_{𝔽_{ε}}(Q_{ε}/A_{ε}, 𝔽_{ε}) which carries the trivial Galois action and is canonically isomorphic to the module of continuous differential forms Ω̂_{A_{ε}/𝔽_{ε}}^{1} under the map

Ω̂_{A_{ε}/𝔽_{ε}}^{1} ^{∼} ⟶ Hom_{𝔽_{ε}}(Q_{ε}/A_{ε}, 𝔽_{ε}) , ω ⟼ (a ↦ Res_{ε}(aω)) .

After choosing a uniformizing parameter z of A_{ε} we can identify Ω̂_{A_{ε}/𝔽_{ε}}^{1} ≅ 𝔽_{ε}[[z]]dz and the inverse map is given by Hom_{𝔽_{ε}}(Q_{ε}/A_{ε}, 𝔽_{ε}) → 𝔽_{ε}[[z]]dz, λ ↦ ∑_{i=0}^{∞} λ(z^{−1−i})z^{i}dz. In particular Hom_{𝔽_{ε}}(Q_{ε}/A_{ε}, 𝔽_{ε}) is non-canonically isomorphic to A_{ε}.

The following result is well known. For a proof see [And93, § 4.2] in case G is a formal Lie group, or [HS15, Theorem 12.2] in general.

Proposition 4.7. Let \underline{M̂} be an effective local shtuka over R, let G = Drq̂(\underline{M̂}) be the associated z-divisible local Anderson module from Section 3 and view \underline{M̂} as lim_{⟵} Hom_{R-groups,𝔽_{ε}-lin}(G[ε^{n}] , 𝔾_{a,R}). Then there is a Gal(K^{sep}/K)-equivariant perfect pairing of A_{ε}-modules

T_{ε}G × Ť_{ε}\underline{M̂} ⟶ Hom_{𝔽_{ε}}(Q_{ε}/A_{ε}, 𝔽_{ε}) ≅ Ω̂_{A_{ε}/𝔽_{ε}}^{1} , (f, m) ⟼ m ∘ f .  (4.4)

which identifies T_{ε}G with the contragredient Gal(K^{sep}/K)-representation Hom_{A_{ε}}(Ť_{ε}\underline{M̂}, A_{ε}) of Ť_{ε}\underline{M̂}. In particular T_{ε}G is a free A_{ε}-module of rank equal to rk \underline{M̂} that carries a continuous action of Gal(K^{sep}/K).

Example 4.8. We describe the ε-adic (dual) Tate module Ť_{ε}\underline{M} = Ť_{ε}\underline{M̂}_{ε}(\underline{M}) of the Carlitz motive \underline{M} = (R[t], t − ϑ) from Example 2.7 by using the local shtuka \underline{M̂} := \underline{M̂}_{ε}(\underline{M}) = (R[[z]], z − ζ) computed there. For all i ∈ ℕ_{0} let ℓ_{i} ∈ K^{sep} be solutions of the equations ℓ_{0}^{q̂−1} = −ζ and ℓq̂ ; i + ζℓ_{i} = ℓ_{i−1}. This implies |ℓ_{i}| = |ζ|^{q̂^{−i}/(q̂−1)} < 1. Define the power series ℓ_{+} = ∑_{i=0}^{∞} ℓ_{i}z^{i} ∈ O_{K^{sep}}[[z]]. It satisfies σ̂(ℓ_{+}) = (z − ζ)·ℓ_{+}, but depends on the choice of the ℓ_{i}. A different choice yields a different power series ℓ̃_{+} which satisfies ℓ̃_{+} = uℓ_{+} for a unit u ∈ (K^{sep}[[z]]^{×})^{σ̂= id} = Aε^{×}, because σ̂(u) = \frac{σ̂(ℓ̃_{+})}{σ̂(ℓ_{+})} = \frac{ℓ̃_{+}}{ℓ_{+}} = u. The field extension 𝔽_{ε}((ζ))(ℓ_{i}: i ∈ ℕ_{0}) of 𝔽_{ε}((ζ)) is the function field analog of the cyclotomic tower ℚ_{p}( \sqrt[p^{i}]{1}: i ∈ ℕ_{0}); see [Har09, § 1.3 and § 3.4]. There is an isomorphism of topological groups called the ε-adic cyclotomic character

χ_{ε}: Gal(𝔽_{ε}((ζ))(ℓ_{i}: i ∈ ℕ_{0})/𝔽_{ε}((ζ))) ^{∼} ⟶ Aε^{×},

which satisfies g(ℓ_{+}) := ∑_{i=0}^{∞} g(ℓ_{i})z^{i} = χ_{ε}(g)·ℓ_{+} in K^{sep}[[z]] for g in the Galois group. It is independent of the choice of the ℓ_{i}. The ε-adic (dual) Tate module Ť_{ε}\underline{M̂} of \underline{M̂} and \underline{M} is generated by ℓ−1_{+} on which the Galois group acts by the inverse of the cyclotomic character. According to Proposition 4.7, Gal(K^{sep}/K) acts on the ε-adic Tate module T_{ε} \underline{C} of the Carlitz module \underline{C} from Example 3.5 by χ_{ε}. So using the notation of Tate twists we may write Ť_{ε}\underline{M} = A_{ε}(−1) and V̌_{ε}\underline{M} = Q_{ε}(−1), as well as T_{ε} \underline{C} = A_{ε}(1) and V_{ε} \underline{C} = Q_{ε}(1). The isomorphisms (4.1) and (4.2) for s > 1/q̂ are given by sending the generator ℓ−1_{+} of Ť_{ε}\underline{M̂} to ℓ−1_{+} ∈ K^{sep}⟨\frac{z}{ζ^{s}} ⟩^{×}.

We compute T_{ε} \underline{C} explicitly. By Example 3.5 the group scheme \underline{C}[ε^{n}] equals the kernel of the endomorphism φ_{z}^{n} of 𝔾_{a,R} = Spec R[x] where φ_{z}^{∗}(x) = ζx + xq̂. By definition φ_{z}^{∗}(ℓ_{0}) = 0 and φ_{z}^{∗}(ℓ_{i}) = ℓ_{i−1} for all i. This means that \underline{C}[ε^{n}](K^{sep}) = A/ε^{n} · ℓ_{n−1} and T_{ε} \underline{C} = A_{ε} · (ℓ_{n−1})_{n} with the action of g ∈ Gal(K^{sep}/K) given by g(ℓ_{n−1})_{n} = χ_{ε}(g)·(ℓ_{n−1})_{n}. In this respect the power series ℓ_{+} = ∑_{n=0}^{∞} ℓ_{n}z^{n} is the Anderson generating function from [And93, § 4.2] of the z-division tower (ℓ_{n−1})_{n}. The pairing

(4.4) sends (ℓ_{n−1})_{n} × ℓ−1_{+} to 1·dz.

Definition 4.9. Let \underline{M̂} be a local shtuka over a valuation ring R as in Notation 1.1. We denote by K[[z − ζ]] the power series ring over K in the “variable” z − ζ and by K((z − ζ)) its fraction field. We consider the ring homomorphism R[[z]] ↪ K[[z − ζ]], z ↦ z = ζ + (z − ζ) and define the de Rham realization of \underline{M̂} as

H_{dR}^{1}(\underline{M̂}, K[[z − ζ]]) := σ̂^{∗}M̂ ⊗_{R[[z]]} K[[z − ζ]] ,
H_{dR}^{1}(\underline{M̂}, K((z − ζ))) := σ̂^{∗}M̂ ⊗_{R[[z]]} K((z − ζ))  and
H_{dR}^{1}(\underline{M̂}, K) := σ̂^{∗}M̂ ⊗_{R[[z]], z↦ζ} K
= H_{dR}^{1}(\underline{M̂}, K[[z − ζ]]) ⊗_{K[[z−ζ]]} K[[z − ζ]]/(z − ζ) .

The de Rham realization H_{dR}^{1}(\underline{M̂}, K((z−ζ))) contains a full K[[z−ζ]]-lattice 𝔮 := τ −1 ; M̂(M̂⊗_{R[[z]]}K[[z−ζ]]), which is called the Hodge-Pink lattice of \underline{M̂}. The de Rham realization H_{dR}^{1}(\underline{M̂}, K) carries a descending separated and exhausting filtration F^{•} by K-subspaces called the Hodge-Pink filtration of \underline{M̂}. It is defined via 𝔭 := H_{dR}^{1}(\underline{M̂}, K[[z − ζ]]) and (for i ∈ ℤ)

F^{i} H_{dR}^{1}(\underline{M̂}, K) := (𝔭 ∩ (z − ζ)^{i}𝔮)/((z − ζ)𝔭 ∩ (z − ζ)^{i}𝔮) ⊂ H_{dR}^{1}(\underline{M̂}, K) .

If we equip H_{dR}^{1}(\underline{M̂}, K((z − ζ))) with the descending filtration F^{i} H_{dR}^{1}(\underline{M̂}, K((z − ζ))) := (z − ζ)^{i}𝔮 by K[[z −ζ]]-submodules, then F^{i} H_{dR}^{1}(\underline{M̂}, K) is the image of H_{dR}^{1}(\underline{M̂}, K[[z −ζ]]) ∩ F^{i} H_{dR}^{1}(\underline{M̂}, K((z −ζ))) in H_{dR}^{1}(\underline{M̂}, K).

Definition 4.10. If \underline{M} = (M, τ_{M}) is an A-motive over R as in Example 2.2 we use A_{K} := A ⊗_{𝔽_{q}} K and A_{K}/J = K, as well as the identification lim_{⟵} A_{K}/J^{n} = K[[z − ζ]] from [HJ16, Lemma 1.5]. Then the de Rham realization of \underline{M} is defined as

H_{dR}^{1}(\underline{M}, K[[z − ζ]]) := σ^{∗}M ⊗_{A_{R}} lim_{⟵} A_{K}/J^{n} ,
H_{dR}^{1}(\underline{M}, K((z − ζ))) := H_{dR}^{1}(\underline{M}, K[[z − ζ]]) ⊗_{K[[z−ζ]]} K((z − ζ))  and
H_{dR}^{1}(\underline{M}, K) := σ^{∗}M ⊗_{A_{R}} A_{K}/J .

For a justification of this definition see [HJ16, §§ 3.4 and 5.5]. The Hodge-Pink lattice of \underline{M} is defined as 𝔮 := τ_{M}^{−1}(M ⊗_{A_{R}} lim_{⟵} A_{K}/J^{n}) ⊂ H_{dR}^{1}(\underline{M}, K((z − ζ))), and the Hodge-Pink filtration of \underline{M} is defined

via 𝔭 := H_{dR}^{1}(\underline{M}, K[[z − ζ]]) and

F^{i} H_{dR}^{1}(\underline{M}, K) := (𝔭 ∩ (z − ζ)^{i}𝔮)/((z − ζ)𝔭 ∩ (z − ζ)^{i}𝔮)
= image of (σ^{∗}M ∩ τ_{M}^{−1}(J^{i}M)) ⊗_{R} K ⊂ H_{dR}^{1}(\underline{M}, K) ;

see [Gos94, § 2.6].

Remark 4.11. If \underline{M̂} := \underline{M̂}_{ε}(\underline{M}) is the associated local shtuka of \underline{M} and f = [𝔽_{ε} : 𝔽_{q}] as in Example 2.2, the map

σ^{∗}τ_{M}^{f−1} = σ^{∗}τ_{M} ∘ σ^{2∗}τ_{M} ∘ ⋯ ∘ σ^{(f−1)∗}τ_{M}: σ^{f∗}M ⊗_{A_{R}} A_{ε,R}/𝔞_{0} ^{∼} ⟶ σ^{∗}M ⊗_{A_{R}} A_{ε,R}/𝔞_{0}

is an isomorphism, because τ_{M} is an isomorphism over A_{ε,R}/𝔞_{i} for all i ≠ 0. Therefore it defines canonical isomorphisms σ^{∗}τ_{M}^{f−1} : H_{dR}^{1}(\underline{M̂}, K[[z − ζ]]) ^{∼} ⟶ H_{dR}^{1}(\underline{M}, K[[z − ζ]]) and σ^{∗}τ_{M}^{f−1} : H_{dR}^{1}(\underline{M̂}, K) ^{∼} ⟶ H_{dR}^{1}(\underline{M}, K), which are compatible with Hodge-Pink lattice and filtration because σ^{∗}τ_{M}^{f−1} ∘ τ −1 ; M̂ = τ_{M}^{−1}.

For the next theorem note that there is a ring homomorphism Q_{ε} = 𝔽_{ε}((z)) ↪ K[[z − ζ]] sending z to z = ζ + (z − ζ), because ζ + (z − ζ) is invertible in K[[z − ζ]].

Theorem 4.12. Let \overline{K} be the completion of an algebraic closure K^{alg} of K. There is a canonical comparison isomorphism

h_{ε,dR} : H_{ε}^{1}(\underline{M̂}, Q_{ε}) ⊗_{Q_{ε}} \overline{K}((z − ζ)) ^{∼} ⟶ H_{dR}^{1}(\underline{M̂}, K((z − ζ))) ⊗_{K((z−ζ))} \overline{K}((z − ζ)) ,

which satisfies h_{ε,dR}(H_{ε}^{1}(\underline{M̂}, Q_{ε}) ⊗_{Q_{ε}} \overline{K}[[z − ζ]]) = 𝔮 ⊗_{K[[z−ζ]]} \overline{K}[[z − ζ]] and which is equivariant for the action of Gal(K^{sep}/K). However, if K is not perfect, it does not allow to recover H_{dR}^{1}(\underline{M̂}, K((z−ζ))) or H_{dR}^{1}(\underline{M̂}, K) from H_{ε}^{1}(\underline{M̂}, Q_{ε}) because the field of Galois invariants \overline{K}^{Gal(K^{sep}/K)} equals the completion { \atop K^{perf}} of the perfect closure of K by the Ax-Sen-Tate Theorem [Ax70, p. 417] and \overline{K}((z −ζ))^{Gal(K^{sep}/K)}

= { \atop K^{perf}}((z − ζ)).

Proof. Note that the map z ↦ ζ + (z − ζ) induces ring homomorphisms R[[z]][\frac{1}{z−ζ}] ↪ \overline{K}((z − ζ))

and K^{sep}⟨\frac{z}{ζ} ⟩ ↪ \overline{K}[[z − ζ]], ∑_{n=0}^{∞} b_{n}z^{n} ↦ ∑_{n=0}^{∞} b_{n}(∑_{i=0}^{n} \binom{n}{i}ζ^{n−i}(z − ζ)^{i}) = ∑_{i=0}^{∞} ζ^{−i}( ∑_{n=i}^{∞} \binom{n}{i}b_{n}ζ^{n})(z − ζ)^{i}. The

series ∑_{n=i}^{∞} \binom{n}{i}b_{n}ζ^{n} converges in \overline{K} because |\binom{n}{i}b_{n}ζ^{n}| ≤ |b_{n}ζ^{n}| → 0 for n → ∞. Thus we can take the

isomorphism h from Remark 4.3 and define

h_{ε,dR} := (τ −1 ; M̂ ∘ h) ⊗ id_{\overline{K}((z−ζ))} : Ť_{ε}\underline{M̂} ⊗_{A_{ε}} \overline{K}((z − ζ)) ^{∼} ⟶ σ̂^{∗}M̂ ⊗_{R[[z]]} \overline{K}((z − ζ)) .

Clearly 𝔮 ⊗_{K[[z−ζ]]} \overline{K}[[z − ζ]] = τ −1 ; M̂(M̂ ⊗_{R[[z]]} \overline{K}[[z − ζ]]) = τ −1 ; M̂ ∘ h(Ť_{ε}\underline{M̂} ⊗_{A_{ε}} \overline{K}[[z − ζ]]).

Remark 4.13. This comparison isomorphism is the function field analog for the comparison isomorphism H_{ét}^{i}(X ×_{L} L^{alg}, ℚ_{p}) ⊗_{ℚ_{p}} B_{dR} ^{∼} ⟶ H_{dR}^{i}(X/L) ⊗_{L} B_{dR} for a smooth proper scheme X over a complete discretely valued field L of characteristic 0 with perfect residue field of characteristic p. The existence of the latter comparison isomorphism was conjectured by Fontaine [Fon82, A.6] and proved by Faltings [Fal89]. It is equivariant for the action of Gal(L^{sep}/L) and allows to compute H_{dR}^{i}(X/L) = (H_{ét}^{i}(X ×_{L} L^{alg}, ℚ_{p}) ⊗_{ℚ_{p}} B_{dR})^{Gal(L^{sep}/L)}, because B_{dR}^{Gal(L^{sep}/L)} = L. Note that it does not allow to reconstruct H_{ét}^{i}(X ×_{L}L^{alg}, ℚ_{p}) from H_{dR}^{i}(X/L). In our comparison isomorphism \overline{K}((z−ζ)) is the analog of B_{dR}; see [Har09, § 2.9].

The entries of a matrix representing the comparison isomorphism with respect to some bases are called the periods of \underline{M̂}, respectively of X. The transcendence degree of the periods of \underline{M̂} was computed by Mishiba [Mis12].

Example 4.14. For the Carlitz motive from Example 4.8 we have H_{ε}^{1}(\underline{M}, Q_{ε}) = Q_{ε} · ℓ−1_{+} ≅ Q_{ε} and H_{dR}^{1}(\underline{M}, K[[z − ζ]]) = K[[z − ζ]] =: 𝔭. The Hodge-Pink lattice is 𝔮 = (z − ζ)^{−1}𝔭 and the Hodge filtration satisfies F^{1} = H_{dR}^{1}(\underline{M}, K) ⊃ F^{2} = (0). With respect to the bases ℓ−1_{+} of H_{ε}^{1}(\underline{M}, Q_{ε}) and 1 of H_{dR}^{1}(\underline{M}, K[[z − ζ]]) the comparison isomorphism h_{ε,dR} from Theorem 4.12 is given by the ε-adic Carlitz period (z − ζ)^{−1}ℓ−1_{+} = σ̂(ℓ_{+})^{−1}. It has a pole of order one at z = ζ because ℓ_{+} ∈ K^{sep}⟨\frac{z}{ζ} ⟩^{×} ⊂ \overline{K}[[z−ζ]]^{×}. So h_{ε,dR}(H_{ε}^{1}(\underline{M}, Q_{ε}) ⊗_{Q_{ε}} \overline{K}[[z − ζ]]) = (z − ζ)^{−1}\overline{K}[[z − ζ]] = 𝔮 ⊗_{K[[z−ζ]]} \overline{K}[[z − ζ]]. Mishiba [Mis12, Example 6.6] shows that σ̂(ℓ_{+})^{−1} is transcendental over 𝔽_{q}(ζ)((z)).

We next study properties of the (dual) Tate module functors.

Theorem 4.15. Assume that R is discretely valued. Then the functor Ť_{ε}: \underline{M̂} ↦ Ť_{ε}\underline{M̂} from the category of local shtukas over R to the category Rep_{A_{ε}} Gal(K^{sep}/K) of representations of Gal(K^{sep}/K) on finite free A_{ε}-modules and the functor V̌_{ε}: \underline{M̂} ↦ V̌_{ε}\underline{M̂} from the category of local shtukas over R with quasi-morphisms to the category Rep_{Q_{ε}} Gal(K^{sep}/K) of representations of Gal(K^{sep}/K) on finite dimensional Q_{ε}-vector spaces are fully faithful.

We will give a proof at the end of the section.

If we additionally require that τM̂ is topologically nilpotent in the sense of Remark 3.3, then the above theorem was previously obtained by Anderson [And93, §4.5, Theorem 1] by a different method from ours. Note that if τM̂ is “topologically nilpotent”, then the action of the (not necessarily commutative) polynomial ring R[[z]]\{τ̂\} on \underline{M̂} extends to the action of the formal power series ring R[[z]]\{\{τ̂\}\}, which is also a (not necessarily commutative) local ring.

The theorem allows to make the following

Definition 4.16. Let R be discretely valued. The full subcategory of Rep_{Q_{ε}} Gal(K^{sep}/K) which is the essential image of the functor V̌_{ε} is called the category of equal characteristic crystalline representations.

We explain the motivation for this definition in Remarks 5.13 and 6.16 below.

Proposition 4.17. Let R be discretely valued, let \underline{M̂} be a local shtuka over R, and set V̌ := V̌_{ε}\underline{M̂}. Then the map

\{ \begin{matrix}\underline{M̂}′ ⊂ \underline{M̂}[\frac{1}{z}]: local shtukas \\ over R of full rank \end{matrix} \} ⟶^{Ť_{ε}} \{ \begin{matrix}Ť′ ⊂ V̌ : Gal(K^{sep}/K)-stable \\ full A_{ε}-lattices \end{matrix} \}

is a bijection.

Let us now prove Theorem 4.15 and Proposition 4.17. We begin with a few lemmas.

Lemma 4.18. Assume that R is discretely valued. Let M̂ be a finitely generated torsion-free R[[z]]- module (not necessarily free). We set M̂′ := M̂[\frac{1}{z}] ∩ (M̂ ⊗_{R[[z]]} K[[z]]), where the intersection is taken inside M̂ ⊗_{R[[z]]} K((z)). Then M̂′ is free over R[[z]] and we have M̂[\frac{1}{z−ζ}] = M̂′[\frac{1}{z−ζ}]. In particular, if M̂ is equipped with an isomorphism τM̂ : σ̂^{∗}M̂[\frac{1}{z−ζ}] ^{∼} ⟶ M̂[\frac{1}{z−ζ}], then \underline{M̂}′ := (M̂′, τM̂′) is a local shtuka, where τM̂′ = τM̂. Furthermore, we have M̂′ = M̂ if M̂ is already free.

Proof. Note that R[[z]][\frac{1}{z}] is a principal ideal domain, being a 1-dimensional factorial ring, so the torsion-free module M̂′[\frac{1}{z}] is free over R[[z]][\frac{1}{z}]. Likewise, M̂ ⊗_{R[[z]]} K[[z]] is free over K[[z]].

Clearly M̂ ⊂ M̂′ and M̂[\frac{1}{z}] = M̂′[\frac{1}{z}]. Choose isomorphisms α: M̂[\frac{1}{z}] ^{∼} ⟶ R[[z]][\frac{1}{z}]^{⊕r} and β: M̂ ⊗_{R[[z]]} K[[z]] ^{∼} ⟶ K[[z]]^{⊕r}. After multiplying α with a high enough power of z we may assume that all entries of the matrix A := αβ^{−1} ∈ GL_{r}(K((z))) lie in K[[z]]. Then every m′ ∈ M̂′ satisfies α(m′) = A·β(m′) ∈ (R[[z]][\frac{1}{z}] ∩ K[[z]])^{⊕r} = R[[z]]^{⊕r}. Let M̂′′ := α^{−1}(R[[z]]^{⊕r}) so that M̂′ ⊂ M̂′′. Since R[[z]] is noetherian, M̂′ is finitely generated over R[[z]]. Together with M̂[\frac{1}{z}] = M̂′[\frac{1}{z}] this implies that there is a power of z which annihilates M̂′/M̂. From M̂′ ⊗_{R[[z]]} K[[z]] ⊂ M̂′′ ⊗_{R[[z]]} K[[z]] ≅ K[[z]]^{⊕r} it follows that M̂′ ⊗_{R[[z]]} K[[z]] has no z-torsion, whence M̂′ ⊗_{R[[z]]} K[[z]] ⊂ M̂′ ⊗_{R[[z]]} K((z)) = M̂ ⊗_{R[[z]]} K((z)). This implies M̂′ ⊗_{R[[z]]} K[[z]] = M̂ ⊗_{R[[z]]} K[[z]] and (M̂′/M̂) ⊗_{R} K = (M̂′/M̂) ⊗_{R[[z]]} K[[z]] = (0). So M̂′/M̂ is annihilated by a power of ζ and thus by some power of z − ζ, which shows that M̂[\frac{1}{z−ζ}] = M̂′[\frac{1}{z−ζ}].

So it remains to show that M̂′ is free over R[[z]]. Note that by construction M̂′ is a reflexive R[[z]]- module; i.e., M̂′ is naturally isomorphic to the R-linear double dual (M̂′)^{∨}^{∨}; indeed, this follows from the fact that M̂′[\frac{1}{z}] and M̂′ ⊗_{R[[z]]} K[[z]] are free over R[[z]][\frac{1}{z}] and K[[z]], respectively, and the equality R[[z]] = R[[z]][\frac{1}{z}] ∩ K[[z]] with the intersection taking place in K((z)). Now, it is well known that a reflexive module over a regular 2-dimensional local ring is necessarily free; cf. [Ser58, §6, Lemme 6]. The last assertion follows from the equation R[[z]] = R[[z]][\frac{1}{z}] ∩ K[[z]] in K((z)).

Lemma 4.19. Assume that ⋂_{n} σ̂^{n}(𝔪_{R}) = (0). Then the base change functor \underline{M̂} ↦ \underline{M̂} ⊗_{R} k from the category of local shtukas over R to the categories of local shtukas over k is faithful.

Remark. The assumption ⋂_{n} σ̂^{n}(𝔪_{R}) = (0) is satisfied if R is discretely valued.

Proof. For local shtukas \underline{M̂} and \underline{M̂}′ over R, let f: \underline{M̂} → \underline{M̂}′ be a morphism which becomes zero over k; i.e., f(\underline{M̂}) = (0) in \underline{M̂}′ ⊗_{R} R/𝔪_{R}. Since we have τM̂′ ∘ σ̂^{∗}f = f ∘ τM̂, it follows inductively that f(\underline{M̂}[\frac{1}{z}]) = (0) in \underline{M̂}′[\frac{1}{z−ζ}] ⊗_{R} R/σ̂^{n}(𝔪_{R}) = \underline{M̂}′[\frac{1}{z}] ⊗_{R} R/σ̂^{n}(𝔪_{R}) for any n ⩾ 1. Now the assumption ⋂_{n} σ̂^{n}(𝔪_{R}) = (0) forces f = 0.

Lemma 4.20. Assume that R is discretely valued, and let \underline{M̂} and \underline{M̂}′ be local shtukas over R. Let f: \underline{M̂} → \underline{M̂}′ be a morphism such that Ť_{ε}f: Ť_{ε}\underline{M̂} → Ť_{ε}\underline{M̂}′ is an isomorphism. Then f is an isomorphism.

Proof. To show that f is an isomorphism, it suffices to show that the determinant of f is an isomorphism. Therefore we may assume that \underline{M̂} and \underline{M̂}′ are of rank 1 by replacing \underline{M̂} and \underline{M̂}′ with their respective top exterior powers.

Let us first show that f is an isogeny; i.e., f[\frac{1}{z}] is an isomorphism. By Nakayama’s lemma, it suffices to show the surjectivity of the following map induced by f[\frac{1}{z}]:

M̂ ⊗_{R[[z]]} k((z)) → M̂′ ⊗_{R[[z]]} k((z)).

Indeed, since both the source and the target are 1-dimensional vector spaces over k((z)), it suffices to show that the above map is non-zero, which is immediate from Lemma 4.19.

Let us now show that f is an isomorphism. By Proposition 4.2 it follows that f induces an isomorphism

f ⊗ 1: M̂ ⊗_{R[[z]]} K[[z]] ^{∼} ⟶ M̂′ ⊗_{R[[z]]} K[[z]].

Since f[\frac{1}{z}] is also an isomorphism, it follows from Lemma 4.18 that f is an isomorphism.

Proof of Theorem 4.15. Assume that R is discretely valued, and let \underline{M̂} and \underline{M̂}′ be local shtukas over R. To prove Theorem 4.15, it suffices to show that for any Gal(K^{sep}/K)-equivariant morphism g: Ť_{ε}\underline{M̂} → Ť_{ε}\underline{M̂}′ there exists a unique morphism f: \underline{M̂} → \underline{M̂}′ with Ť_{ε}f = g.

Since we have M̂ ⊗_{R[[z]]} K[[z]] = (Ť_{ε}\underline{M̂} ⊗_{A_{ε}} K^{sep}[[z]])^{Gal(K^{sep}/K)} which matches τM̂ ⊗ 1 and 1 ⊗ σ̂ by Proposition 4.2, it follows that g induces a uniquely determined morphism

f_{K}: M̂ ⊗_{R[[z]]} K[[z]] → M̂′ ⊗_{R[[z]]} K[[z]]

satisfying (τM̂′ ⊗ 1) ∘ σ̂^{∗}f_{K} = f_{K} ∘ (τM̂ ⊗ 1). Furthermore, any morphism f: \underline{M̂} → \underline{M̂}′ with Ť_{ε}f = g has to satisfy f = f_{K}|M̂. Therefore, it suffices to show that f_{K}(M̂) ⊂ M̂′ for any g.

We define M̂′′_{1}, M̂′′_{2} ⊂ M̂′ ⊗_{R[[z]]} K[[z]] as follows:

M̂′′_{1} := f_{K}(M̂)[\frac{1}{z}] ∩ (f_{K}(M̂) ⊗_{R[[z]]} K[[z]])
M̂′′_{2} := (f_{K}(M̂) ∩ M̂′)[\frac{1}{z}] ∩ ((f_{K}(M̂) ∩ M̂′) ⊗_{R[[z]]} K[[z]]) .

By Lemma 4.18 applied to the torsion free modules f_{K}(M̂) and (f_{K}(M̂)∩M̂′), we obtain local shtukas \underline{M̂}′′_{1} and \underline{M̂}′′_{2} with underlying R[[z]]-modules M̂′′_{1} and M̂′′_{2}, respectively, and the natural maps \underline{M̂} → \underline{M̂}′′_{1}, \underline{M̂}′′_{2} ↪ \underline{M̂}′′_{1}, and \underline{M̂}′′_{2} ↪ \underline{M̂}′ are morphisms of local shtukas. Clearly, we have M̂′′_{1} ⊗_{R[[z]]} K[[z]] = M̂′′_{2} ⊗_{R[[z]]} K[[z]] since both are equal to the image of f_{K}, so the natural inclusion \underline{M̂}′′_{2} ↪ \underline{M̂}′′_{1} induces an isomorphism Ť_{ε}\underline{M̂}′′_{2} ^{∼} ⟶ Ť_{ε}\underline{M̂}′′_{1}. We can now apply Lemma 4.20 to show that \underline{M̂}′′_{1} = \underline{M̂}′′_{2}. Therefore, we obtain a map f: \underline{M̂} → \underline{M̂}′ by the following composition

\underline{M̂} → \underline{M̂}′′_{1} = \underline{M̂}′′_{2} ↪ \underline{M̂}′,

which clearly extends to f_{K}.

Proof of Proposition 4.17. Let \underline{M̂} be a local shtuka over R. We want to show that Ť_{ε} induces a bijection from the set of local shtukas \underline{M̂}′ ⊂ \underline{M̂}[\frac{1}{z}] to the set of Galois-stable A_{ε}-lattices Ť′ ⊂ V̌_{ε}\underline{M̂}. The injectivity of Ť_{ε} is clear from Theorem 4.15, so it suffices to show the surjectivity.

Let Ť′ ⊂ V̌_{ε}\underline{M̂} be a Galois-stable A_{ε}-lattice. We want to show that there exists a local shtuka \underline{M̂}′ ⊂ \underline{M̂}[\frac{1}{z}] with Ť_{ε}\underline{M̂}′ = Ť′. We set N := (Ť′ ⊗_{A_{ε}} K^{sep}[[z]])^{Gal(K^{sep}/K)}, which can be viewed as a K[[z]]-lattice in M̂[\frac{1}{z}] ⊗_{R[[z]]} K[[z]] by Proposition 4.2, and set M̂′ := M̂[\frac{1}{z}] ∩ N. By construction, M̂′ is finitely generated over R[[z]], and we have M̂′ = M̂′[\frac{1}{z}] ∩ (M̂′ ⊗_{R[[z]]} K[[z]]). So by Lemma 4.18, \underline{M̂}′ := (M̂′, τM̂′) is a local shtuka where τM̂′ is the restriction of τM̂.

It remains to show that Ť_{ε}\underline{M̂}′ = Ť′. By Proposition 4.2, it suffices to show that M̂′⊗_{R[[z]]}K[[z]] = N, which follows from the left exactness of the following sequence:

\xymatrix{0 \ar[r] & M̂′ ⊗_{R[[z]]} K[[z]] \ar[r] & N[\frac{1}{z}] \ar[r] & N[\frac{1}{z}]/N.}  (4.5)

Indeed, this sequence is the flat scalar extension of the left exact sequence 0 → M̂′ → M̂[\frac{1}{z}] → N[\frac{1}{z}]/N; note that we have N[\frac{1}{z}] = M̂[\frac{1}{z}] ⊗_{R[[z]]} K[[z]] and

(N[\frac{1}{z}]/N) ⊗_{R[[z]]} K[[z]] ≅ \underrightarrow{lim}_{r} (z^{−r}N/N ⊗_{R[[z]]/(z^{r})} K[[z]]/(z^{r})) ≅ \underrightarrow{lim}_{r}(z^{−r}N/N)[\frac{1}{ζ}] ≅ N[\frac{1}{z}]/N.

This concludes the proof.

5 Hodge-Pink structures

From now on we work over the base scheme S = Spec R where R is a valuation ring as in Notation 1.1 with fraction field K and residue field k. In particular ζ = 0 in k. We assume that there is a section k ↪ R and we fix one. As mentioned in the introduction we want to describe the analog of Fontaine’s classification of crystalline p-adic Galois representations by filtered isocrystals. In this function field analog we make the following

Definition 5.1. A z-isocrystal over k is a pair (D, τ_{D}) consisting of a finite dimensional k((z))-vector space together with a k((z))-isomorphism τ_{D}: σ̂^{∗}D ^{∼} ⟶ D. A morphism (D, τ_{D}) → (D′, τ_{D′}) is a k((z))-homomorphism f: D → D′ satisfying τ_{D′} ∘ σ̂^{∗}f = f ∘ τ_{D}.

Here again σ̂^{∗}D := D ⊗_{k((z)), σ̂} k((z)) and we set σ̂_{D}^{∗}x := x ⊗ 1 ∈ σ̂^{∗}D for x ∈ D. By the assumed existence of the fixed section k ↪ R there is a ring homomorphism

k((z)) ⟶ K[[z − ζ]] , z ⟼ ζ + (z − ζ) , ∑_{i} b_{i}z^{i} ⟼ ∑_{j=0}^{∞}(z − ζ)^{j} · ∑_{i} \binom{i}{j}b_{i}ζ^{i−j} .  (5.1)

We always consider K[[z − ζ]] and its fraction field K((z − ζ)) as k((z))-vector spaces via (5.1).

Definition 5.2. A z-isocrystal with Hodge-Pink structure over R is a triple \underline{D} = (D, τ_{D}, 𝔮_{D}) consisting of a z-isocrystal (D, τ_{D}) over k and a K[[z − ζ]]-lattice 𝔮_{D} in D ⊗_{k((z))} K((z − ζ)) 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 rk \underline{D}.

A morphism (D, τ_{D}, 𝔮_{D}) → (D′, τ_{D′}, 𝔮_{D′}) is a k((z))-homomorphism f: D → D′ satisfying τ_{D′} ∘ σ̂^{∗}f = f ∘ τ_{D} and (f ⊗ id)(𝔮_{D}) ⊂ 𝔮_{D′}.

A strict subobject \underline{D}′ ⊂ \underline{D} is a z-isocrystal with Hodge-Pink structure of the form \underline{D}′ = (D′, τ_{D}|_{σ̂^{∗}D′}, 𝔮_{D} ∩ D′ ⊗_{k((z))} K((z − ζ))) where D′ ⊂ D is a k((z))-subspace with τ_{D}(σ̂^{∗}D′) = D′.

Remark 5.3. This definition slightly deviates from [GL11, § 3 and § 7] and [Har11, § 2.3] where the Hodge-Pink lattice 𝔮_{D} was defined to be a K[[z − ζ]]-lattice in σ̂^{∗}D ⊗_{k((z))} K((z − ζ)). The reason for our definition here is explained in Example 5.7 below.

Definition 5.4. On a z-isocrystal with Hodge-Pink structure \underline{D} there always is the tautological lattice 𝔭_{D} := D ⊗_{k((z))} K[[z − ζ]]. Since K[[z − ζ]] is a principal ideal domain the elementary divisor theorem provides basis vectors v_{i} ∈ 𝔭_{D} such that 𝔭_{D} = ⨁_{i=1}^{r} K[[z −ζ]]·v_{i} and 𝔮_{D} = ⨁_{i=1}^{r} K[[z −ζ]]·(z −ζ)^{μ_{i}} ·v_{i} for integers μ_{1} ≥ … ≥ μ_{r}. We call (μ_{1}, …, μ_{r}) the Hodge-Pink weights of \underline{D}. Alternatively if e is large enough such that 𝔮_{D} ⊂ (z −ζ)^{−e}𝔭_{D} or (z −ζ)^{e}𝔭_{D} ⊂ 𝔮_{D} then the Hodge-Pink weights are characterized by

(z − ζ)^{−e}𝔭_{D}/𝔮_{D} ≅ ⨁_{i=1}^{n} K[[z − ζ]]/(z − ζ)^{e+μ_{i}} ,

or 𝔮_{D}/(z − ζ)^{e}𝔭_{D} ≅ ⨁_{i=1}^{n} K[[z − ζ]]/(z − ζ)^{e−μ_{i}}

The category of z-isocrystals with Hodge-Pink structure possesses tensor products and duals

\underline{D} ⊗ \underline{D}′ = (D ⊗_{k((z))} D′, τ_{D} ⊗ τ_{D′}, 𝔮_{D} ⊗_{K[[z−ζ]]} 𝔮_{D′}) ,
\underline{D}^{∨} = (D^{∨} := Hom_{k((z))}(D, k((z))), (τ_{D}^{−1})^{∨}, Hom_{K[[z−ζ]]}(𝔮_{D}, K[[z − ζ]])) ,

internal Hom’s, and the unit object (k((z)), τ_{D} = 1, 𝔮_{D} = 𝔭_{D}). The endomorphism ring of the unit object is 𝔽_{ε}((z)). For every n ∈ ℤ we consider the Tate object

\underline{1l}(n) := (D = k((z)) , τ_{D} = z^{−n} , 𝔮_{D} = (z − ζ)^{n}𝔭_{D}) .

The Hodge-Pink lattice 𝔮_{D} induces a descending filtration of D_{K} := D ⊗_{k((z)), z↦ζ} K by K-subspaces as follows. Consider the natural projection

𝔭_{D} → → 𝔭_{D}/(z − ζ)𝔭_{D} = D_{K} .

The Hodge-Pink filtration F^{•}D_{K} = (F^{i}D_{K})_{i∈ℤ} is defined by letting F^{i}D_{K} be the image in D_{K} of 𝔭_{D} ∩ (z − ζ)^{i}𝔮_{D} for all i ∈ ℤ. This means, F^{i}D_{K} = (𝔭_{D} ∩ (z − ζ)^{i}𝔮_{D})/((z − ζ)𝔭_{D} ∩ (z − ζ)^{i}𝔮_{D}).

Example 5.5. The Hodge-Pink lattice contains finer information than the Hodge-Pink filtration. For example let D = k((z))^{⊕r} and 𝔮_{D} = (z−ζ)^{2}𝔭_{D}+K[[z−ζ]]·(v_{0}+(z−ζ)v_{1}) ⊂ 𝔭_{D} for vectors v_{0}, v_{1} ∈ K^{⊕r}. Then F^{−2}D_{K} = D_{K} ⊃ F^{−1}D_{K} = K · v_{0} = F^{0}D_{K} ⊃ F^{1}D_{K} = (0). So the information about v_{1} is not contained in the Hodge-Pink filtration.

Remark 5.6. In comparison with the Hodge-Pink structures at ∞ from [HJ16, § 2] our Frobenius τ_{D} replaces the weight filtration W_{•}H from [HJ16, Definition 2.3] and the Hodge-Pink weights from [HJ16, Remark 2.4] are the negatives of the Hodge-Pink weights we defined in Definition 5.4. Observe that the Frobenius τ_{D} induces on D an increasing filtration given by Newton slopes similarly to [Zin84].

At the heart of the theory is the following

Example 5.7. Let \underline{M̂} = (M̂, τM̂) be a local shtuka over Spec R. We set (D̃, τD̃) := (M̂, τM̂)⊗_{R[[z]]}k((z)) and take (D, τ_{D}) := σ̂^{∗}(D̃, τD̃) as the underlying z-isocrystal. This deviates from [GL11, § 3 and § 7] and [Har11, § 2.3] where instead (D̃, τD̃) was taken as the underlying z-isocrystal. Note however, that there is a canonical isomorphism τD̃: (D, τ_{D}) ^{∼} ⟶ (D̃, τD̃). The reason for our definition here is that we also define an integral structure H_{cris}^{1}(\underline{M̂}, k[[z]]) in Definition 5.14 below. In order to construct 𝔮_{D} we will need a “comparison isomorphism”. This does not exist in general. However, if R is discretely valued it is constructed as follows. Consider the R-algebra

R[[z, z^{−1}} := { ∑_{i=−∞}^{∞} b_{i}z^{i}: b_{i} ∈ R , |b_{i}| |ζ|^{ri} → 0 (i → −∞) for all r > 0 } .  (5.2)

It is a subring of K[[z −ζ]] via the expansion ∑_{i=−∞}^{∞} b_{i}z^{i} = ∑_{j=0}^{∞} ζ^{−j}( ∑_{i=−∞}^{∞} \binom{i}{j}b_{i}ζ^{i})(z −ζ)^{j}. The homomor-

phism (5.1) factors through R[[z, z^{−1}}. We view the elements of R[[z, z^{−1}} as functions that converge on the punctured open unit disc {0 < |z| < 1}. An example of such a function is

ℓ_{−} := ∏_{i∈ℕ_{0}}(1 − \frac{ζ^{q̂^{i}}}{z}) ∈ 𝔽_{ε}[[ζ]][[z, z^{−1}\} ⊂ R[[z, z^{−1}\} ,  (5.3)

because the coefficient of z^{−n} is ∑_{0≤i_{1}<...<i_{n}}(−ζ^{q̂^{i_{1}}} ) ⋯ (−ζ^{q̂^{in}}) which has absolute value |ζ|^{1+q̂+...+q̂^{n−1}} =

|ζ|^{(q̂^{n}−1)/(q̂−1)}. Note that ℓ_{−} = (1 − \frac{ζ}{z})·σ̂(ℓ_{−}). If we evaluate σ̂(ℓ_{−}) at z = ζ we obtain σ̂(ℓ_{−})|_{z=ζ} = ∏_{i∈ℕ_{>0}}(1−ζ^{q̂^{i}−1}). If we were working at the place ∞ on the projective line ℙ^{1} instead of at ε ⊂ A, and hence took ζ = ϑ^{−1} and q̂ = q, the number \sqrt[q−1]{−ζ^{q}} ·σ̂(ℓ_{−})|_{z=ζ} would be the inverse of the period of the Carlitz module which is considered to be the function field analog of 2πi. Moreover, \sqrt[q−1]{−ζ^{q}} · σ̂(ℓ_{−}) would equal Papanikolas function Ω ∈ ℂ_{∞}[[\frac{1}{z}]]; see [Pap08, 3.3.4].

Further note that in the reduction we have R[[z, z^{−1}} ⊗_{R} k = k((z)) and ℓ_{−} ≡ 1 mod 𝔪_{R}.

The following lemma of A. Genestier and V. Lafforgue [GL11, Lemma 7.4] is used for the construction of the Hodge-Pink lattice 𝔮_{D} attached to the local shtuka \underline{M̂}.

Lemma 5.8. If R is discretely valued then there is a unique functorial isomorphism

δM̂: M̂ ⊗_{R[[z]]} R[[z, z^{−1}}[ℓ−1_{−}] ^{∼} ⟶ D̃ ⊗_{k((z))} R[[z, z^{−1}}[ℓ−1_{−}]

which satisfies δM̂ ∘ τM̂ = τD̃ ∘ σ̂^{∗}δM̂ and δM̂ ≡ idD̃ mod 𝔪_{R}.

Proof. We refer to [GL11, Lemma 7.4] and [Har11, Lemma 2.3.1] for a proof and just present the idea.

We write τM̂ as a matrix T ∈ GL_{r}(R[[z]][\frac{1}{z−ζ}]) and set S := T mod 𝔪_{R} ∈ GL_{r}(k((z))). Under the fixed section k ↪ R we view S as an element of GL_{r}(R[[z]][z^{−1}]). We set

C_{m} := S · σ̂(S) · . . . · σ̂^{m}(S) · σ̂^{m}(T^{−1}) · . . . · T^{−1} .  (5.4)

Then C_{m} · T = S · σ̂^{∗}C_{m−1} and T^{−1} − S^{−1} ≡ 0 mod 𝔪_{R} implies that

C_{m} − C_{m−1} = S · . . . · σ̂^{m}(S) · σ̂^{m}(T^{−1} − S^{−1}) · σ̂^{m−1}(T^{−1}) · . . . · T^{−1} ≡ 0 mod 𝔪_{R}^{q^{m}} .

From this one derives that δM̂ := lim_{m→∞} C_{m} converges in GL_{r}(R[[z, z^{−1}}[ℓ−1_{−}]) and satisfies δM̂ · T = S ·

σ̂^{∗}δM̂. For example if T = (z−ζ)^{d} and hence S = z^{d}, one obtains δM̂ = lim_{m→∞} ∏_{i=0}^{m}(1− \frac{ζ^{q̂^{i}}}{z})^{−d} = ℓ−d_{−}.

If R is not discretely valued the isomorphism δM̂ might not exist, or if it exists it might not be unique; see the discussion on [Har11, pp. 1293 f ]. Therefore we include it into the data.

Definition 5.9. Let R be an arbitrary valuation ring as in Notation 1.1. A rigidified local σ̂-shtuka over R is a triple (M̂, τM̂, δM̂) where (M̂, τM̂) is a local σ̂-shtuka and δM̂ is an isomorphism as in Lemma 5.8.

Remark 5.10. The proof of Lemma 5.8 works not only if R is discretely valued, but more generally if T − S ≡ 0 mod (π) and T^{−1} − S^{−1} ≡ 0 mod (π) for an element π ∈ 𝔪_{R}. In particular if \underline{M̂} = \underline{M̂}_{ε}(\underline{M}) for an A-motive \underline{M} = (M, τ_{M}) over R, then this is satisfied. Namely, after choosing bases of the locally free A_{R}-modules M and σ^{∗}M on a neighborhood Spec A_{R}[f^{−1}] with f ∈ A_{R} of the point (𝔪_{R}, J) ∈ Spec A_{R}, the matrix T of τ_{M} with respect to these bases contains only finitely many elements of R and therefore the differences T − S and T^{−1} − S^{−1} likewise contain only finitely many elements of 𝔪_{R}. All these elements lie in (π) for a suitable π ∈ 𝔪_{R}. This implies that there exists a canonical rigidification on \underline{M̂} = \underline{M̂}_{ε}(\underline{M}) even if R is an arbitrary valuation ring as in Notation 1.1.

After these preparations we can complete Example 5.7 by defining the Hodge-Pink lattice 𝔮_{D} associated with a rigidified local shtuka (M̂, τM̂, δM̂) as the image

𝔮_{D} := σ̂^{∗}δM̂ ∘ τ −1 ; M̂(M̂ ⊗_{R[[z]]} K[[z − ζ]]) ,  (5.5)

under σ̂^{∗}δM̂ ∘ τ −1 ; M̂ = τ −1 ; D̃ ∘ δM̂: M̂ ⊗_{R[[z]]} K((z − ζ)) ^{∼} ⟶ D ⊗_{k((z))} K((z − ζ)) = 𝔭_{D}[\frac{1}{z−ζ}].

Definition 5.11. The functor

\underline{M̂} = (M̂, τM̂, δM̂) ⟼ ℍ(\underline{M̂}) := \underline{D} := (D, τ_{D}, 𝔮_{D}) with  (5.6)
(D, τ_{D}) = σ̂^{∗}(M̂, τM̂) ⊗_{R[[z]]} k((z)) and 𝔮_{D} from (5.5)

from the category of rigidified local shtukas with quasi-morphisms over R to the category of zisocrystals with Hodge-Pink structure over R is called the mysterious functor. Clearly rk ℍ(\underline{M̂}) = rk \underline{M̂}.

We quote the following result from [Har11, Proposition 2.4.10].

Proposition 5.12. Let R be an arbitrary valuation ring as in Notation 1.1. The functor ℍ from (5.6) is fully faithful.

Remark 5.13. Let R be discretely valued. In view of Theorem 4.15 we like to view the essential image of V̌_{ε} as the (analog of the) category of crystalline Galois representations and the mysterious functor ℍ as the analog of Fontaine’s functor from crystalline Galois representations to filtered isocrystals:

[figure: {crystalline Gal(K^{sep}/K)-representations}; Rep_{Q_{ε}} Gal(K^{sep}/K); _{{local shtukas over R with quasi-morphisms}}; V̌_{ε}; ≅; ℍ; {z-isocrystals with Hodge-Pink structure}.]

(5.7)

The name “mysterious functor” derives from the fact that for an abelian variety X over a p-adic field L with good reduction X_{0}, Tate [Tat66] and Grothendieck [Gro74] had constructed a functor relating the p-adic étale cohomology H_{ét}^{i}(X ×_{L} L^{alg}, ℚ_{p}) to the crystalline cohomology H_{cris}^{i}(X_{0}/L_{0}) with its Hodge filtration. Here L_{0} is the fraction field of the ring of p-typical Witt vectors with coefficients in the (perfect) residue field of L and H_{cris}^{i}(X_{0}/L_{0}) is a “filtered isocrystal”; see Remark 5.21 below. Grothendieck then posed the problem to extend this functor (the “mysterious functor”) to general proper smooth schemes X over L with good reduction. For those X the problem was solved by Fontaine [Fon79, Fon82, Fon90, Fon94], 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 H_{ét}^{i}(X ×_{L}L^{alg}, ℚ_{p}) is crystalline. After contributions by Grothendieck, Tate, Fontaine, Lafaille, Messing, Hyodo, Kato and many others, Fontaine’s conjecture was proved independently by Faltings [Fal89], Niziol [Niz98] and Tsuji [Tsu99]. Tate’s original construction [Tat66] proceeds similarly to our diagram (5.7), as it starts with the p-divisible group X[p^{∞}] of the abelian variety X and associates with it on the one hand the p-adic Tate module of X, which is the dual of H_{ét}^{1}(X ×_{L} L^{alg}, ℚ_{p}), and on the other hand the filtered isocrystal which is the dual of H_{cris}^{1}(X_{0}/L_{0}). In our diagram the category of local shtukas over R with quasi-morphisms takes the place of the p-divisible group in Tate’s construction. This is a perfect analog, because its full subcategory of effective local shtukas is anti-equivalent to the category of z-divisible local Anderson modules (see Remark 3.2). The latter arise from Drinfeld modules and abelian Anderson A-modules in the same way as p-divisible groups arise from abelian varieties (see Example 3.4). The reader should also note that in Fontaine’s theory the information given by the Hodge filtration is fine enough to determine the Galois representation. In the function field analog this is not the case and one needs the finer information contained in the Hodge-Pink lattice; compare Example 5.5. These observations are the reason for Definition 4.16; see also Remark 6.16. They suggest the following

Definition 5.14. Let R be an arbitrary valuation ring as in Notation 1.1 and let \underline{M̂} be a rigidified local shtuka over R. Then H_{cris}^{1}(\underline{M̂}, k((z))) := σ̂^{∗}(M̂, τM̂)⊗_{R[[z]]}k((z)) is called the crystalline realization of \underline{M̂}.

If \underline{M} is an A-motive over R as in Example 2.2 and \underline{M̂}_{ε}(\underline{M}) is its associated rigidified local shtuka (see Remark 5.10), then H_{cris}^{1}(\underline{M}, k((z))) := H_{cris}^{1}(\underline{M̂}_{ε}(\underline{M}), k((z))) is called the crystalline realization of \underline{M}.

We also define the integral structure H_{cris}^{1}(\underline{M̂}, k[[z]]) := σ̂^{∗}(M̂, τM̂) ⊗_{R[[z]]} k[[z]] ⊂ H_{cris}^{1}(\underline{M̂}, k((z))) and H_{cris}^{1}(\underline{M}, k[[z]]) := H_{cris}^{1}(\underline{M̂}_{ε}(\underline{M}), k[[z]]).

Example 5.15. We continue with the discussion of the Carlitz motive \underline{M} = (R[t], t − ϑ) from Example 2.7 and its local shtuka \underline{M̂} := \underline{M̂}_{ε}(\underline{M}) = (R[[z]], z − ζ). The associated z-isocrystal is (k((z)), z). The isomorphism δM̂ from Lemma 5.8 is given by multiplication with the element ℓ−1_{−} from (5.3) because ℓ−1_{−} · (z − ζ) = z · σ̂(ℓ−1_{−}). Since σ̂(ℓ_{−}) converges at z = ζ to the non-zero element σ̂(ℓ_{−})|_{z=ζ} = ∏_{i∈ℕ_{>0}}(1 − ζ^{q̂^{i}−1}) ∈ 𝔽_{ε}[[ζ]]^{×}, we have σ̂(ℓ_{−}) ∈ 𝔽_{ε}((ζ))[[z − ζ]]^{×} and 𝔮_{D} = σ̂(ℓ−1_{−})(z − ζ)^{−1} · K[[z − ζ]] = (z − ζ)^{−1}𝔭_{D}. So the Hodge-Pink weight is −1 and

ℍ(\underline{M̂}) = (k((z)), z, (z − ζ)^{−1}𝔭_{D}) = \underline{1l}(−1) .

In order to describe comparison isomorphisms which relate the crystalline realization H_{cris}^{1}(\underline{M̂}, k((z))) of a rigidified local shtuka with the other realizations we let \overline{K} be the completion of an algebraic closure of K, and we recall the definition of O_{\overline{K}}[[z, z^{−1}\} from (5.2) and the elements ℓ_{−} ∈ 𝔽_{ε}[[ζ]][[z, z^{−1}\} from (5.3) and ℓ_{+} ∈ O_{\overline{K}}[[z]] from Example 4.8, which also satisfies ℓ_{+} ∈ \overline{K}⟨\frac{z}{ζ} ⟩^{×} and σ̂(ℓ_{+}) = (z − ζ)·ℓ_{+}. We set

ℓ := ℓ_{+}ℓ_{−} ∈ O_{\overline{K}}[[z, z^{−1}\} .  (5.8)

Then σ̂(ℓ) = z ·ℓ because ℓ_{−} = (1 − \frac{ζ}{z})·σ̂(ℓ_{−}), and g(ℓ) = χ_{ε}(g)·ℓ for g ∈ Gal(K^{sep}/K) where χ_{ε} is the cyclotomic character from Example 4.8. With these properties ℓ is the function field analog of Fontaine’s period log[\underline{e}] ∈ B̃_{rig} of the multiplicative group, where \underline{e} = (e_{i}: i ∈ ℕ_{0}) is a compatible system of primitive p^{i}-th roots of unity e_{i} and [\underline{e}] is its Teichmüller lift; see [Har09, § 2.7].

The following lemma is the function field analog of the fundamental exact sequence [CF00, Proposition 1.3(v)] of ℚ_{p}-vector spaces

\xymatrix{0 \ar[r] & ℚ_{p} \ar[r] & B̃_{rig}^{φ= id} ⊕ B_{dR}^{+} \ar[r] & B_{dR} \ar[r] & 0}  (5.9)

in which B̃_{rig}^{φ= id} = B_{cris}^{φ= id}; see for example [Ber04b, § II.3.4].

Lemma 5.16. The following sequence of Q_{ε}-vector spaces is exact

[figure: 0; Q_{ε}; O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}]^{σ̂= id} ⊕ \overline{K}[[z − ζ]]; \overline{K}((z − ζ)); 0; a; (a, a)  ,  (f, g); f − g .]

Remark. Note that in [Har09, last line on p. 1745] the exactness of a similar sequence was stated. That sequence contains an error, as in the middle term “[ℓ^{−1}]” is missing.

We prove the lemma simultaneously with the following

Corollary 5.17. For every c ∈ \overline{K} there is an f = ∑_{i∈ℤ} b_{i}z^{i} ∈ O_{\overline{K}}[[z, z^{−1}\} with f = z^{−1}σ̂(f) and c = f(ζ) := ∑_{i∈ℤ} b_{i}ζ^{i}.

Proof of Lemma 5.16 and Corollary 5.17. In Lemma 5.16 the exactness on the left is clear because the maps into each of the summands are ring homomorphisms.

To prove exactness in the middle let g = \frac{b}{ℓ^{n}} ∈ O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}]^{σ̂= id} ∩\overline{K}[[z −ζ]] with b ∈ O_{\overline{K}}[[z, z^{−1}\}. Since σ̂(ℓ^{n}) = z^{n}·ℓ^{n} we must have b = z^{−n}σ̂(b). It follows from [Har11, Proposition 1.4.4] that b ∈ Q_{ε}·ℓ^{n}. This can also be seen directly as follows. For s ≥ r > 0 consider the ring

\overline{K}⟨\frac{z}{ζ^{r}} , \frac{ζ^{s}}{z} ⟩ := \{ ∑_{i=−∞}^{∞}b_{i}z^{i} : |b_{i}| |ζ|^{ri} → 0 (i → ∞), |b_{i}| |ζ|^{si} → 0 (i → −∞) \} ,

which is a principal ideal domain by [Laz62, Proposition 4]. Fix a real number r with 1/q̂ < r < 1. Then 1 < rq̂ < q̂. Therefore the elements ℓ_{+} and σ̂(ℓ_{−}) are units in \overline{K}⟨\frac{z}{ζ^{r}} , \frac{ζ^{rq̂}}{z}⟩ and the element

(1− \frac{ζ}{z})^{n}g = ℓ−n_{+}σ̂(ℓ_{−})^{−n}b ∈ \overline{K}⟨\frac{z}{ζ^{r}} , \frac{ζ^{rq̂}}{z}⟩ has a zero of order n at z = ζ because g ∈ \overline{K}[[z −ζ]]. By [Laz62,

Lemme 2] this shows that it is divisible by (1 − \frac{ζ}{z})^{n} in \overline{K}⟨\frac{z}{ζ^{r}} , \frac{ζ^{rq̂}}{z}⟩. So g = ∑_{i=−∞}^{∞} b_{i}z^{i} ∈ \overline{K}⟨\frac{z}{ζ^{r}} , \frac{ζ^{rq̂}}{z}⟩

satisfies g = σ̂(g) = ∑_{i=−∞}^{∞} bq̂ ; iz^{i} in \overline{K}⟨\frac{z}{ζ^{rq̂}}, \frac{ζ^{rq̂}}{z}⟩. It follows that b_{i} = bq̂ ; i, whence b_{i} ∈ 𝔽_{ε}. Now the

convergence condition |b_{i}| |ζ|^{rq̂i} → 0 for i → −∞ implies that g ∈ 𝔽_{ε}((z)) = Q_{ε} as desired. Thus the sequence is exact in the middle.

We next prove Corollary 5.17. The condition f = z^{−1}σ̂(f) = ∑_{i∈ℤ} bq̂ ; iz^{i−1} implies b_{i} = b_{i−1}^{q̂^{−1}} = b_{0}^{q̂^{−i}}, whence f = ∑_{i∈ℤ} b_{0}^{q̂^{−i}}z^{i}. The convergence condition on f for i → −∞ further implies that |b_{0}| < 1. Conversely this guarantees that f ∈ O_{\overline{K}}[[z, z^{−1}\}. Thus we must find an element b_{0} ∈ \overline{K} with |b_{0}| < 1 and

∑_{i=−∞}^{∞} b_{0}^{q̂^{−i}}ζ^{i} = c .  (5.10)

We consider the valuation v on \overline{K} with v(x) := log |x|/ log |ζ|, which satisfies |x| = |ζ|^{v(x)} and v(ζ) = 1. If f ∈ O_{\overline{K}}[[z, z^{−1}\} satisfies f = z^{−1}σ̂(f) and f(ζ) = c then z^{n}f ∈ O_{\overline{K}}[[z, z^{−1}\} satisfies z^{n}f = z^{−1}σ̂(z^{n}f) and (z^{n}f)(ζ) = ζ^{n}c for all n ∈ ℤ. Therefore we may multiply c with an integral power of ζ and assume that \frac{1}{q̂−1} < v(c) ≤ 1+ \frac{1}{q̂−1}, that is |ζ| > |c|^{q̂−1} ≥ |ζ|q̂. To solve equation (5.10) in this case we iteratively try to find u_{n} ∈ \overline{K} such that

∑_{i=−n}^{n} u_{n}^{q̂^{−i}}ζ^{i} = c  (5.11)

for each n ∈ ℕ with n ≥ 2. Writing u_{n} = ∑_{i=2}^{n} x_{n} and i = n − j, multiplying (5.11) with ζ^{n} and raising it to the q̂^{n}-th power we have to solve the equations

∑_{j=0}^{2n} ζ^{(2n−j)q̂^{n}}(x_{n})^{q̂^{j}} = (x_{n})^{q̂^{2n}} + … + ζ^{(2n−1)q̂^{n}}(x_{n})q̂ + ζ^{2nq̂^{n}}x_{n} = c_{n}  (5.12)

for x_{n} ∈ \overline{K} where c_{2} = ζ^{2q̂^{2}}c^{q̂^{2}} and c_{n} = −(u_{n−1})^{q̂^{2n}} − ζ^{2nq̂^{n}}u_{n−1} for n ≥ 3. We claim that there are solutions x_{n} ∈ \overline{K} with |x_{2}| = |c| and |x_{n}|q̂ = |c| |ζ|^{q̂^{n}} < |c| |ζ|q̂ ≤ |c|q̂ for n ≥ 3. From this claim it follows that |u_{n}| = |c| and that the limit b_{0} := lim_{n→∞} u_{n} exists in \overline{K} with |b_{0}| = |c| < 1. In the expression

∑_{|i|>n} b_{0}^{q̂^{−i}}ζ^{i} + ∑_{i=−n}^{n}(b_{0} − u_{n})^{q̂^{−i}}ζ^{i} = ∑_{i=−∞}^{∞} b_{0}^{q̂^{−i}}ζ^{i} − ∑_{i=−n}^{n} u_{n}^{q̂^{−i}}ζ^{i} = ∑_{i=−∞}^{∞} b_{0}^{q̂^{−i}}ζ^{i} − c

the first sum goes to zero for n → ∞ because f ∈ O_{\overline{K}}[[z, z^{−1}\}. Since v(b_{0} − u_{n}) = v(x_{n+1}) = q̂^{−1}v(c)+q̂^{n} the i-th summand in the second sum has valuation v((b_{0}−u_{n})^{q̂^{−i}}ζ^{i}) = q̂^{−i−1}v(c)+q̂^{n−i}+i > (q̂ − 1)(n − i) + 1 + i ≥ n + 1 by Bernoulli’s inequality. So the second sum likewise goes to zero and b_{0} solves equation (5.10).

To prove the claim we use the Newton polygon of (5.12), which is defined as the lower convex hull in the plane ℝ^{2} of the points

(0, v(c_{n})) , (1, 2nq̂^{n}) , (q̂, (2n − 1)q̂^{n}) , …, (q̂^{j}, (2n − j)q̂^{n}) , …, (q̂^{2n}, 0) .

The piecewise linear function passing through these points has slope −\frac{q̂^{n−j}}{q̂−1} on the interval [q̂^{j}, q̂^{j+1}] for all j = 0, …, 2n−1. In particular, these slopes are strictly increasing. If n = 2 and v(c_{2}) = q̂^{2}v(c)+2q̂^{2} the Newton polygon starts with the line segment of slope −v(c) on the interval [0, q̂^{2}] because v(c) ≤ 1+ \frac{1}{q̂−1} ≤ 2 implies v(c_{2})−1·v(c) = (q̂^{2}−1)v(c)+2q̂^{2} < 4q̂^{2} and v(c_{2})−q̂·v(c) = (q̂^{2}−q̂)v(c)+2q̂^{2} ≤ 3q̂^{2}, and because on the next interval [q̂^{2}, q̂^{3}] the slope is −\frac{1}{q̂−1} > −v(c). So by the theory of the Newton polygon, (5.12) has a solution x_{2} ∈ \overline{K} with v(x_{2}) = v(c).

If n ≥ 3 then n < 3(n − 2) + 1 ≤ (q̂^{2} − 1)(n − 2) + (q̂ − 1) implies ∑_{j=n+1}^{2n}(q̂^{n} − q̂^{2n−j}) ≤ nq̂^{n} < (q̂^{n+2} − q̂^{n})(n − 2) + (q̂^{n+1} − q̂^{n}) ≤ ∑_{j=1}^{n}(q̂^{2n−j} − q̂^{n}), whence 2nq̂^{n} < ∑_{j=1}^{2n} q̂^{2n−j} = \frac{q̂^{2n}−1}{q̂−1}. Therefore the induction hypothesis v(u_{n−1}) = v(c) > \frac{1}{q̂−1} yields

v(u_{n−1}^{q̂^{2n}}) = q̂^{2n}v(c) > v(c) + \frac{q̂^{2n}−1}{q̂−1} > v(c) + 2nq̂^{n} = v(ζ^{2nq̂^{n}}u_{n−1})

and hence v(c_{n}) = v(c)+2nq̂^{n}. It follows that the Newton polygon starts with the line segment of slope −\frac{v(c)}{q̂} −q̂^{n−1} on the interval [0, q̂] because v(c)+2nq̂^{n}+1·(−\frac{v(c)}{q̂} −q̂^{n−1}) = \frac{(q̂−1)v(c)}{q̂} −q̂^{n−1}+2nq̂^{n} < 2nq̂^{n},

and because on the next interval [q̂, q̂^{2}] the slope is −\frac{q̂^{n−1}}{q̂−1} ≥ −q̂^{n−1} > −\frac{v(c)}{q̂} − q̂^{n−1}. Again by the

theory of the Newton polygon, (5.12) has a solution x_{n} ∈ \overline{K} with v(x_{n}) = \frac{v(c)}{q̂} + q̂^{n−1} for n ≥ 3. This proves our claim and hence also the corollary.

Finally, to prove exactness on the right in Lemma 5.16 let c (z − ζ)^{−n} ∈ \overline{K}((z − ζ)) for c ∈ \overline{K} and fix an n-th root \sqrt[n]{c} ∈ \overline{K}. By the corollary there is an f ∈ O_{\overline{K}}[[z, z^{−1}\} with f = z^{−1}σ̂(f) and f(ζ) = \sqrt[n]{c} · (\frac{ℓ}{z−ζ})|_{z=ζ} = \sqrt[n]{c} · (z^{−1}ℓ_{+}σ̂(ℓ_{−}))|_{z=ζ}. Therefore f^{n}/ℓ^{n} ∈ O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}]^{σ̂= id} and f/ℓ ≡ \sqrt[n]{c} (z − ζ)^{−1} mod \overline{K}[[z − ζ]] implies c (z − ζ)^{−n} − f^{n}/ℓ^{n} ∈ (z − ζ)^{1−n}\overline{K}[[z − ζ]]. In this way we can successively get rid of all denominators in \overline{K}((z − ζ)) and this proves exactness on the right.

Theorem 5.18. Let \underline{M̂} be a rigidified local shtuka over R. There are canonical comparison isomorphisms between the de Rham and crystalline realizations

h_{dR,cris} : H_{dR}^{1}(\underline{M̂}, K[[z − ζ]]) ^{∼} ⟶ H_{cris}^{1}(\underline{M̂}, k((z))) ⊗_{k((z))} K[[z − ζ]] and
h_{dR,cris} :  H_{dR}^{1}(\underline{M̂}, K)  ^{∼} ⟶ H_{cris}^{1}(\underline{M̂}, k((z))) ⊗_{k((z)), z↦ζ} K ,

which are compatible with the Hodge-Pink lattices and Hodge-Pink filtrations. They usually are not compatible with the integral structures H_{dR}^{1}(\underline{M̂}, R) := σ̂^{∗}M̂/(z−ζ)σ̂^{∗}M̂ ⊂ H_{dR}^{1}(\underline{M̂}, K) and H_{cris}^{1}(\underline{M̂}, k[[z]]).

Proof. Taking the isomorphism σ̂^{∗}δM̂ from Lemma 5.8, which is an isomorphism over σ̂^{∗}R[[z, z^{−1}}[ℓ−1_{−}] = R[[z, z^{−1}}[σ̂(ℓ_{−})^{−1}], and using the inclusion R[[z, z^{−1}}[σ̂(ℓ_{−})^{−1}] ⊂ K[[z−ζ]], see Example 5.15, we define

h_{dR,cris} := σ̂^{∗}δM̂ ⊗ id_{K[[z−ζ]]} : σ̂^{∗}M̂ ⊗_{R[[z]]} K[[z − ζ]] ^{∼} ⟶ D ⊗_{k((z))} K[[z − ζ]] .

We will see in Example 5.19 below that h_{dR,cris} does not need to be compatible with the integral structures.

Example 5.19. To give an example in which h_{dR,cris} is not compatible with the integral structures, we let A_{ε} = 𝔽_{q}[[z]] and assume that q̂ = q ≥ 3. As our base ring we take R = k[[π]] with ζ = π^{q+1}. So the ramification index of γ: A_{ε} ↪ R is q + 1 > q. Over R we consider the local shtuka \underline{M̂} = (R[[z]]^{⊕2}, τM̂ = ( \begin{matrix}0 & z−ζ \\ 1 & π−1 \end{matrix} )). It is the local shtuka at ε = (t) ⊂ 𝔽_{q}[t] associated with the Drinfeld 𝔽_{q}[t]- module \underline{E} = (𝔾_{a,R}, φ) with φ: 𝔽_{q}[t] → R\{τ\}, φ_{t} = ζ +(1−π)τ +τ^{2} which has good ordinary reduction. We compute its rigidification δ_{M}, or rather δ_{M}^{−1} by following the proof of Lemma 5.8. We have

T = ( \begin{matrix}0 & z−ζ \\ 1 & π−1 \end{matrix} ) and therefore S = ( \begin{matrix}0 & z \\ 1 & −1 \end{matrix} ) and S^{−1} = z^{−1} ( \begin{matrix}1 & z \\ 1 & 0 \end{matrix} ). We obtain C_{0}^{−1} = T S^{−1} = ( \begin{matrix}1−ζ/z & 0 \\ π/z & 1 \end{matrix} )

and T − S = ( \begin{matrix}0 & −ζ \\ 0 & π \end{matrix} ). The C_{m} from (5.4) satisfy the recursion formula

C_{m}^{−1} − C_{m−1}^{−1} = T · . . . · σ̂^{m−1}(T) · σ̂^{m}(T − S) · σ̂^{m}(S^{−1}) · . . . · S^{−1} ∈ M_{2}(\frac{π^{q^{m}}}{z^{m+1}} R[[z]]) .

We have δ_{M}^{−1} = lim_{m→∞} C_{m}^{−1} and we want to evaluate σ̂^{∗}δ_{M}^{−1} at z = ζ. We observe σ̂(C_{0}^{−1})|_{z=ζ} =

( \begin{matrix}1−ζ^{q}/ζ & 0 \\ π^{q}/ζ & 1 \end{matrix} ) = ( \begin{matrix}1−ζ^{q−1} & 0 \\ π^{−1} & 1 \end{matrix} ) ∉ M_{2}(R) and σ̂(C_{m}^{−1} − C_{m−1}^{−1})|_{z=ζ} ∈ M_{2}(\frac{π^{q^{m+1}}}{z^{m+1}} R[[z]])|_{z=ζ} ⊂ M_{2}(R) for

m ≥ 1, because \frac{π^{q^{m+1}}}{ζ^{m+1}} = π^{q^{m+1}−(q+1)(m+1)} ∈ R as q^{m+1} − (q + 1)(m + 1) ≥ 0 for q ≥ 3 and m ≥ 1. This shows that σ̂(δ_{M}^{−1})|_{z=ζ} ∉ GL_{2}(R), that is h_{dR,cris} is not compatible with the integral structures.

Note that this example is the function field analog of the example [BO83, Remark 2.10] which shows that also for a proper smooth scheme X over O_{L} the comparison isomorphism H_{dR}^{i}(X/O_{L}) ⊗_{O_{L}} L ^{∼} ⟶ H_{cris}^{i}(X_{0}/W) ⊗_{W} L does not need to be compatible with the integral structures; see Remark 5.21.

Theorem 5.20. Let \underline{M̂} be a rigidified local shtuka over R. There is a canonical comparison isomorphism between the ε-adic and crystalline realizations

h_{ε,cris} : H_{ε}^{1}(\underline{M̂}, Q_{ε}) ⊗_{Q_{ε}} O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}] ^{∼} ⟶ H_{cris}^{1}(\underline{M̂}, k((z))) ⊗_{k((z))} O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}] .

The isomorphism h_{ε,cris} is Gal(K^{sep}/K)- and τ̂-equivariant, where on the left module Gal(K^{sep}/K) acts on both factors and τ̂ is id ⊗σ̂, and on the right module Gal(K^{sep}/K) acts only on O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}] and τ̂ is (τ_{D} ∘ σ̂_{D}^{∗}) ⊗ σ̂. In other words h_{ε,cris} = τ_{D} ∘ σ̂^{∗}h_{ε,cris}. Moreover, h_{ε,cris} satisfies h_{ε,dR} = (h_{dR,cris}^{−1} ⊗ id_{\overline{K}((z−ζ))}) ∘ (h_{ε,cris} ⊗ id_{\overline{K}((z−ζ))}). It allows to recover H_{ε}^{1}(\underline{M̂}, Q_{ε}) from H_{cris}^{1}(\underline{M̂}, k((z))) as the intersection inside H_{cris}^{1}(\underline{M̂}, k((z))) ⊗_{k((z))} \overline{K}((z − ζ))

h_{ε,cris}(H_{ε}^{1}(\underline{M̂}, Q_{ε})) = (H_{cris}^{1}(\underline{M̂}, k((z))) ⊗_{k((z))} O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}])^{τ̂= id} ∩ 𝔮_{D} ⊗_{K[[z−ζ]]} \overline{K}[[z − ζ]] ,

where 𝔮_{D} ⊂ H_{cris}^{1}(\underline{M̂}, k((z)))⊗_{k((z))}K((z−ζ)) is the Hodge-Pink lattice of \underline{M̂}. Since k((z)) ⊊ R[[z, z^{−1}\} ⊂ (O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}])^{Gal(K^{sep}/K)}, it does not allow to recover H_{cris}^{1}(\underline{M̂}, k((z))) from H_{ε}^{1}(\underline{M̂}, Q_{ε}).

Remark 5.21. The comparison isomorphisms h_{dR,cris} are the function field analogs for the comparison isomorphism H_{dR}^{i}(X/O_{L})⊗_{O_{L}} L ^{∼} ⟶ H_{cris}^{i}(X_{0}/W)⊗_{W} L between de Rham and crystalline cohomology of a proper smooth scheme X over a complete discrete valuation ring O_{L} with perfect residue field κ of characteristic p and fraction field L of characteristic 0, where W is the ring of p-typical Witt vectors with coefficients in κ and X_{0} = X ⊗_{O_{L}} κ; see [BO83, Corollary 2.5]. If L_{0} is the fraction field of W then H_{cris}^{i}(X_{0}/L_{0}) = H_{cris}^{i}(X_{0}/W) ⊗_{W} L_{0} is an F -isocrystal, that is a finite dimensional L_{0}-vector space equipped with a Frobenius linear automorphism, and the Hodge filtration on H_{dR}^{i}(X/O_{L})⊗_{O_{L}} L makes it into a filtered isocrystal via the comparison isomorphism.

Less straightforward is the comparison isomorphism h_{ε,cris} which is the function field analog of Fontaine’s comparison isomorphism H_{ét}^{i}(X ×_{L} L^{alg}, ℚ_{p}) ⊗_{ℚ_{p}} B̃_{rig} ^{∼} ⟶ H_{cris}^{i}(X_{0}/L_{0}) ⊗_{L_{0}} B̃_{rig}. Namely, with a Gal(L^{alg}/L)-representation V such as H_{ét}^{i}(X×_{L}L^{alg}, ℚ_{p}) Fontaine associates a filtered isocrystal D_{cris}(V) := (V ⊗_{ℚ_{p}} B_{cris})^{Gal(L^{alg}/L)} and he calls V “crystalline” if dim_{L_{0}} D_{cris}(V) = dim_{ℚ_{p}} V. In this case there is a comparison isomorphism V ⊗_{ℚ_{p}} B_{cris} ^{∼} ⟶ D_{cris}(V) ⊗_{L_{0}} B_{cris}, which is already defined over the subring B̃_{rig} ⊂ B_{cris}; see [Ber04b, II.3.5] and [Ber02, p. 228]. It was then conjectured by Fontaine [Fon82, A.11] and proved by Faltings [Fal89] that H_{ét}^{i}(X ×_{L} L^{alg}, ℚ_{p}) is indeed crystalline and D_{cris}(H_{ét}^{i}(X ×_{L} L^{alg}, ℚ_{p})) = H_{cris}^{i}(X_{0}/L_{0}). The fundamental exact sequence (5.9) allows to also recover

H_{ét}^{i}(X ×_{L} L^{alg}, ℚ_{p}) ≅ F^{0}(H_{cris}^{i}(X_{0}/L_{0}) ⊗_{L_{0}} B̃_{rig})^{Frob = id}

from H_{cris}^{i}(X_{0}/L_{0}).

In our comparison isomorphism, ℓ is the analog of log[\underline{e}] ∈ B̃_{rig} where \underline{e} = (e_{i}: i ∈ ℕ_{0}) is a compatible system of primitive p^{i}-th roots of unity e_{i} and [\underline{e}] is its Teichmüller lift; see Example 5.15. Our ring O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}] is the analog of B̃_{rig}; see [Har09, § 2.5 and § 2.7]. In that sense we could write F^{0} H_{dR}^{1}(\underline{M̂}, K((z − ζ))) := 𝔮 and

F^{0} H_{cris}^{1}(\underline{M̂}, k((z))) ⊗_{k((z))} O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}] :=
H_{cris}^{1}(\underline{M̂}, k((z))) ⊗_{k((z))} O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}] ∩ 𝔮_{D} ⊗_{K[[z−ζ]]} \overline{K}[[z − ζ]] .

Proof of Theorem 5.20. To construct h_{ε,cris} we use for s ∈ ℝ_{>0} the notation

K⟨\frac{z}{ζ^{s}} , z^{−1}\} := \{ ∑_{i=−∞}^{∞}b_{i}z^{i}: b_{i} ∈ K , |b_{i}| |ζ|^{ri} → 0 (i → ±∞) for all r ≥ s \}  (5.13)

for the ring of rigid analytic functions with coefficients in K on the punctured closed disc {0 < |z| ≤ |ζ|^{s}\} of radius |ζ|^{s}. The ring K⟨\frac{z}{ζ} , z^{−1}\} contains both the rings K⟨\frac{z}{ζ} ⟩ and R[[z, z^{−1}\}, and so ℓ_{+} ∈ \overline{K}⟨\frac{z}{ζ} , z^{−1}\}^{×}. In addition to δM̂ we take the isomorphism h from Remark 4.3 and define

h_{ε,cris} := (σ̂^{∗}δM̂ ∘ τ −1 ; M̂ ∘ h) ⊗ id_{\overline{K}⟨ \frac{z}{ζ} ,z^{−1}\}[ℓ^{−1}]}

h_{ε,cris} : Ť_{ε}\underline{M̂} ⊗_{A_{ε}} \overline{K}⟨\frac{z}{ζ} , z^{−1}\}[ℓ^{−1}] ^{∼} ⟶ H_{cris}^{1}(\underline{M̂}, k((z))) ⊗_{k((z))} \overline{K}⟨\frac{z}{ζ} , z^{−1}\}[ℓ^{−1}] .

It satisfies h_{ε,cris} = τ_{D}∘σ̂^{∗}h_{ε,cris}. If we choose an A_{ε}-basis of Ť_{ε}\underline{M̂} and a k((z))-basis of H_{cris}^{1}(\underline{M̂}, k((z))) = D := σ̂^{∗}M̂ ⊗_{R[[z]]}k((z)), we may write h_{ε,cris} and its inverse as matrices H, H^{−1} ∈ GL_{r}(\overline{K}⟨\frac{z}{ζ} , z^{−1}\}[ℓ^{−1}]). Let also T ∈ GL_{r}(k((z))) ⊂ GL_{r}(O_{\overline{K}}[[z, z^{−1}\}) be the matrix of τ_{D} with respect to the second basis. Then the equivariance of h_{ε,cris} with respect to τ̂ implies that T · σ̂(H) = H. After multiplying H and H^{−1} each with a high enough power of ℓ we have ℓ^{n}H, ℓ^{m}H^{−1} ∈ M_{r}(\overline{K}⟨\frac{z}{ζ} , z^{−1}\}) with σ̂(ℓ^{n}H) = z^{n}T^{−1}ℓ^{n}H and σ̂(ℓ^{m}H^{−1}) = z^{m}ℓ^{m}H^{−1}T. From Lemma 5.22 below it follows that ℓ^{n}H, ℓ^{m}H^{−1} ∈ M_{r}(O_{\overline{K}}[[z, z^{−1}\}) and H ∈ GL_{r}(O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}]). So h_{ε,cris} comes from an isomorphism

h_{ε,cris} : H_{ε}^{1}(\underline{M̂}, Q_{ε}) ⊗_{Q_{ε}} O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}] ^{∼} ⟶ H_{cris}^{1}(\underline{M̂}, k((z))) ⊗_{k((z))} O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}] .

By definition of h_{ε,dR} := (τ −1 ; M̂ ∘ h) ⊗ id_{\overline{K}((z−ζ))} in Theorem 4.12 and h_{dR,cris} := σ̂^{∗}δM̂ ⊗ id_{K[[z−ζ]]} in Theorem 5.18 we obtain h_{ε,dR} = (h_{dR,cris}^{−1} ⊗ id_{\overline{K}((z−ζ))}) ∘ (h_{ε,cris} ⊗ id_{\overline{K}((z−ζ))}).

To prove the remaining statement we use the exact sequence of Q_{ε}-vector spaces from Lemma 5.16 and tensor it with H_{ε}^{1}(\underline{M̂}, Q_{ε}) to obtain the left column in the diagram

[figure: 0; 0; H_{ε}^{1}(\underline{M̂}, Q_{ε}); h_{ε,cris}; ≅; h_{ε,cris}(H_{ε}^{1}(\underline{M̂}, Q_{ε})); H_{ε}^{1}(\underline{M̂}, Q_{ε})⊗_{Q_{ε}} O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}]^{σ̂= id} ; ⨁; h_{ε,cris}; ≅; (H_{cris}^{1}(\underline{M̂}, k((z))) ⊗_{k((z))} O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}])^{τ̂= id} ; ⨁; H_{ε}^{1}(\underline{M̂}, Q_{ε}) ⊗_{Q_{ε}} \overline{K}[[z − ζ]]; h_{ε,cris}; ≅; 𝔮_{D} ⊗_{K[[z−ζ]]} \overline{K}[[z − ζ]]; H_{ε}^{1}(\underline{M̂}, Q_{ε}) ⊗_{Q_{ε}} \overline{K}((z − ζ)); h_{ε,cris}; ≅; H_{cris}^{1}(\underline{M̂}, k((z))) ⊗_{k((z))} \overline{K}((z − ζ)) .]

The second horizontal map is an isomorphism because h_{ε,cris} is τ̂-equivariant. The third horizontal map is an isomorphism because h_{ε,dR}(H_{ε}^{1}(\underline{M̂}, Q_{ε})⊗_{Q_{ε}} \overline{K}[[z −ζ]]) = 𝔮⊗_{K[[z−ζ]]} \overline{K}[[z −ζ]] by Theorem 4.12 and because h_{dR,cris} is compatible with the Hodge-Pink lattices by Theorem 5.18. From the right column it follows that

h_{ε,cris}(H_{ε}^{1}(\underline{M̂}, Q_{ε})) = (H_{cris}^{1}(\underline{M̂}, k((z))) ⊗_{k((z))} O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}])^{τ̂= id} ∩ 𝔮_{D} ⊗_{K[[z−ζ]]} \overline{K}[[z − ζ]] .

The following lemma is the function field analog of [Ber04a, Proposition I.4.1 and Corollary I.4.2].

Lemma 5.22. Let U ∈ M_{r}(K⟨\frac{z}{ζ^{s}} , z^{−1}\}) for some s > 0, and let V, W ∈ M_{r}(R[[z, z^{−1}\}) such that σ̂(U) = V UW. Then U ∈ M_{r}(R[[z, z^{−1}}).

Remark 5.23. The lemma and its proof are valid more generally if R is replaced by an admissible formal R-algebra B^{∘} in the sense of Raynaud, if K is replaced by B := B^{∘} ⊗_{R} K and | . | is a K-Banach norm on B with B^{∘} = {b ∈ B: |b| ≤ 1}; see [BL93].

Proof of Lemma 5.22. For the elements ∑_{i=−∞}^{∞} b_{i}z^{i} ∈ K⟨\frac{z}{ζ^{s}} , z^{−1}\} we consider the norm ∥∑_{i} b_{i}z^{i}∥_{s} :=

max_{i∈ℤ} |b_{i}| |ζ|^{si} and we extend this to a norm on the matrix U by taking the maximum of the norms of the entries of U. It satisfies ∥V U∥_{s} ≤ ∥V ∥_{s} · ∥U∥_{s}. Since ∥z∥_{s} = |ζ|^{s} < 1 there is an integer c such that ∥z^{c}U∥_{s} ≤ 1. Likewise there is an integer k such that ∥z^{k}V ∥_{s}, ∥z^{k}W ∥_{s} ≤ 1. The entries of V and W are of the form ∑_{i} b_{i}z^{i} with |b_{i}| ≤ 1. For those we have |b_{i}|q̂ ≤ |b_{i}| and this implies ∥σ̂^{n}(∑_{i} b_{i}z^{i})∥_{s} = ∥ ∑_{i} b_{i}^{q̂^{n}}z^{i}∥_{s} ≤ ∥ ∑_{i} b_{i}z^{i}∥_{s}, and hence ∥σ̂^{n}(z^{k}V)∥_{s}, ∥σ̂^{n}(z^{k}W)∥_{s} ≤ 1 for all n ∈ ℕ_{0}. We obtain

∥σ̂^{n+1}(z^{c+2k(n+1)}U)∥_{s} = ∥σ̂^{n}(z^{k}V)σ̂^{n}(z^{c+2kn}U)σ̂^{n}(z^{k}W)∥_{s} ≤ ∥σ̂^{n}(z^{c+2kn}U)∥_{s} ≤ 1

for all n ∈ ℕ_{0} by induction. Let ∑_{i} b_{i}z^{i} be an entry of U. Then ∑_{i} b_{i}^{q̂^{n}}z^{i+c+2kn} is the corresponding entry of σ̂^{n}(z^{c+2kn}U). The estimate 1 ≥ ∥∑_{i} b_{i}^{q̂^{n}}z^{i+c+2kn}∥_{s} = max_{i} |b_{i}|^{q̂^{n}}|ζ|^{s(i+c+2kn)} implies |b_{i}| ≤ |ζ|^{−s(i+c+2kn)q̂^{−n}} for all i. For fixed i and n → +∞ the exponent goes to 0 and so we find |b_{i}| ≤ 1, that is b_{i} ∈ R for all i ∈ ℤ. In order that ∑_{i} b_{i}z^{i} ∈ R[[z, z^{−1}} we have to verify that for all r > 0 the condition |b_{i}| |ζ|^{ri} → 0 for i → −∞ holds. From ∑_{i} b_{i}z^{i} ∈ K⟨\frac{z}{ζ^{s}} , z^{−1}\} we already know that this condition holds for all r ≥ s. If r < s and i < 0 then |b_{i}| |ζ|^{ri} < |b_{i}| |ζ|^{si} and so the condition also holds for the remaining r < s. This proves that U ∈ M_{r}(R[[z, z^{−1}}) as desired.

Example 5.24. For the Carlitz motive from Examples 4.14 and 5.15 the comparison isomorphisms from Theorems 5.18 and 5.20 between H_{dR}^{1}(\underline{M}, K[[z − ζ]]) = K[[z − ζ]] with basis 1 and H_{ε}^{1}(\underline{M}, Q_{ε}) = Q_{ε} · ℓ−1_{+} with basis ℓ−1_{+} on the one hand and H_{cris}^{1}(\underline{M}, k((z))) = (k((z)), z, (z − ζ)^{−1}𝔭_{D}) with basis 1 on the other hand are given explicitly as follows. With respect to these bases h_{dR,cris} is given by σ̂(ℓ−1_{−}) ∈ K[[z − ζ]]^{×} and h_{ε,cris} is given by σ̂(ℓ^{−1}) = σ̂(ℓ_{−})^{−1}σ̂(ℓ_{+})^{−1} = z^{−1}ℓ^{−1}. In particular, h_{dR,cris} modulo z − ζ is given by ∏_{i∈ℕ_{>0}}(1 − ζ^{q̂^{i}−1})^{−1} ∈ OK^{×}. So in this example h_{dR,cris} is compatible with the integral structures.

6 Admissibility and weak admissibility

It turns out that not all z-isocrystals with Hodge-Pink structure can arise via the functor ℍ from (5.6). Namely a necessary condition is weak admissibility as in the following

Definition 6.1. Let R be an arbitrary valuation ring as in Notation 1.1, let \underline{D} = (D, τ_{D}, 𝔮_{D}) be a z-isocrystal with Hodge-Pink structure over R and set r = dim_{k((z))} D.

(a) Choose a k((z))-basis of D and let det τ_{D} be the determinant of the matrix representing τ_{D} with respect to this basis. The number t_{N}(\underline{D}) := ord_{z}(det τ_{D}) is independent of this basis and is called the Newton slope of \underline{D}.

(b) The integer t_{H}(\underline{D}) := −μ_{1} − … − μ_{r}, where μ_{1}, …, μ_{r} are the Hodge-Pink weights of \underline{D} from Definition 5.4, satisfies ∧^{r}𝔮_{D} = (z − ζ)^{−t_{H}(\underline{D})} ∧^{r} 𝔭_{D} and is called the Hodge slope of \underline{D}.

(c) \underline{D} is called weakly admissible (or semi-stable of slope 0) if

t_{H}(\underline{D}) = t_{N}(\underline{D})  and  t_{H}(\underline{D}′) ≤ t_{N}(\underline{D}′) for every strict subobject \underline{D}′ ⊂ \underline{D}.

(d) \underline{D} is called admissible if there exists a rigidified local σ̂-shtuka \underline{M̂} over R with \underline{D} = ℍ(\underline{M̂}).

Remark 6.2. This definition parallels Fontaine’s definition [Fon79] of (weak) admissibility of filtered isocrystals. Namely, Fontaine calls a filtered isocrystal admissible if it comes from a crystalline Galois representation; compare Remark 5.13.

In comparison with the Hodge-Pink structures at ∞ from [HJ16, § 2] the Hodge slope t_{H}(\underline{D}) corresponds to deg_{𝔮}, the Newton slope t_{N}(\underline{D}) corresponds to deg^{W}, and our notion of weak admissibility corresponds to Pink’s notion of “local semi-stability”; see [HJ16, Definition 2.7].

Example 6.3. For the Carlitz motive from Example 5.15 and its z-isocrystal with Hodge-Pink structure \underline{D} = (k((z)), z, (z − ζ)^{−1}𝔭_{D}) one has t_{N}(\underline{D}) = 1 = t_{H}(\underline{D}). It is (weakly) admissible.

To obtain a criterion for (weak) admissibility we need to recall a little bit of the theory of σ-bundles from [HP04], see also [HJ16, § 7]. Here we consider a variant which also works if K is not algebraically closed. Namely, we do not consider the punctured open unit disc {0 < |z| < 1} but instead the punctured closed disc \{0 < |z| ≤ |ζ|\} of radius |ζ|, and the ring K⟨\frac{z}{ζ} , z^{−1}\} of rigid analytic functions with coefficients in K on it from (5.13). There are two morphisms from K⟨\frac{z}{ζ} , z^{−1}\} to K⟨\frac{z}{ζq̂}, z^{−1}\}, namely

ι: K⟨\frac{z}{ζ} , z^{−1}\} ⟶ K⟨\frac{z}{ζq̂}, z^{−1}\}, ∑_{i} b_{i}z^{i} ⟼ ∑_{i} b_{i}z^{i}  and
σ̂: K⟨\frac{z}{ζ} , z^{−1}\} ⟶ K⟨\frac{z}{ζq̂}, z^{−1}\}, ∑_{i} b_{i}z^{i} ⟼ ∑_{i} bq̂ ; iz^{i} .

For a K⟨\frac{z}{ζ} , z^{−1}\}-module F we define the two K⟨\frac{z}{ζq̂}, z^{−1}\}-modules σ̂^{∗}F := F ⊗_{K⟨ \frac{z}{ζ} ,z^{−1}\}, σ̂} K⟨\frac{z}{ζq̂}, z^{−1}\} and ι^{∗}F := F ⊗_{K⟨ \frac{z}{ζ} ,z^{−1}\}, ι} K⟨\frac{z}{ζq̂}, z^{−1}\}.

Definition 6.4. A σ̂-bundle (on \{0 < |z| ≤ |ζ|\}) of rank r over K is a pair \underline{F} = (F, τ_{F}) consisting of a free K⟨\frac{z}{ζ} , z^{−1}\}-module F of rank r together with an isomorphism τ_{F}: σ̂^{∗}F ^{∼} ⟶ ι^{∗}F of K⟨\frac{z}{ζq̂}, z^{−1}\}- modules.

A homomorphism f: (F, τ_{F}) → (G, τ_{G}) between σ̂-bundles is a homomorphism f: F → G of K⟨\frac{z}{ζ} , z^{−1}\}-modules which satisfies τ_{F} ∘ σ̂^{∗}f = ι^{∗}f ∘ τ_{G}.

The τ̂-invariants of (F, τ_{F}) are defined as \underline{F} τ̂ := \{ f ∈ F : τ_{F}(σ̂^{∗}f) = f \}. It is a vector space over \{f ∈ K⟨\frac{z}{ζ} , z^{−1}\} : σ̂(f) = f \} = 𝔽_{ε}((z)) = Q_{ε}.

Example 6.5. 1. The trivial σ̂-bundle over K is \underline{F}_{0,1} := (K⟨\frac{z}{ζ} , z^{−1}\}, id_{K⟨ \frac{z}{ζq̂} ,z^{−1}\}}).

2. More generally, for relatively prime integers d, r with r > 0 we let \underline{F}_{d,r} be the σ̂-bundle over K consisting of F_{d,r} = K⟨\frac{z}{ζ} , z^{−1}\}^{⊕r} with

τ_{F_{d,r}} := ( 0 1 0 0 0 0  1 z^{−d} 0  0 ) .

3. We already saw in equation (5.8) that ℓ ∈ \underline{F}_{1,1}τ̂.

Lemma 6.6. If K is algebraically closed the evaluation at z = ζ induces an exact sequence of 𝔽_{ε}((z))- vector spaces

\xymatrix{0 \ar[r] & 𝔽_{ε}((z)) · ℓ \ar[r] & \underline{F}_{1,1}τ̂ \ar[r] & K \ar[r] & 0 , \\  &  & f(z) \ar@{|->}[r] & f(ζ)}

where K is an 𝔽_{ε}((z))-vector space via the inclusion 𝔽_{ε}((z)) ↪ K, z ↦ ζ.

Proof. The sequence is obviously exact on the left. Furthermore, ℓ maps to zero under the right morphism. To prove exactness in the middle let f ∈ \underline{F}_{1,1}τ̂ vanish at z = ζ. Then it vanishes at z = ζ^{q̂^{i}} for all i ∈ ℕ_{0} and g := f/ℓ ∈ K⟨\frac{z}{ζ} , z^{−1}\} satisfies g = σ̂^{∗}(g) ∈ 𝔽_{ε}((z)) as desired. The exactness on the right was established in Corollary 5.17.

The structure theory of σ̂-bundles over an algebraically closed complete field (for example over the completion \overline{K} of an algebraic closure of K) was developed in [HP04] and [Har11, § 1.4].

Theorem 6.7. We assume that K is algebraically closed (or we work over \overline{K}).

(a) Any σ̂-bundle \underline{F} over K is isomorphic to ⨁_{i} \underline{F}_{d_{i},r_{i}} for pairs of relatively prime integers d_{i}, r_{i} with r_{i} > 0, which are uniquely determined by \underline{F} up to permutation. They satisfy rk \underline{F} = ∑_{i} r_{i} and we define the degree of \underline{F} as deg \underline{F} := ∑_{i} d_{i}.

(b) There is a non-zero morphism \underline{F}_{d′,r′} → \underline{F}_{d,r} if and only if \frac{d′}{r′} ≤ \frac{d}{r}.

(c) Any σ̂-sub-bundle \underline{F}′ ⊂ \underline{F}_{d,r}^{⊕n} satisfies deg \underline{F}′ ≤ \frac{d}{r} · rk \underline{F}′.

Proof. Statements (a) and (b) are [Har11, Theorem 1.4.2 and Proposition 1.4.5]. Statement (c) follows easily from (a) and (b). Namely, F′ ≅ ⨁_{i} \underline{F}_{d_{i},r_{i}} by (a) with \frac{d_{i}}{r_{i}} ≤ \frac{d}{r} by (b) yields (c).

Now let (\underline{M̂}, δM̂) = (M̂, τM̂, δM̂) be a rigidified local shtuka over R as in Definition 5.9, and let \underline{D} = (D, τ_{D}, 𝔮_{D}) be the associated z-isocrystal with Hodge-Pink structure over R. We define two σ̂-bundles on {0 < |z| ≤ |ζ|}

\underline{E}(\underline{M̂}) := (E(\underline{M̂}), τ_{E}) := σ̂^{∗}\underline{M̂} ⊗_{R[[z]]} k((z)) ⊗_{k((z))} K⟨\frac{z}{ζ} , z^{−1}\} = (D, τ_{D}) ⊗_{k((z))} K⟨\frac{z}{ζ} , z^{−1}\} and
\underline{F}(\underline{M̂}) := (F(\underline{M̂}), τ_{F}) := \underline{M̂} ⊗_{R[[z]]} K⟨\frac{z}{ζ} , z^{−1}\} .

There is a canonical isomorphism σ̂^{∗}δM̂ ∘ τ −1 ; M̂: \underline{F}(\underline{M̂})[ℓ−1_{−}] ^{∼} ⟶ \underline{E}(\underline{M̂})[ℓ−1_{−}]. We use this isomorphism to view F(\underline{M̂}) as a K⟨\frac{z}{ζ} , z^{−1}\}-submodule of E(\underline{M̂})[ℓ−1_{−}]. If \underline{M̂} is effective, that is if τM̂(σ̂^{∗}M̂) ⊂ M̂, we have E(\underline{M̂}) ⊂ σ̂^{∗}δM̂ ∘ τ −1 ; M̂(F(\underline{M̂})). Assume that τM̂(σ̂^{∗}M̂) ⊊ M̂. Then we visualize these σ̂-bundles by the following diagram, in which the thick lines represent K⟨\frac{z}{ζ} , z^{−1}\}-modules:

F(\underline{M̂}) E(\underline{M̂}) \underline{M̂} ⊗_{R[[z]]} K⟨\frac{z}{ζ} , z^{−1}\} σ̂^{∗}\underline{M̂} ⊗_{R[[z]]} K⟨\frac{z}{ζ} , z^{−1}\} {|ζ| ≥ |z| > 0} z = ζ z = ζq̂ …

Modules drawn higher contain the ones drawn below. All K⟨\frac{z}{ζ} , z^{−1}\}-modules coincide outside the discrete set ⋃_{i∈ℕ_{0}}{z = ζ^{q̂^{i}}} ⊂ {|ζ| ≥ |z| > 0}. At those points in ⋃_{i∈ℕ_{0}}{z = ζ^{q̂^{i}}} where two modules are drawn at almost the same height, they also coincide. Indeed, F(\underline{M̂}) equals M̂. Moreover, it contains τM̂(σ̂^{∗}M̂) and differs from it only at z = ζ by our assumption. Also E(\underline{M̂}) coincides with (σ̂^{∗}M̂) ⊗_{R[[z]]} K⟨\frac{z}{ζ} , z^{−1}\} at z = ζ because σ̂^{∗}δM̂ is an isomorphism of K⟨\frac{z}{ζ} , z^{−1}\}[σ̂^{∗}ℓ−1_{−}]-modules and σ̂^{∗}ℓ_{−} is a unit in K[[z − ζ]]. Finally, the strict inclusion τM̂(σ̂^{∗}M̂) ⊊ M̂ at z = ζ is transported by τ_{F} and τ_{E} to strict inclusions E(\underline{M̂}) ⊂ F(\underline{M̂}) at ⋃_{i∈ℕ_{0}}\{z = ζ^{q̂^{i}}\}.

The stalks at z = ζ, or more precisely the tensor products with K[[z − ζ]] satisfy

E(\underline{M̂}) ⊗_{K⟨ \frac{z}{ζ} ,z^{−1}\}} K[[z − ζ]] = 𝔭_{D}  (6.1)
σ̂^{∗}δM̂ ∘ τ −1 ; M̂(F(\underline{M̂})) ⊗_{K⟨ \frac{z}{ζ} ,z^{−1}\}} K[[z − ζ]] = 𝔮_{D} ⊂ E(\underline{M̂}) ⊗_{K⟨ \frac{z}{ζ} ,z^{−1}\}} K((z − ζ)) = 𝔭_{D}[\frac{1}{z−ζ}] .

We thus can view F(\underline{M̂}) as the modification of E(\underline{M̂}) at ⋃_{i∈ℕ_{0}}\{z = ζ^{q̂^{i}}\} defined by the inclusion 𝔮_{D} ⊂ 𝔭_{D}[\frac{1}{z−ζ}], that is

σ̂^{∗}δM̂ ∘ τ −1 ; M̂(F(\underline{M̂})) = \underline{E}(\underline{M̂})[ℓ−1_{−}] ∩ ⋂_{j∈ℕ_{0}} τ_{E}^{j}(𝔮_{D} ⊗_{K[[z−ζ]], σ̂^{j}} K[[z − ζ^{q̂^{j}}]])

= \{ f ∈ E(\underline{M̂})[ℓ−1_{−}]: τ_{E}^{i}(σ̂^{i}(f)) ∈ 𝔮_{D} for all i ≤ 0 \} .

From this description it becomes clear that \underline{E}(\underline{M̂}) and \underline{F}(\underline{M̂}) only depend on the z-isocrystal with Hodge-Pink structure \underline{D} = (D, τ_{D}, 𝔮_{D}) = ℍ(\underline{M̂}) associated with \underline{M̂}. Namely, we can define

\underline{E}(\underline{D}) := (E(\underline{D}), τ_{E}) := (D, τ_{D}) ⊗_{k((z))} K⟨\frac{z}{ζ} , z^{−1}\}  and  (6.2)
F(\underline{D}) = (F(\underline{D}), τ_{F}) := \underline{E}(\underline{D})[ℓ−1_{−}] ∩ ⋂_{j∈ℕ_{0}} τ_{E}^{j}(𝔮_{D} ⊗_{K[[z−ζ]], σ̂^{j}} K[[z − ζ^{q̂^{j}}]]) .  (6.3)

Definition 6.8. We call (\underline{E}(\underline{D}), \underline{F}(\underline{D})) from (6.2) and (6.3) the pair of σ̂-bundles associated with \underline{D}.

The following proposition is analogous to [HJ16, Proposition 7.5] and was proved in [Har11, Lemma 2.4.5], where we used the notation P = \underline{E}(\underline{D}) and Q = \underline{F}(\underline{D}).

Proposition 6.9. The degree (defined in Theorem 6.7(a)) satisfies

t_{N}(\underline{D}) = − deg \underline{E}(\underline{D})  and  t_{H}(\underline{D}) = deg \underline{F}(\underline{D}) − deg \underline{E}(\underline{D}) .

The criterion for (weak) admissibility is now the following

Theorem 6.10. Let \underline{D} be a z-isocrystal with Hodge-Pink structure over R and let (\underline{E}(\underline{D}), \underline{F}(\underline{D})) be the associated pair of σ̂-bundles. Then

(a) \underline{D} is admissible if and only if over \overline{K} there is an isomorphism \underline{F}(\underline{D}) ≅ \underline{F}_{0,1}^{⊕ rk \underline{D}}.

(b) \underline{D} is weakly admissible if and only if deg \underline{F}(\underline{D}) = 0 ≥ deg \underline{F}(\underline{D}′) for every strict subobject \underline{D}′ ⊂ \underline{D}.

Proof. (b) directly follows from Proposition 6.9.

(a) was proved in [Har11, Theorem 2.4.7]. We explain some parts of the proof.

The implication “⟹” is easy to see. Choose a rigidified local shtuka \underline{M̂} := (M̂, τM̂, δM̂) over R with \underline{D} = ℍ(\underline{M̂}). We saw in Remark 4.3 that there is an isomorphism h: Ť_{ε}\underline{M̂} ⊗_{A_{ε}} K^{sep}⟨\frac{z}{ζ} ⟩ ^{∼} ⟶ \underline{M̂} ⊗_{R[[z]]} K^{sep}⟨\frac{z}{ζ} ⟩, which is Gal(K^{sep}/K)- and τ̂-equivariant. Since K^{sep}⟨\frac{z}{ζ} ⟩ ⊂ \overline{K}⟨\frac{z}{ζ} , z^{−1}\} we obtain an isomorphism of σ̂-bundles \underline{F}_{0,1}^{⊕ rk \underline{M}} ^{∼} ⟶ Ť_{ε}\underline{M̂} ⊗_{A_{ε}} \underline{F}_{0,1} ^{∼} ⟶ \underline{F}(\underline{M̂}) = \underline{F}(\underline{D}) over \overline{K}.

The converse implication “⟸” is more complicated, as one has to construct the local shtuka out of \underline{F}(\underline{D}) ≅ \underline{F}_{0,1}^{⊕ rk \underline{D}}; see [Har11, Theorem 2.4.7 and Proposition 2.4.9].

Corollary 6.11. A z-isocrystal with Hodge-Pink structure over R which is admissible is also weakly admissible.

Proof. This follows from the characterization of (weak) admissibility in Theorem 6.10 together with Theorem 6.7(c) applied to the σ̂-sub-bundle \underline{F}(\underline{D}′) ⊂ \underline{F}(\underline{D}).

Whether the converse of Corollary 6.11 holds, depends on the field K. Before we state cases in which it holds, we discuss the following

Example 6.12. For the Carlitz motive and its local shtuka \underline{M̂} = (R[[z]], z − ζ) from Example 6.3 we have \underline{E}(\underline{M̂}) = \underline{F}_{−1,1} and \underline{F}(\underline{M̂}) = (K⟨\frac{z}{ζ} , z^{−1}\}, z − ζ) which is isomorphic over \overline{K}⟨\frac{z}{ζ} , z^{−1}\} to \underline{F}_{0,1} via an isomorphism given by multiplication with ℓ_{+}.

Example 6.13. We show that “weakly admissible does not imply admissible” if the field K is algebraically closed (actually perfect suffices) and complete. Let D = k((z))^{⊕2}, τ_{D} = ( \begin{matrix}0 & 1 \\ z^{−3} & 0 \end{matrix} ). We search a Hodge-Pink lattice 𝔮_{D} ⊂ 𝔭_{D} with dim_{K} 𝔭_{D}/𝔮_{D} = 3 for which \underline{D} = (D, τ_{D}, 𝔮_{D}) is not admissible. This means the Hodge-Pink weights are (2, 1) or (3, 0). Note that any such \underline{D} is weakly admissible, because t_{N}(\underline{D}) = −3 = t_{H}(\underline{D}) and there are no proper subobjects \underline{D}′ ⊂ \underline{D}, as the z-isocrystal (D, τ_{D}) is simple.

We find 𝔮_{D} by imposing the condition that there is a σ̂-sub-bundle \underline{F}_{1,1} ⊂ \underline{F}(\underline{D}) ⊂ \underline{E}(\underline{D}) = \underline{F}_{3,2}. Then \underline{F}(\underline{D}) ≅ \underline{F̸}_{0,1}^{⊕2} because there is no non-zero homomorphism \underline{F}_{1,1} → \underline{F}_{0,1} by Theorem 6.7(b). Therefore Theorem 6.10 will imply that \underline{D} is not admissible. One easily computes

Hom(\underline{F}_{1,1}, \underline{E}(\underline{D})) = \{ f = ∑_{i∈ℤ} z^{i} · ( \begin{matrix}u^{q̂^{−2i}} \\ u^{q̂^{−2i−3}} \end{matrix} ) : u ∈ K, |u| < 1 \} ,  (6.4)

because f = ( \begin{matrix}f_{1} \\ f_{2} \end{matrix} ) must satisfy the equation τ_{D} · σ̂(f) = f · z^{−1}. This implies σ̂(f_{2}) = z^{−1}f_{1} and

z^{−3}σ̂(f_{1}) = z^{−1}f_{2}, whence f_{1} = z^{−1}σ̂^{2}(f_{1}). Writing f_{1} = ∑_{i∈ℤ} b_{i}z^{i}, we must have b_{i} = (b_{i+1})^{q̂^{2}}. So for b_{0} = u ∈ K we obtain f_{1} = ∑_{i∈ℤ} u^{q̂^{−2i}}z^{i}. Now the convergence condition for f ∈ K⟨\frac{z}{ζ} , z^{−1}\} implies |u| < 1.

Thus, if we take any u ∈ K with 0 < |u| < 1 and the corresponding f ≠ 0 from (6.4) and 𝔮_{D} := (z − ζ)^{3}𝔭_{D} + K[[z − ζ]] · f the homomorphism f: \underline{F}_{1,1} → \underline{E}(\underline{D}) factors through \underline{F}(\underline{D}) ⊂ \underline{E}(\underline{D}), as can be seen from (6.3). This implies that \underline{D} is not admissible.

On the positive side there is the following

Theorem 6.14. If the field K satisfies the following condition

(a) [GL11, Théorème 7.3]: K is discretely valued, or

(b) [Har11, Theorem 2.5.3]: the value group of K is finitely generated, or

(c) [Har11, Theorem 2.5.3]: the value group of K does not contain an element x ≠ 1 such that x^{q̂^{−n}} ∈ |K^{×}| for all n ∈ ℕ_{0}, or

(d) [Har11, Theorem 2.5.3]: the completion { \atop K·k^{alg}} of the compositum K ·k^{alg} inside K^{alg} does not contain an element a with 0 < |a| < 1 such that a^{q̂^{−n}} ∈ { \atop K·k^{alg}} for all n ∈ ℕ_{0},

then every weakly admissible z-isocrystal with Hodge-Pink structure over R is admissible.

Proof. Clearly the conditions satisfy (a)⟹(b)⟹(c)⟹(d), because the value groups of K and { \atop K·k^{alg}} coincide. Under condition (d) the theorem was proved in [Har11, Theorem 2.5.3]. In terms of Example 6.13 the idea is that in this case one cannot have f ∈ 𝔮_{D}.

Remark 6.15. This theorem is the analog of the Theorem of Colmez and Fontaine [CF00, Théorème A], which states that in the theory of p-adic Galois representations every weakly admissible filtered isocrystal \underline{D} = (D, τ_{D}, F^{•}D_{L}) over a discretely valued extension L of ℚ_{p} with perfect residue field, comes from a representation of Gal(L^{alg}/L) on a finite dimensional ℚ_{p}-vector space; compare Remark 5.21. In the special case where the Hodge filtration satisfies F^{0}D_{L} = D_{L} ⊃ F^{1}D_{L} ⊃ F^{2}D_{L} = (0) and all Newton slopes of the isocrystal (D, τ_{D}) lie in the interval [0, 1] it was proved by Kisin [Kis06, Theorem 0.3] and Breuil [Bre00, Theorem 1.4] that \underline{D} is (weakly) admissible if and only if \underline{D} comes from a p-divisible group. Also in this situation there are weakly admissible filtered isocrystals \underline{D} over ℂ_{p} which do not come from a p-divisible group; see [Har13, Example 6.7].

Remark 6.16. Note that the Theorem of Colmez and Fontaine [CF00, Théorème A] from the previous remark actually says that a continuous representation of Gal(L^{sep}/L) in a finite dimensional ℚ_{p}-vector space is crystalline if and only if it isomorphic to F^{0}(\underline{D}⊗_{L_{0}} B̃_{rig})^{Frob = id} for a weakly admissible filtered isocrystal \underline{D} = (D, τ_{D}, F^{•}D_{L}) over L. In the function field case, when K is discretely valued, we could therefore define the category of equal characteristic crystalline representations of Gal(K^{sep}/K) as the essential image of the functor

\underline{D} = (D, τ_{D}, 𝔮_{D}) ⟼ (\underline{D} ⊗_{k((z))} O_{\overline{K}}[[z, z^{−1}\}[ℓ^{−1}])^{τ̂= id} ∩ 𝔮_{D} ⊗_{K[[z−ζ]]} \overline{K}[[z − ζ]]  (6.5)

from weakly admissible z-isocrystals with Hodge-Pink structure \underline{D} to continuous representations of Gal(K^{sep}/K) in finite dimensional Q_{ε}-vector spaces. By Theorems 6.14, 5.20 and 4.15 and Proposition 5.12 this functor is fully faithful and this definition coincides with our Definition 4.16 above.

Remark 6.17. The functor (6.5) can also be described via the associated σ̂-bundles. Namely if \underline{D} is admissible, that is arises from a (rigidified) local shtuka \underline{M̂} over R with \underline{D} = ℍ(\underline{M̂}), then \underline{F}(\underline{D}) = \underline{F}(\underline{M̂}) = \underline{M̂} ⊗_{R[[z]]} K⟨\frac{z}{ζ} , z^{−1}\}, and hence

(\underline{F}(\underline{D}) ⊗_{K⟨ \frac{z}{ζ} ,z^{−1}\}} \overline{K}⟨\frac{z}{ζ} , z^{−1}\})τ̂ = (\underline{M̂} ⊗_{R[[z]]} \overline{K}⟨\frac{z}{ζ} , z^{−1}\})τ̂ = (Ť_{ε}\underline{M̂} ⊗_{A_{ε}} \underline{F}_{0,1})τ̂ = V̌_{ε}\underline{M̂} ;

see the proof of Theorem 6.10(a). In the analogous situation of p-adic Galois representations, mentioned in the previous remarks, there is a similar description due to Fargues and Fontaine [FF13]. Namely, consider the punctured open unit disc \{0 < |z| < 1\} over \overline{K}. Then every σ̂-bundle on {0 < |z| ≤ |ζ|} extends canonically to a σ̂-bundle over {0 < |z| < 1}; see [Har11, Proposition 1.4.1(b)]. The quotient {0 < |z| < 1}/σ̂^{ℤ} of {0 < |z| < 1} modulo the action of σ̂ exists in the category of Huber’s [Hub96] adic spaces and the category of vector bundles on {0 < |z| < 1}/σ̂^{ℤ} is equivalent to the category of σ̂-bundles over {0 < |z| < 1}. Under this equivalence the Q_{ε}-vector space of global sections of the vector bundle on {0 < |z| < 1}/σ̂^{ℤ} equals the τ̂-invariants of the associated σ̂-bundle.

In the theory of p-adic Galois representations the analog of the quotient {0 < |z| < 1}/σ̂^{ℤ} over ℚ_{p} is the adic curve X_{F,E}^{ad} constructed by Fargues [Far14]. Every weakly admissible filtered isocrystal \underline{D} = (D, τ_{D}, F^{•}D_{L}) over L gives rise to two vector bundles E^{∞} and E_{∞} on X_{F,E}^{ad} which agree outside the point corresponding to B_{dR}^{+}. These are analogous to the vector bundles on {0 < |z| < 1}/σ̂^{ℤ} corresponding to our σ̂-bundles \underline{E}(\underline{D}) and \underline{F}(\underline{D}), which agree outside the image of the point z = ζ on \{0 < |z| < 1\}/σ̂^{ℤ} corresponding to our analog \overline{K}[[z − ζ]] of B_{dR}^{+}. By [Far14, Théorèmes 4.43 and 3.5] the p-adic Galois representation associated with \underline{D} equals the global sections of E_{∞} over X_{F,E}^{ad}.

7 Torsion local shtukas and torsion Galois representations

As a preparation for the equi-characteristic deformation theory, which will be discussed in Section 8, we need a “torsion version” of equi-characteristic Fontaine’s theory – or rather, a suitable function field analog of finite flat group schemes of p-power order. Let R be an arbitrary valuation ring as in Notation 1.1. From Lemma 7.4 onwards we will assume that R is discretely valued. The analogy between p-divisible groups and local shtukas suggests the following

Definition 7.1. A torsion local shtuka (over R) is a pair \underline{M̂} = (M̂, τM̂) consisting of a finitely presented R[[z]]-module M̂ which is z-power torsion and free (necessarily of finite rank) as an R-module, and an isomorphism τM̂: σ̂^{∗}M̂[\frac{1}{z−ζ}] ^{∼} ⟶ M̂[\frac{1}{z−ζ}]. If τM̂(σ̂^{∗}M̂) ⊂ M̂ then \underline{M̂} is called effective, and if (z − ζ)^{d}M̂ ⊂ τM̂(σ̂^{∗}M̂) ⊂ M̂ then we say that \underline{M̂} is of height ⩽ d. If τM̂(σ̂^{∗}M̂) = M̂ then \underline{M̂} is called étale.

A morphism of torsion local shtukas f: (M̂, τM̂) → (M̂′, τM̂′) over R is a morphism of R[[z]]-modules f: M̂ → M̂′ which satisfies τM̂′ ∘ σ̂^{∗}f = f ∘ τM̂.

Remark 7.2. (a) Since ζ is M̂-regular and M̂ is z-power torsion, M̂ injects into M̂[\frac{1}{z−ζ}].

(b) Since M̂ is finite free over R the pair (M̂, τM̂) with its R[[z]]-module structure can be defined in terms of finitely many parameters over R. Therefore it is defined over R′[[z]] for some finitely generated and hence discretely valued A_{ε}-subalgebra R′ ⊂ R. So we may assume that R is discretely valued.

(c) The definition of torsion local shtukas over S ∈ Nilp_{A_{ε}} is not completely straightforward, because we have M̂[\frac{1}{z−ζ}] = 0 for any finitely generated O_{S}[[z]]-module M̂ killed by some power of z. Although it is possible to define torsion local shtukas over S ∈ Nilp_{A_{ε}} (cf. Remark 7.6), we will only consider torsion local shtukas over R.

The following example (together with Lemma 7.5) explains why we can regard torsion local shtukas over R as the function field analog of finite flat group schemes of p-power order.

Example 7.3. Let f: \underline{M̂}_{1} → \underline{M̂}_{0} be an isogeny of local shtukas over R; i.e., a morphism which is also a quasi-isogeny. Let M̂ := coker f. Since f is a morphism of local shtukas, τM̂_{0} induces τM̂ : σ̂^{∗}M̂[\frac{1}{z−ζ}] ^{∼} ⟶ M̂[\frac{1}{z−ζ}]. We claim that \underline{M̂} := (M̂, τM̂) is a torsion local shtuka over R, which is effective (respectively, of height ⩽ d) if \underline{M̂}_{0} is effective (respectively, if (z−ζ)^{d}M̂_{0} ⊂ τM̂_{0}(σ̂^{∗}M̂_{0}) ⊂ M̂_{0}).

Since R[[z]] ⊂ R[[z]][\frac{1}{z}] the map f is injective and there is an n such that z^{n}M̂_{0} ⊂ f(M̂_{1}). In particular M̂ is killed by z^{n}. Tensoring with the residue field k over R yields an exact sequence

0 ⟶ Tor_{1}^{R}(k, M̂) ⟶ M̂_{1} ⊗_{R[[z]]} k[[z]] ⟶^{f⊗ id_{k}} M̂_{0} ⊗_{R[[z]]} k[[z]] ⟶ M̂ ⊗_{R} k ⟶ 0 .

As M̂_{1} and M̂_{0} are free we have M̂_{i} ⊗_{R[[z]]} k[[z]] ≅ k[[z]]^{⊕ rk \underline{M̂}_{i}} for i = 0, 1 with rk \underline{M̂}_{0} = rk \underline{M̂}_{1}. Since M̂ ⊗_{R} k is killed by z^{n} the map f ⊗ id_{k} is injective by the elementary divisor theorem. So Tor_{1}^{R}(k, M̂) = (0) and since M̂ = coker(f: M̂_{1}/z^{n}M̂_{1} → M̂_{0}/z^{n}M̂_{0}) is finitely presented over R it is free of finite rank by Nakayama’s Lemma; e.g. [Eis95, Exercise 6.2].

From now on we assume that R is discretely valued. We will need the following

Lemma 7.4. Assume that R is discretely valued, and let M̂ be a finitely generated module over R[[z]]/(z^{n}) for some n. Then the following are equivalent

(a) M̂ is flat over R;

(b) The kernel of any surjective map M̂_{0} → → M̂, where M̂_{0} is a free R[[z]]-module of finite rank, is free over R[[z]].

Proof. Since R[[z]] is regular, whence Cohen-Macaulay, the theorem of Auslander and Buchsbaum [Eis95, Theorem 19.9] tells us that

proj.dim_{R[[z]]}M̂ + depth_{R[[z]]}M̂ = depth(R[[z]]) = dim(R[[z]]) = 2

for any finitely generated R[[z]]-module M̂. If M̂ is flat over R, then ζ is M̂-regular so the depth of M̂ is at least 1. Therefore, the projective dimension of M̂ is at most 1; in other words, M̂ admits a minimal projective resolution consisting of two terms, say F̂_{1} → F̂_{0}. If M̂_{0} → → M̂ is an epimorphism for a free R[[z]]-module M̂_{0} then M̂_{0} ≅ F̂_{0} ⊕ N̂_{0} for a free R[[z]]-module N̂_{0} and ker(M̂_{0} → → M̂) ≅ F̂_{1} ⊕ N̂_{0} by [Eis95, Theorem 20.2 and Lemma 20.1].

Conversely, assume that M̂ is not flat over R and killed by z^{n}. Then the 𝔪_{R}-torsion of M̂ is of finite positive length since it is non-zero and killed by some powers of z and ζ, so the depth of M̂ as a R[[z]]-module is zero. So the projective dimension of M̂ is 2; in other words, any projective resolution of M̂ consists of at least three terms.

The following lemma states that any torsion local shtuka over a discretely valued R arises from the construction in Example 7.3. This is analogous to Raynaud’s theorem [BBM82, 3.1.1], which states that any finite flat group scheme can be embedded, Zariski-locally on the base, in some abelian scheme.

Lemma 7.5. Let \underline{M̂} be a torsion local shtuka over R. Then there exist local shtukas \underline{M̂}_{0} and \underline{M̂}_{1}, and an isogeny f: \underline{M̂}_{1} → \underline{M̂}_{0} such that we have a short exact sequence

0 ⟶ \underline{M̂}_{1} ⟶^{f} \underline{M̂}_{0} ⟶ \underline{M̂} ⟶ 0 ,

where each arrow is τ̂-equivariant. If \underline{M̂} is effective of height ⩽ d (i.e., it satisfies (z − ζ)^{d}M̂ ⊂ τM̂(σ̂^{∗}M̂) ⊂ M̂), then we may take \underline{M̂}_{0} and \underline{M̂}_{1} to satisfy (z − ζ)^{d}M̂_{i} ⊂ τM̂_{i}(σ̂^{∗}M̂_{i}) ⊂ M̂_{i}.

Proof. The proof is analogous to the proof of [Kis06, 2.3.4]. By replacing τM̂ with (z − ζ)^{n}τM̂ for some n, we may assume that \underline{M̂} satisfies (z − ζ)^{d}M̂ ⊂ τM̂(σ̂^{∗}M̂) ⊂ M̂. We now choose a finite free R[[z]]/(z − ζ)^{d}-module L which surjects onto M̂/τM̂(σ̂^{∗}M̂), and a finite free R[[z]]-module M̂_{0} which fits into the following diagram

\xymatrix{0 \ar[r] & ker(M̂_{0} → → L) \ar[d] \ar[r] & M̂_{0} \ar@{->>}[d] \ar[r] & L \ar@{->>}[d] \ar[r] & 0 \\ 0 \ar[r] & _{_{σ̂^{∗}}M̂} \ar[r]^{τM̂} & M̂ \ar[r] & M̂/τM̂(σ̂^{∗}M̂) \ar[r] & 0} .

Furthermore, by enlarging M̂_{0} we may arrange so that the left vertical map is also surjective; this can be done by replacing M̂_{0} with M̂_{0} ⊕ M̂′, where M̂′ is a finite free R[[z]]-module which surjects onto the kernel of L → → M̂/τM̂(σ̂^{∗}M̂) and maps to zero in M̂.

Applying Lemma 7.4 to M̂_{0} → → L, ker(M̂_{0} → → L) is free over R[[z]]. We decompose ker(M̂_{0} → → L) = N̂ ⊕ N̂′ into two free modules so that N̂ ⊗_{R[[z]]} k = (σ^{∗}M̂) ⊗_{R[[z]]} k, and hence N̂ → → σ^{∗}M̂ is surjective by Nakayama. Then we may lift σ^{∗}M̂_{0} → → σ^{∗}M̂ to a map σ^{∗}M̂_{0} → N̂ which is also surjective by Nakayama and therefore a direct summand. We can now lift the latter map to an isomorphism σ^{∗}M̂_{0} ⟶^{∼} N̂ ⊕ N̂′ = ker(M̂_{0} → → L), as both modules are free of same finite rank over R[[z]]. Therefore, we obtain a map

τM̂_{0} : σ̂^{∗}M̂_{0} ^{∼} ⟶ ker(M̂_{0} → → L) ↪ M̂_{0}

lifting τM̂: σ̂^{∗}M̂ ^{∼} ⟶ τM̂(σ̂^{∗}M̂). Clearly, \underline{M̂}_{0} := (M̂_{0}, τM̂_{0}) is an effective local shtuka over R with (z − ζ)^{d}M̂_{0} ⊂ τM̂_{0}(σ̂^{∗}M̂_{0}) ⊂ M̂_{0}. Applying Lemma 7.4 to M̂_{0} → → M̂, M̂_{1} := ker(M̂_{0} → → M̂) is free over R[[z]] so \underline{M̂}_{1} := (M̂_{1}, τM̂_{1} := τM̂_{0}|_{M̂_{1}[ \frac{1}{z−ζ}]}) is an effective local shtuka with (z − ζ)^{d}M̂_{1} ⊂ τM̂_{1}(σ̂^{∗}M̂_{1}) ⊂

M̂_{1}. Let f: \underline{M̂}_{1} → \underline{M̂}_{0} be the morphism induced by the natural inclusion, then \underline{M̂} is the cokernel of f.

Remark 7.6. We have seen that the definition of torsion local shtukas over R does not straightforwardly apply to base schemes S ∈ Nilp_{A_{ε}}. On the other hand, Lemma 7.5 and the anti-equivalence between effective local shtukas and z-divisible local Anderson modules (Remark 3.2) suggest the following notion as the function filed analog of finite flat group schemes of p-power order.

We define a torsion local Anderson module over S ∈ Nilp_{A_{ε}} to be a finite locally free A_{ε}-module scheme H over S such that for some Zariski covering {S_{α}} of S there exist a z-divisible local Anderson module G_{α} over S_{α} and a monomorphism H_{S_{α}} ↪ G_{α} of fppf sheaves of A_{ε}-modules for each α.

Note that the anti-equivalence of categories Drq̂ in Remark 3.2 can be extended to bounded effective local shtukas over R by limit as in Remark 2.6. If M̂ is a torsion local shtuka over R obtained as the cokernel of an isogeny f: \underline{M̂}_{1} → \underline{M̂}_{0} of effective local shtukas, then we set

Drq̂(\underline{M̂}) := ker(Drq̂ f: Drq̂(\underline{M̂}_{0}) → Drq̂(\underline{M̂}_{1})).

Then Drq̂(\underline{M̂}) is independent of the choice of f up to isomorphism, and this induces an anti-equivalence of categories between effective torsion local shtukas and torsion local Anderson modules over R. Indeed, it is possible to define Drq̂ directly without choosing f: \underline{M̂}_{1} → \underline{M̂}_{0} as

Drq̂(\underline{M̂}) := Spec (Sym_{R} M̂)/(m^{⊗q̂} − τM̂(σ̂∗ ; M̂m): m ∈ M̂);

like in (3.1) in Section 3. And if H = Drq̂(\underline{M̂}) for some effective torsion local shtuka \underline{M̂} over R, then we can recover \underline{M̂} as Hom_{R-groups,𝔽_{ε}-lin}(H , 𝔾_{a,R}) by [Abr06, Theorem 2] or [HS15, Theorem 5.2].

To a torsion local shtuka over R, we associate a torsion Galois representation as follows.

Definition 7.7. Let \underline{M̂} be a torsion local shtuka over R. We define the (dual) Tate module of \underline{M̂} to be

Ť_{ε}\underline{M̂} := (M̂ ⊗_{R[[z]]} K^{sep}[[z]])τ̂ := \{m ∈ M̂ ⊗_{R[[z]]} K^{sep}[[z]]: τM̂(σ̂∗ ; M̂m) = m\}.

Proposition 7.8. For a torsion local shtuka \underline{M̂} over R, Ť_{ε}\underline{M̂} is a torsion A_{ε}-module of length equal to rk_{R}(M̂), which carries a discrete action of Gal(K^{sep}/K). Moreover the inclusion Ť_{ε}\underline{M̂} ⊂ M̂ ⊗_{R[[z]]} K^{sep}[[z]] defines a canonical isomorphism of K^{sep}[[z]]-modules

Ť_{ε}\underline{M̂} ⊗_{A_{ε}} K^{sep}[[z]] ^{∼} ⟶ \underline{M̂} ⊗_{R[[z]]} K^{sep}[[z]]  (7.1)

which is Gal(K^{sep}/K)- and τ̂-equivariant, where on the left module Gal(K^{sep}/K) acts on both factors and τ̂ is id ⊗σ̂, and on the right module Gal(K^{sep}/K) acts only on K^{sep}[[z]] and τ̂ is (τM̂ ∘ σ̂∗ ; M̂) ⊗ σ̂.

In particular one can recover \underline{M̂} ⊗_{R[[z]]} K[[z]] = (Ť_{ε}\underline{M̂} ⊗_{A_{ε}} K^{sep}[[z]])^{Gal(K^{sep}/K)} as the Galois invariants.

If we have a τ̂-equivariant short exact sequence 0 → \underline{M̂}′ → \underline{M̂} → \underline{M̂}′′ → 0 where each term is either a local shtuka or a torsion local shtuka over R, then we have a Gal(K^{sep}/K)-equivariant short exact sequence

0 ⟶ Ť_{ε}\underline{M̂}′ ⟶ Ť_{ε}\underline{M̂} ⟶ Ť_{ε}\underline{M̂}′′ ⟶ 0.

Proof. Let \underline{M̂} be a torsion local shtuka over R. To see that Ť_{ε}\underline{M̂} carries a natural discrete action of Gal(K^{sep}/K), note that M̂ is killed by some power of z, say by z^{n}, so the natural Gal(K^{sep}/K)-action on

M̂ ⊗_{R[[z]]} K^{sep}[[z]] = M̂ ⊗_{R[[z]]} K^{sep}[[z]]/(z^{n})

is discrete and commutes with τM̂.

Let us consider a τ̂-equivariant short exact sequence 0 → \underline{M̂}_{1} ⟶^{f} \underline{M̂}_{0} → \underline{M̂} → 0, where \underline{M̂} is a torsion local shtuka over R and \underline{M̂}_{0} and \underline{M̂}_{1} are local shtukas over R. We obtain a Gal(K^{sep}/K)- equivariant sequence

0 ⟶ Ť_{ε}\underline{M̂}_{1} ⟶^{Ť_{ε}f} Ť_{ε}\underline{M̂}_{0} ⟶ Ť_{ε}\underline{M̂} ⟶ 0 ,  (7.2)

which by definition is exact on the left and in the middle. To see that it is also exact on the right, we obtain from Proposition 4.2

coker(Ť_{ε}f) ⊗_{A_{ε}} K^{sep}[[z]] ^{∼} ⟶ M̂ ⊗_{R[[z]]} K^{sep}[[z]] ↩ Ť_{ε}\underline{M̂} ⊗_{A_{ε}} K^{sep}[[z]].

Combining these, it follows that the natural inclusion induces a Gal(K^{sep}/K)- and τ̂-equivariant isomorphism

Ť_{ε}\underline{M̂} ⊗_{A_{ε}} K^{sep}[[z]] ^{∼} ⟶ M̂ ⊗_{R[[z]]} K^{sep}[[z]].

Taking τ̂-invariants yields Ť_{ε}\underline{M̂} = coker(Ť_{ε}f) and so the sequence (7.2) is also exact on the right. The claim on the A_{ε}-length of Ť_{ε}\underline{M̂} follows from this isomorphism.

It now remains to show that Ť_{ε} on the category of torsion local shtukas over R is exact. Indeed, it is clearly left exact by definition, and the right exactness follows from the length consideration.

Recall from Remark 7.6 that to an effective torsion local shtuka \underline{M̂} over R we associated a torsion local Anderson module H := Drq̂(\underline{M̂}) over R. Hence we have a finite-length A_{ε}-module H(K^{sep}) equipped with a discrete Gal(K^{sep}/K)-action. We will now discuss the comparison between H(K^{sep}) and Ť_{ε}\underline{M̂} in a way that is analogous to and compatible with Proposition 4.7. For this, we need some preparations.

Lemma 7.9. Let H be the kernel of an isogeny G_{0} → G_{1} of z-divisible local Anderson modules over R. Then there exists a natural Galois-equivariant short exact sequence

0 ⟶ T_{ε}G_{0} ⟶ T_{ε}G_{1} ⟶ H(K^{sep}) ⟶ 0 .

The surjective map on the right can be defined as follows. Given ξ: Q_{ε}/A_{ε} → G_{1}(K^{sep}) (which is an element of T_{ε}G_{1}), choose a lift ξ̃: Q_{ε} → G_{0}(K^{sep}). Then we have ξ̃(1) ∈ H(K^{sep}), and it only depends on ξ (not on the choice of ξ̃). The surjective map on the right is given by ξ ↦ ξ̃(1).

Proof. For any n, consider the exact sequence 0 → A_{ε} → ε^{−n}A_{ε} → ε^{−n}A_{ε}/A_{ε} → 0, where the first two terms are projective over A_{ε}. By applying to it the long exact sequence for Hom_{A_{ε}}( • , H(K^{sep})), we obtain the following natural Galois-equivariant isomorphism

\xymatrix{Ext_{A_{ε}}^{1}(ε^{−n}A_{ε}/A_{ε}, H(K^{sep})) ≅ coker[Hom_{A_{ε}}(ε^{−n}A_{ε}, H(K^{sep})) \ar[rr] \ar[dr]|{ξ↦ξ(z^{−n})}^{≅} &  & Hom_{A_{ε}}(A_{ε}, H(K^{sep}))] \ar[dr]^{ξ↦ξ(1)}_{≅} \\  & ≅ coker[  H(K^{sep}) \ar[rr]^{[z^{n}]} &  & H(K^{sep})  ],}  (7.3)

where we let Gal(K^{sep}/K) act trivially on ε^{−n}A_{ε}/A_{ε}. (Indeed, this isomorphism is independent of the choice of z.) In particular, if we choose n so that z^{n} kills H, then we have a natural isomorphism Ext_{A_{ε}}^{1}(ε^{−n}A_{ε}/A_{ε}, H(K^{sep})) ⟶^{∼} H(K^{sep}), and the natural inclusion ε^{−n}A_{ε}/A_{ε} ↪ ε^{−n−1}A_{ε}/A_{ε} induces an isomorphism Ext_{A_{ε}}^{1}(ε^{−n−1}A_{ε}/A_{ε}, H(K^{sep})) ⟶^{∼} Ext_{A_{ε}}^{1}(ε^{−n}A_{ε}/A_{ε}, H(K^{sep})), which induces the identity map on H(K^{sep}).

Recall that we have the following Galois-equivariant short exact sequence of A_{ε}-modules

0 ⟶ H(K^{sep}) ⟶ G_{0}(K^{sep}) ⟶ G_{1}(K^{sep}) ⟶ 0 .

If z^{n} kills H, then we have the following natural Galois-equivariant exact sequence via the Hom-Ext long exact sequence (independent of the choice of z):

0 → Hom_{A_{ε}}(ε^{−n}A_{ε}/A_{ε}, H(K^{sep})) → Hom_{A_{ε}}(ε^{−n}A_{ε}/A_{ε}, G_{0}(K^{sep}))

⟶^{g} Hom_{A_{ε}}(ε^{−n}A_{ε}/A_{ε}, G_{1}(K^{sep})) → Ext_{A_{ε}}^{1}(ε^{−n}A_{ε}/A_{ε}, H(K^{sep})) → 0 . (7.4)

The right exactness follows from the fact that Ext_{A_{ε}}^{1}(ε^{−n}A_{ε}/A_{ε}, G_{i}(K^{sep})) = 0 by the analog of (7.3) for G_{i} using that [z^{n}]: G_{i}(K^{sep}) → G_{i}(K^{sep}) is surjective as G_{i} is z-divisible. Now by taking the projective limit of the sequence (7.4) as we increase n and observing that both the kernel and the cokernel of g satisfy the Mittag-Leffler condition, we obtain

0 → Hom_{A_{ε}}(Q_{ε}/A_{ε}, H(K^{sep})) → Hom_{A_{ε}}(Q_{ε}/A_{ε}, G_{0}(K^{sep}))

⟶^{g} Hom_{A_{ε}}(Q_{ε}/A_{ε}, G_{1}(K^{sep})) → H(K^{sep}) → 0 .

Since Hom_{A_{ε}}(Q_{ε}/A_{ε}, H(K^{sep})) = 0, we obtain the short exact sequence as in the statement.

It remains to explicitly describe the surjective map T_{ε}G_{1} → H(K^{sep}). Choosing n so that z^{n} kills H, the map, by construction, factors as follows:

T_{ε}G_{1} = Hom_{A_{ε}}(Q_{ε}/A_{ε}, G_{1}(K^{sep})) → Hom_{A_{ε}}(ε^{−n}A_{ε}/A_{ε}, G_{1}(K^{sep}))

→ Ext_{A_{ε}}^{1}(ε^{−n}A_{ε}/A_{ε}, H(K^{sep})) ⟶^{∼} H(K^{sep}),

where the first arrow is the restriction map and the second arrow is the connecting homomorphism in (7.4). Keeping in mind the description of Ext_{A_{ε}}^{1}(ε^{−n}A_{ε}/A_{ε}, H(K^{sep})) in (7.3), it is straightforward that the description given in the statement matches with the connecting homomorphism of (7.4) given by the snake’s lemma.

Remark 7.10. One can prove Lemma 7.9 by directly checking that the map T_{ε}G_{1} → H(K^{sep}) is well defined and gives the desired short exact sequence. We instead appealed to the isomorphism Ext_{A_{ε}}^{1}(ε^{−n}A_{ε}/A_{ε}, H(K^{sep})) ⟶^{∼} H(K^{sep}) for the sake of conceptual clarity.

In order to relate (7.2) with the exact sequence in Lemma 7.9, we need a little digression on (some variant of) Pontryagin duality. Consider a short exact sequence of A_{ε}-modules

0 ⟶ T_{0} ⟶ T_{1} ⟶ T ⟶ 0 ,

where T_{0} and T_{1} are finitely generated free A_{ε}-modules, and T is of finite length. Then by applying to it the long exact sequence for Hom_{A_{ε}}( • , Ω̂_{A_{ε}/𝔽_{ε}}^{1}), we get

0 ⟶ Hom_{A_{ε}}(T_{1}, Ω̂_{A_{ε}/𝔽_{ε}}^{1}) ⟶ Hom_{A_{ε}}(T_{0}, Ω̂_{A_{ε}/𝔽_{ε}}^{1}) ⟶ Ext_{A_{ε}}^{1}(T, Ω̂_{A_{ε}/𝔽_{ε}}^{1}) ⟶ 0 ,  (7.5)

since Hom_{A_{ε}}(T, Ω̂_{A_{ε}/𝔽_{ε}}^{1}) = 0 and Ext_{A_{ε}}^{1}(T_{i}, Ω̂_{A_{ε}/𝔽_{ε}}^{1}) = 0 for i = 0, 1 (as T_{i} are projective). Now, consider the following exact sequence

0 ⟶ Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⟶ Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⊗_{A_{ε}} Q_{ε} ⟶ Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⊗_{A_{ε}} Q_{ε}/A_{ε} ⟶ 0 .

Since Ω̂_{A_{ε}/𝔽_{ε}}^{1} is a rank-1 free module over A_{ε}, the last two terms are injective over A_{ε}. Using that T is torsion and Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⊗_{A_{ε}} Q_{ε} is torsion free, we get the following natural isomorphism

Hom_{A_{ε}}(T, Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⊗_{A_{ε}} Q_{ε}/A_{ε}) ^{∼} ⟶ Ext_{A_{ε}}^{1}(T, Ω̂_{A_{ε}/𝔽_{ε}}^{1}).

Combining this isomorphism with the exact sequence (7.5), we obtain the following exact sequence:

0 ⟶ Hom_{A_{ε}}(T_{1}, Ω̂_{A_{ε}/𝔽_{ε}}^{1}) ⟶ Hom_{A_{ε}}(T_{0}, Ω̂_{A_{ε}/𝔽_{ε}}^{1}) ⟶ Hom_{A_{ε}}(T, Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⊗_{A_{ε}} Q_{ε}/A_{ε}) ⟶ 0 . (7.6)

We can make explicit the surjective map in (7.6) in a similar way as Lemma 7.9. Namely, we view

ξ: T_{0} → Ω̂_{A_{ε}/𝔽_{ε}}^{1} as a map T_{1} → Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⊗_{A_{ε}} Q_{ε} (using the isomorphism T_{0} ⊗_{A_{ε}} Q_{ε} ⟶^{∼} T_{1} ⊗_{A_{ε}} Q_{ε}),

and take ξ̄: T = T_{1}/T_{0} → Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⊗_{A_{ε}} Q_{ε}/A_{ε} to be its reduction. Then the surjective map in (7.6) is defined by ξ ↦ ξ̄.

Applying (7.6) to (7.2) and the exact sequence in Lemma 7.9, we obtain the following commutative diagrams with exact columns:

\xymatrix{0 \ar[d] & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ T_{ε}G_{0} \ar[r]^{≅} \ar[d] & Hom_{A_{ε}}(Ť_{ε}\underline{M̂}_{0}, Ω̂_{A_{ε}/𝔽_{ε}}^{1}) \ar[d] & _{Ť_{ε}\underline{M̂}_{1}} \ar[r]^{≅} \ar[d] & Hom_{A_{ε}}(T_{ε}G_{1}, Ω̂_{A_{ε}/𝔽_{ε}}^{1}) \ar[d] \\ T_{ε}G_{1} \ar[r]^{≅} \ar[d] & Hom_{A_{ε}}(Ť_{ε}\underline{M̂}_{1}, Ω̂_{A_{ε}/𝔽_{ε}}^{1}) \ar[d] & _{Ť_{ε}\underline{M̂}_{0}} \ar[r]^{≅} \ar[d] & Hom_{A_{ε}}(T_{ε}G_{0}, Ω̂_{A_{ε}/𝔽_{ε}}^{1}) \ar[d] \\ H(K^{sep}) \ar[r]^{≅}_{(∗′)} \ar[d] & Hom_{A_{ε}}(Ť_{ε}\underline{M̂}, Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⊗_{A_{ε}} Q_{ε}/A_{ε}); \ar[d] & _{Ť_{ε}\underline{M̂}} \ar[r]^{≅}_{(∗)} \ar[d] & Hom_{A_{ε}}(H(K^{sep}), Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⊗_{A_{ε}} Q_{ε}/A_{ε}); \ar[d] \\ 0 & 0 & 0 & 0}

(7.7)

where the horizontal isomorphisms on the first two rows are defined in Proposition 4.7.

We now give an intrinsic description of the isomorphism (∗) and (∗′) in the above diagram, not depending on the choice of the isogeny \underline{M̂}_{1} → \underline{M̂}_{0}. Assume that H is killed by z^{n}. Then we can identify the Galois-equivariant A/ε^{n}-isomorphism

Hom_{𝔽_{ε}}(A/ε^{n}, H(K^{sep})) ^{∼} ⟶ H(K^{sep})),

defined by ξ̄ ↦ ξ̄(1). This isomorphism is compatible with increasing n with respect to the natural projections A/ε^{n+n′} → → A/ε^{n}.

Let us consider the following A/ε^{n}-linear isomorphism

Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⊗_{A_{ε}} ε^{−n}A_{ε}/A_{ε} ^{∼} ⟶ Hom_{𝔽_{ε}}(A/ε^{n}, 𝔽_{ε}), ω ↦ (a ↦ Res_{ε}(ãω̃)),

where ω̃ ∈ ε^{−n}Ω̂_{A_{ε}/𝔽_{ε}}^{1} and ã ∈ A_{ε} are some lifts of ω and a, respectively. Note that the residue Res_{ε}(ãω̃) modulo ε^{n} only depends on a and ω. If we choose a uniformizing parameter z ∈ A_{ε}, then this map can be identified with the reduction of the isomorphism Ω̂_{A_{ε}/𝔽_{ε}}^{1} ^{∼} ⟶ Hom_{𝔽_{ε}}(Q_{ε}/A_{ε}, 𝔽_{ε}) (introduced above Proposition 4.7) via the isomorphism A/ε^{n} ^{∼} ⟶ ε^{−n}A_{ε}/A_{ε} sending 1 to \frac{1}{z^{n}}, and we can write down the inverse map as λ ↦ dz ⊗ (∑_{i=1}^{n} λ(z^{i−1})z^{−i}).

Let us now state the following comparison result analogous to Proposition 4.7:

Proposition 7.11. We use the notation as above, and view \underline{M̂} as Hom_{R-groups,𝔽_{ε}-lin}(H , 𝔾_{a,R}); cf. Remark 7.6. Then the following Galois-equivariant pairing of A_{ε}-modules

H(K^{sep}) × Ť_{ε}\underline{M̂} ⟶ Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⊗_{A_{ε}} Q_{ε}/A_{ε} ,
(f, m) ↦ m ∘ f ∈ Hom_{𝔽_{ε}}(A/ε^{n}, 𝔽_{ε}) ≅ Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⊗_{A_{ε}} ε^{−n}A_{ε}/A_{ε} ⊂ Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⊗_{A_{ε}} Q_{ε}/A_{ε};

is perfect; in other words, it induces the following Galois-equivariant A_{ε}-linear isomorphisms:

H(K^{sep}) ^{∼} ⟶ Hom_{A_{ε}}(Ť_{ε}\underline{M̂}, Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⊗_{A_{ε}} Q_{ε}/A_{ε}) and Ť_{ε}\underline{M̂} ^{∼} ⟶ Hom_{A_{ε}}(H(K^{sep}), Ω̂_{A_{ε}/𝔽_{ε}}^{1} ⊗_{A_{ε}} Q_{ε}/A_{ε}).

Furthermore, if we choose an isogeny \underline{M̂}_{1} → \underline{M̂}_{0} whose cokernel is \underline{M̂}, then the above isomorphism coincides with the isomorphisms (∗) and (∗′) in the diagram (7.7).

If we choose a uniformizing parameter z ∈ A_{ε} and identify Ω̂_{A_{ε}/𝔽_{ε}}^{1} ≅ A_{ε}dz, then the above pairing non-canonically identifies H(K^{sep}) with the Pontryagin dual of Ť_{ε}\underline{M̂} as a torsion Gal(K^{sep}/K)- representation.

Proof. By Lemma 7.5 we can find an isogeny of effective local shtukas \underline{M̂}_{1} → \underline{M̂}_{0} whose cokernel is the given effective torsion local shtuka \underline{M̂}. Therefore, it suffices to show that the pairing in the statement induces the isomorphisms (∗) and (∗′) in the diagram (7.7). Now, since the surjective vertical arrows in the diagram (7.7) are made completely explicit (cf. Lemma 7.9 and the discussion below (7.6)), we can directly check the commutativity of (7.7) with horizontal maps given by the pairings in Propositions 4.7 and 7.11.

Remark 7.12. As suggested in the proof of Proposition 7.11, one can easily show that Propositions 4.7 and 7.11 are equivalent. Indeed, it is possible to deduce Proposition 4.7 by directly proving Proposition 7.11 and apply it to H = G[z^{n}] for a z-divisible local Anderson module G over R.

Definition 7.13. Let Rep_{A_{ε}}^{tors} Gal(K^{sep}/K) denote the category of finite-length torsion A_{ε}-modules equipped with a discrete action of Gal(K^{sep}/K). Then for T ∈ Rep_{A_{ε}}^{tors} Gal(K^{sep}/K), a torsion local shtuka model of T is a torsion local shtuka \underline{M̂} over R equipped with a Gal(K^{sep}/K)-equivariant isomorphism T ≅ Ť_{ε}\underline{M̂}.

Contrary to the case of local shtukas over R, the functor Ť_{ε} from the category of torsion local shtukas over R to Rep_{A_{ε}}^{tors} Gal(K^{sep}/K) is not fully faithful. In particular, it is possible to have more than one torsion local shtuka model for T ∈ Rep_{A_{ε}}^{tors} Gal(K^{sep}/K). For example, for any torsion local shtuka \underline{M̂} over R, the Galois action on Ť_{ε}\underline{M̂} trivializes after replacing R with some finite extension, so as R gets more ramified we obtain more torsion local shtuka models of the trivial torsion Galois module.

Example 7.14. In this example, we find more torsion local shtuka models of A_{ε}/(z^{n}) with trivial Gal(K^{sep}/K)-action. We consider the following torsion local shtuka for any d ∈ ℤ:

\underline{M̂}_{n}^{d} := (R[[z]]/(z^{n}), (z − ζ)^{d}).

Using the notation from Example 4.8, it follows that Ť_{ε}\underline{M̂}_{n}^{d} ⊂ K^{sep}[[z]]/(z^{n}) is an A_{ε}/(z^{n})-lattice generated by (ℓ_{+})^{−d} mod z^{n} with the Galois action given by the character (χ_{ε})^{−d} mod z^{n}. Moreover, if e ∈ ℕ is such that q̂^{e} ≥ n and d = q̂^{e}(q̂ − 1)d′ then we have (ℓ_{+})^{−d} mod z^{n} = ℓ_{0}^{−d} = (−ζ)^{−q̂^{e}d′} ∈ K[[z]]/(z^{n}); in other words, (χ_{ε})^{d} ≡ 1 mod z^{n}. Therefore, in this case \underline{M̂}_{n}^{d} defines a torsion local shtuka model of the trivial Galois module A_{ε}/(z^{n}) for any d ∈ ℤ. Note that all these are pairwise non-isomorphic because (ℓ_{+})^{−d} mod z^{n} is not a unit in R[[z]]/(z^{n}).

Proposition 7.15. The full subcategory of Rep_{A_{ε}}^{tors} Gal(K^{sep}/K), consisting of torsion Galois representations which admit an effective torsion local shtuka model with height ⩽ d, is stable under subquotients, direct products, and twisted duality T ↦ T^{∨}(d) := Hom_{A_{ε}}(T, Q_{ε}/A_{ε}) ⊗ (χ_{ε})^{d}, where χ_{ε} is defined in Example 4.8.

The proposition, especially the assertion about subquotients, is not obvious since Ť_{ε} may not be fully faithful. Recall that the analogous assertion for torsion representations of the Galois group of a p-adic field which admit finite flat group scheme models over the valuation ring can be obtained by working with the scheme-theoretic closure of the generic fiber. The following lemma provides an analog of the scheme-theoretic closure, which is needed for the proof of Proposition 7.15.

Lemma 7.16. Let \underline{M̂} be a torsion local shtuka over R, and consider a Gal(K^{sep}/K)-stable quotient Ť_{ε}\underline{M̂} → → T′. We define M̂′ to be the image of the following map:

M̂ ↪ M̂ ⊗_{R[[z]]} K^{sep}[[z]] ≅ Ť_{ε}\underline{M̂} ⊗_{A_{ε}} K^{sep}[[z]] → → T′ ⊗_{A_{ε}} K^{sep}[[z]].

Then, τM̂ induces an isomorphism τM̂′ : σ̂^{∗}M̂′[\frac{1}{z−ζ}] ^{∼} ⟶ M̂′[\frac{1}{z−ζ}], and \underline{M̂}′ := (M̂′, τM̂′) is a torsion local shtuka over R. If \underline{M̂} is effective (respectively, effective with height ⩽ d), then so is \underline{M̂}′. Furthermore, \underline{M̂}′ is the unique quotient of \underline{M̂} such that T′ = Ť_{ε}\underline{M̂}′ as a quotient of Ť_{ε}\underline{M̂}.

Proof. Note that M̂′ is flat over the discrete valuation ring R and finitely presented over the noetherian ring R[[z]] because it is a submodule of T′ ⊗_{A_{ε}} K^{sep}[[z]] and a quotient of M̂. The isomorphism τM̂′ is obtained from the diagram

[figure: σ̂^{∗}M̂[\frac{1}{z−ζ}]; τM̂; ≅; σ̂^{∗}M̂′[\frac{1}{z−ζ}]; τM̂′; ≅; T′ ⊗_{A_{ε}} K^{sep}[[z]]; M̂[\frac{1}{z−ζ}]; M̂′[\frac{1}{z−ζ}]; T′ ⊗_{A_{ε}} K^{sep}[[z]]]

using that σ̂: R → R is flat. The diagram also shows that \underline{M̂}′ is effective (respectively, effective with height ⩽ d) if \underline{M̂} is. Finally, the uniqueness of \underline{M̂}′ follows from Proposition 7.8.

Remark 7.17. In the setting of Lemma 7.16, assume furthermore that \underline{M̂} is effective. Then for a Galois-stable quotient T′ of Ť_{ε}\underline{M̂}, the associated \underline{M̂}′ is effective. So there is a unique A_{ε}-submodule scheme H′_{K} of the generic fiber of H := Drq̂ \underline{M̂}′ such that H′_{K}(K^{sep}) is the Pontryagin dual of T′. Then Drq̂ \underline{M̂}′ is the scheme-theoretic closure of H′_{K} in H, which can be seen from the uniqueness of the scheme-theoretic closure.

Proof of Proposition 7.15. The claim on direct products is clear as Ť_{ε} commutes with direct products.

Let \underline{M̂} be an effective torsion local shtuka over R with height ⩽ d, and set T := Ť_{ε}\underline{M̂}. Lemma 7.16 shows that any Galois-stable quotient of T admits an effective torsion local shtuka model with height ⩽ d. Let T′ ⊂ T be a Galois-stable A_{ε}-submodule. Then by Lemma 7.16, there exists a quotient \underline{M̂}′′ of \underline{M̂} corresponding to T/T′. Then Proposition 7.8 implies that M̂′ := ker(M̂ → → M̂′′) can naturally be viewed as a torsion local shtuka model of T′, which is effective and of height ⩽ d.

We construct a torsion local shtuka model of T^{∨}(d) which is effective and of height ⩽ d. We first define \underline{M̂}^{∨} := (M̂^{∨}, τM̂^{∨}) as follows:

M̂^{∨} := Hom_{R[[z]]}(M̂, R[[z]][\frac{1}{z}]/R[[z]]) ,  (7.8)
τM̂^{∨} := (τ −1 ; M̂)^{∨}: σ̂^{∗}M̂^{∨}[\frac{1}{z−ζ}] ≅ (σ̂^{∗}M̂)[\frac{1}{z−ζ}]^{∨} ^{∼} ⟶ M̂^{∨}[\frac{1}{z−ζ}] .

Note that if z^{n}M̂ = (0) then

[figure: Hom_{R}(M, R) ≅ Hom_{R}(M, R) ⊗_{R[[z]]/(z^{n})} \frac{1}{z^{n}} R[[z]]/R[[z]]; ∼; M̂^{∨}; h ⊗ a; (m ↦ a · h(m)) .]

So as an R-module it is flat and finitely generated over R. By [Eis95, Proposition 2.10] we have

M̂^{∨} ⊗_{R[[z]]} K^{sep}[[z]] ≅ Hom_{K^{sep}[[z]]}(M̂ ⊗_{R[[z]]} K^{sep} , K^{sep}[[z]][\frac{1}{z}]/K^{sep}[[z]])
≅ Hom_{A_{ε}}(Ť_{ε}M̂, Q_{ε}/A_{ε}) ⊗_{A_{ε}} K^{sep}[[z]] ,

and hence Ť_{ε}M̂^{∨} ≅ Hom_{A_{ε}}(Ť_{ε}M̂, Q_{ε}/A_{ε}). Note that \underline{M̂}^{∨} is not necessarily an effective torsion local shtuka, but we can slightly modify to get

\underline{M̂}^{∨,d} := (M̂^{∨}, (z − ζ)^{d}τM̂^{∨}),  (7.9)

which is effective and of height ⩽ d if the same holds for \underline{M̂}. To see T^{∨}(d) ≅ Ť_{ε}\underline{M̂}^{∨,d}, note that (ℓ_{+})^{d}Ť_{ε}\underline{M̂}^{∨} = Ť_{ε}\underline{M̂}^{∨,d} inside M̂^{∨} ⊗_{R[[z]]} K^{sep}[[z]], using the notation from Example 4.8.

Just as for finite flat group scheme models of torsion Galois representations of a p-adic field, there exists a natural notion of partial ordering on (equivalence classes of) torsion local shtuka models.

Definition 7.18. We fix T ∈ Rep_{A_{ε}}^{tors} Gal(K^{sep}/K), and consider torsion local shtuka models \underline{M̂} and \underline{M̂}′ of T. We write \underline{M̂} ≾ \underline{M̂}′ if there exists a (necessarily unique) map \underline{M̂} → \underline{M̂}′ which respects the identification Ť_{ε}\underline{M̂} ≅ T ≅ Ť_{ε}\underline{M̂}′. We say that \underline{M̂} and \underline{M̂}′ are equivalent if \underline{M̂} ≾ \underline{M̂}′ and \underline{M̂} ≿ \underline{M̂}′; or equivalently, if there exists a necessarily unique isomorphism \underline{M̂} ≅ \underline{M̂}′ which respects the identification Ť_{ε}\underline{M̂} ≅ T ≅ Ť_{ε}\underline{M̂}′.

Lemma 7.19. For T ∈ Rep_{A_{ε}}^{tors} Gal(K^{sep}/K), we have the following:

(a) ≾ defines a partial ordering on the set of equivalence classes of torsion local shtuka models of T.

(b) In the set of equivalence classes of effective torsion local shtuka models of T with height ⩽ d, there exist unique maximal and minimal elements with respect to the partial ordering ≾. Furthermore, the formation of maximal and minimal effective torsion local shtuka models with height ⩽ d is functorial in the sense that any Galois-equivariant morphism of torsion Galois representations comes from a unique map of their maximal (respectively, minimal) effective torsion local shtuka models with height ⩽ d.

(c) The set of equivalence classes of effective torsion local shtuka models of T with height ⩽ d is finite.

Proof. For any torsion local shtuka models \underline{M̂} and \underline{M̂}′ of T, the isomorphism Ť_{ε}\underline{M̂} ≅ T ≅ Ť_{ε}\underline{M̂}′ enables us to identify the equivalence classes of \underline{M̂} and \underline{M̂}′ as R[[z]]-submodules of T ⊗_{R[[z]]} K^{sep}[[z]] which are stable under 1 ⊗ σ̂. One can easily check that \underline{M̂} + \underline{M̂}′ and \underline{M̂} ∩ \underline{M̂}′ can naturally be viewed as torsion local shtuka models of T, where τ̂ is obtained by the restriction of 1⊗σ̂. Namely, \underline{M̂} +\underline{M̂}′ is the torsion local shtuka from Lemma 7.16 associated with the quotient T_{ε}\underline{M̂} = T_{ε}\underline{M̂}′ of T_{ε}(\underline{M̂} ⊕ \underline{M̂}′), and \underline{M̂} ∩\underline{M̂}′ is the kernel of \underline{M̂} ⊕\underline{M̂}′ → → \underline{M̂} +\underline{M̂}′. This shows that ≾ is a partial ordering, i.e., part (a).

We next show that the set of equivalence classes of effective torsion local shtuka models of T admits a unique maximal element (if the set is non-empty). For an effective torsion local shtuka model \underline{M̂} of T we consider

Drq̂(\underline{M̂}) := Spec (Sym_{R} M̂)/(m^{⊗q̂} − τM̂(σ̂∗ ; M̂m): m ∈ M̂);

as in Remark 7.6. If \underline{M̂}′ is another effective torsion local shtuka model of T with \underline{M̂} ≾ \underline{M̂}′, then there exists a finite morphism Drq̂(\underline{M̂}′) → Drq̂(\underline{M̂}) which is an isomorphism over Spec K. On the other hand, the normalization X̃ of Drq̂(\underline{M̂}) in its generic fiber is finite over Drq̂(\underline{M̂}) as the generic fiber is étale over K by the Jacobi criterion or because \underline{M̂} is étale over K. Indeed, the trace pairing

(O_{Drq̂(\underline{M̂})} ⊗_{R} K) × (O_{Drq̂(\underline{M̂})} ⊗_{R} K) → K

is perfect by étaleness [EGA, IV_{4}, Proposition 18.2.3(c)], so the dual of O_{Drq̂(\underline{M̂})} is finite over O_{Drq̂(\underline{M̂})} and contains the normalisation of O_{Drq̂(\underline{M̂})}. Thus the normalization is finite. This shows that the set of equivalence class of effective torsion local shtuka models of T is bounded above with respect to ≾, because every such is contained in the effective torsion local shtuka corresponding to X̃. On the other hand, if \underline{M̂} and \underline{M̂}′ are effective torsion local shtuka models of T, then so is the torsion local shtuka model \underline{M̂} + \underline{M̂}′. This shows the uniqueness of the maximal element.

Let us show there exists a unique minimal effective torsion local shtuka model with height ⩽ d up to equivalence. Let \underline{M̂}′ be a torsion local shtuka model of T^{∨}(d) maximal among those effective and of height ⩽ d. Since \underline{M̂}′ ↦ (\underline{M̂}′)^{∨,d} reverses the partial ordering by (7.8), it follows that (\underline{M̂}′)^{∨,d} is minimal among effective torsion local shtuka models of T with height ⩽ d.

For the functoriality claim on maximal and minimal objects, we may assume that \underline{M̂} and \underline{M̂}′ are maximal effective torsion local shtuka models of T = Ť_{ε}\underline{M̂} and T′ = Ť_{ε}\underline{M̂}′, respectively. (The case of minimal objects is reduced to this since the functor (7.9) switches the maximal and the minimal objects.) Let f: T → T′ be a Gal(K^{sep}/K)-equivariant map. If f is surjective, then the image of M̂ under M̂ ↪ T ⊗_{A_{ε}} K^{sep}[[z]] ⟶^{f⊗1} T′ ⊗_{A_{ε}} K^{sep}[[z]] is an effective local torsion shtuka model of T′ with height ⩽ d, so it is contained in M̂′ by the maximality of \underline{M̂}′. If f is not surjective, then consider a surjective map T ⊕T′ ⟶^{f+ id} T′. Let \underline{M̂}′′ denote a torsion local shtuka model of T ⊕T′ maximal among those effective and of height ⩽ d, then f + id induces a map \underline{M̂}′′ → \underline{M̂}′. By maximality of \underline{M̂}′′, there is a map \underline{M̂} ⊕ \underline{M̂}′ → \underline{M̂}′′, so we obtain

\underline{M̂} ⟶^{( id,0)} \underline{M̂} ⊕ \underline{M̂}′ ⟶ \underline{M̂}′′ ⟶ \underline{M̂}′,

which induces f: T → T′. This proves part (b).

Let \underline{M̂}^{+} and \underline{M̂}^{−} respectively denote the maximal and minimal effective torsion local shtuka models of T with height ⩽ d. Since R is discretely valued, the quotient M̂^{+}/M̂^{−} is of finite length over R[[z]]. Since any equivalence class of effective torsion local shtuka models of T with height ⩽ d gives rise to a unique R[[z]]-submodule of M̂^{+}/M̂^{−}, we obtain the desired finiteness claim (i.e., part (c)).

Lemma 7.20. Let T be a finitely generated free A_{ε}-module equipped with a continuous action of Gal(K^{sep}/K). Then the following are equivalent:

(a) There exists an effective local shtuka \underline{M̂} with (z − ζ)^{d}M̂ ⊂ τM̂(σ̂^{∗}M̂) ⊂ M̂ such that T ≅ Ť_{ε}\underline{M̂}.

(b) For each positive integer n, there exists an effective torsion local shtuka \underline{M̂}_{n} with height ⩽ d (i.e., satisfying (z − ζ)^{d}M̂_{n} ⊂ τM̂_{n}(σ̂^{∗}M̂_{n}) ⊂ M̂_{n}) such that T/(z^{n}) ≅ Ť(\underline{M̂}_{n}).

Proof. It is clear that (a) implies (b), so let us assume (b). By choosing each \underline{M̂}_{n} to be maximal among effective torsion local shtuka models with height ⩽ d, we may assume that for each n there exists a (not necessarily surjective) morphism \underline{M̂}_{n+1} → \underline{M̂}_{n} that induces the natural projection T/(z^{n+1}) → → T/(z^{n}) by Lemma 7.19(b). Set M̂′ := \underleftarrow{lim}_{n} M̂_{n}, equipped with τM̂′ : σ^{∗}M̂′[\frac{1}{z−ζ}] ^{∼} ⟶ M̂′[\frac{1}{z−ζ}] obtained as the limit of {τM̂_{n}}. Note that the projective system (M̂_{n}) satisfies the Mittag-Leffler condition, because by Lemma 7.19(c) for each n the set of images M̂_{n′} → M̂_{n} is finite. Therefore M̂′/zM̂′ is a quotient of M̂_{n} for n ≫ 1, and since M̂′ is z-adically separated and complete, it is finitely generated over R[[z]].

We next consider the following injective map

M̂′ = \underleftarrow{lim}_{n} M̂_{n} → \underleftarrow{lim}_{n}(M̂_{n} ⊗_{R[[z]]} K^{sep}[[z]]) ≅ \underleftarrow{lim}_{n}(T/z^{n}T ⊗_{R[[z]]} K^{sep}[[z]]) ≅ T ⊗_{A_{ε}} K^{sep}[[z]],

where the non-trivial isomorphism is from Proposition 7.8. (The injectivity follows from the left exactness of the projective limit.) In particular, M̂′ is torsion-free. By K^{sep}[[z]]-linearly extending it, we obtain a map

M̂′ ⊗_{R[[z]]} K^{sep}[[z]] → T ⊗_{A_{ε}} K^{sep}[[z]],  (7.10)

which is injective by torsion-freeness of M̂′. We claim that this map is an isomorphism. Indeed, the surjectivity follows since the image of M̂′ in M̂_{n} ⊗_{R[[z]]} K^{sep}[[z]] ≅ T/z^{n}T ⊗_{A_{ε}} K^{sep}[[z]] coincides with the image of M̂_{n′} for n′ ≫ n by Lemma 7.19(c).

One can now apply Lemma 4.18 to (M̂′, τM̂′) to obtain a local shtuka \underline{M̂}′′ := (M̂′′, τM̂′′) with M̂′[\frac{1}{z−ζ}] = M̂′′[\frac{1}{z−ζ}]. So we have (z − ζ)^{d}M̂′′ ⊂ τM̂′′(σ̂^{∗}M̂′′) ⊂ M̂′′. Furthermore, the isomorphism (7.10) implies that V̌_{ε}\underline{M̂}′′ ≅ T [\frac{1}{z}]. Then by Proposition 4.17, there exists a local shtuka \underline{M̂} isogenous to \underline{M̂}′′ with T = Ť_{ε}\underline{M̂}. Finally, the property (z − ζ)^{d}M̂ ⊂ τM̂(σ̂^{∗}M̂) ⊂ M̂ could be checked up to isogeny, which concludes the proof.

Remark 7.21. Using the same notation in the proof of Lemma 7.20, one can also show that \underline{M̂} = \underline{M̂}′′ by using the isomorphism (7.10) and suitably adapting the exact sequence (4.5).

We finish the section by discussing torsion local shtukas with coefficients. Let B be an A_{ε}-algebra with #B < ∞, and set R[[z]]_{B} := R[[z]] ⊗_{A_{ε}} B. We define σ̂ : R[[z]]_{B} → R[[z]]_{B} by B-linearly extending σ̂ : R[[z]] → R[[z]].

Definition 7.22. A torsion local B-shtuka over R is a pair \underline{M̂}_{B} = (M̂_{B}, τM̂_{B}) consisting of a finitely generated free R[[z]]_{B}-module M̂_{B}, and an R[[z]]_{B}-linear isomorphism τM̂_{B}: σ̂^{∗}M̂_{B}[\frac{1}{z−ζ}] ^{∼} ⟶ M̂_{B}[\frac{1}{z−ζ}]. We take the obvious notion of morphisms.

If B′ is a B-algebra with #B′ < ∞ and \underline{M̂}_{B} is a torsion local B-shtuka, then we define a torsion local B′-shtuka \underline{M̂}_{B} ⊗_{B} B′ := (M̂_{B} ⊗_{B} B′, τM̂_{B} ⊗ id_{B′}). We call \underline{M̂}_{B} ⊗_{B} B′ the scalar extension of \underline{M̂}_{B}

We may view a torsion local B-shtuka \underline{M̂}_{B} as a torsion local shtuka by forgetting the B-action. We say that \underline{M̂}_{B} is effective (respectively, effective with height ⩽ d; respectively, étale) if it is so as a torsion local shtuka. These properties are stable under scalar extensions.

Lemma 7.23. (a) For any torsion local B-shtuka \underline{M̂}_{B}, Ť_{ε}\underline{M̂}_{B} is a finitely generated free B-module of rank equal to rk_{R[[z]]_{B}} M̂_{B}, where the B-action on Ť_{ε}\underline{M̂}_{B} is induced from the B-action on \underline{M̂}_{B}.

(b) For any B-algebra B′ with #B′ < ∞, we have a natural Gal(K^{sep}/K)-equivariant isomorphism Ť_{ε}(\underline{M̂}_{B} ⊗_{B} B′) ≅ (Ť_{ε}\underline{M̂}_{B}) ⊗_{B} B′.

Proof. Both claims are straightforward from the isomorphism (7.1).

Let \underline{M̂} = (M̂, τM̂) be a torsion local shtuka equipped with a B-action that commutes with τM̂, and assume that Ť_{ε}\underline{M̂} is free over B. Note that it does not necessarily follow that \underline{M̂} is a torsion local B-shtuka; indeed, if B is non-reduced then one can create an example where M̂ is not free over R[[z]] ⊗_{A_{ε}} B by the same approach as in [Kim11, Remark 1.6.3]. On the other hand, we have the following lemma:

Lemma 7.24. Let 𝔽 be a finite extension of 𝔽_{ε}. Then any local shtuka \underline{M̂} equipped with an 𝔽-action is a torsion local 𝔽-shtuka; i.e., the underlying module M̂ is free over R[[z]]_{𝔽}.

Proof. Since the assertion can be checked after increasing the residue field of R, we may assume that R contains 𝔽. Since z = 0 in 𝔽 we have R[[z]]_{𝔽} ≅ R ⊗_{𝔽_{ε}} 𝔽 ≅ ∏_{i=0}^{[𝔽:𝔽_{ε}]−1} R^{(i)} where R^{(i)} is R viewed as an 𝔽-algebra via σ̂^{i} : 𝔽 → R. Likewise, we obtain the isotypic decomposition M̂ = ∏_{i} M̂^{(i)} for any local shtuka \underline{M̂} = (M̂, τM̂) with 𝔽-action. Since M̂ is ζ-torsion free, each factor M̂^{(i)} is free over R^{(i)}. It remains to show that the R^{(i)}-rank of M̂^{(i)} is constant, which follows since τM̂ restricts to an

isomorphism σ̂^{∗}M̂^{(i−1)}[\frac{1}{z−ζ}] ⟶^{∼} M̂^{(i)}[\frac{1}{z−ζ}] for any 1 ⩽ i ⩽ [𝔽 : 𝔽_{ε}] − 1.

8 Deformation theory of Galois representations

Mazur [Maz89] has introduced the notion of Galois deformation rings (both for number fields and p-adic local fields), as an attempt to see the counterpart on the Galois representation side of p-adic deformations of modular forms; cf. Hida theory. Since Wiles’s proof of Fermat’s last theorem, which opened up its application to modularity of Galois representations, Galois deformation theory has become an indispensable technical tool in number theory.

Among the important actors in modularity lifting theorems are p-adic local deformation rings with a certain condition in terms of Fontaine’s theory, such as flat deformation rings, crystalline deformation rings and potentially semi-stable deformation rings. In the second author’s thesis (cf. [Kim11, §4]), it was shown that there exists an equi-characteristic analog of flat deformation rings (or crystalline deformation rings) by working with torsion local shtukas and Hodge-Pink theory instead of finite flat group schemes and Fontaine’s theory. The existence of such equi-characteristic deformation rings is quite surprising because without imposing any deformation condition, the deformation functor would have infinite-dimensional tangent space; cf. Remark 8.2. Indeed, the main result we obtain is the finiteness of the tangent space when we impose a suitable deformation condition in terms of torsion local shtuka models.

In this section, we indicate the main idea for the existence of the equi-characteristic deformation rings, and list some of their properties, such as smoothness and dimension, which are analogous to Kisin’s study of flat deformation rings; cf. [Kis09a].

Throughout this section, we assume that K is a finite extension of Q_{ε}. We fix a finite extension 𝔽 of 𝔽_{ε} and a continuous homomorphism

ρ̄: Gal(K^{sep}/K) → GL_{r}(𝔽).

Let Art_{𝔽[[z]]} denote the category of Artinian local 𝔽[[z]]-algebras with residue field 𝔽.

Definition 8.1. For B ∈ Art_{𝔽[[z]]}, a framed deformation of ρ̄ over B is a continuous homomorphism ρ_{B}: Gal(K^{sep}/K) → GL_{r}(B) which reduces to ρ̄ modulo the maximal ideal 𝔪_{B} of B. For any morphism B → B′ in Art_{𝔽[[z]]}, we define the scalar extension ρ_{B′} of a framed deformation ρ_{B} to be the composite

ρ_{B′}: Gal(K^{sep}/K) ⟶^{ρ_{B}} GL_{r}(B) → GL_{r}(B′).

Let D□ ; ρ̄ denote the set-valued functor on Art_{𝔽[[z]]}, where D□ ; ρ̄(B) is the set of framed deformations of ρ̄ over B, and for any morphism B → B′ in Art_{𝔽[[z]]} the map D□ ; ρ̄(B) → D□ ; ρ̄(B′) is defined by the scalar extension ρ_{B} ↦ ρ_{B′}.

A deformation of ρ̄ over B is an equivalence class of framed deformations of ρ̄ over B, where two framed deformations ρ_{B} and ρ′_{B} are equivalent if and only if there exists g ∈ ker(GL_{r}(B) → → GL_{r}(𝔽)) which conjugates ρ_{B} to ρ′_{B}. Let Dρ̄ denote a set-valued functor on Art_{𝔽[[z]]}, where Dρ̄(B) is the set of deformations of ρ̄ over B.

Remark 8.2. If K is replaced by a finite extension of ℚ_{p}, then it is a well-known result of Mazur that D□ ; ρ̄ is pro-representable by a complete local noetherian ring (and Dρ̄ is pro-representable under some restriction on ρ̄). For equi-characteristic K, on the other hand, these functors cannot be prorepresented by a complete local noetherian ring because the reduced tangent spaces Dρ̄(𝔽[u]/(u^{2})) and D□ ; ρ̄(𝔽[u]/(u^{2})) are infinite-dimensional. Indeed, we have Dρ̄(𝔽[u]/(u^{2})) ≅ H^{1}(K, ρ̄ ⊗ ρ̄^{∨}) by [Maz89, p. 391], and the Galois cohomology group is infinite-dimensional over 𝔽, because if \overline{1}: Gal(K^{sep}/K) → GL_{1}(𝔽) is the trivial representation H^{1}(K, \overline{1}) ≅ Hom(Gal(K^{sep}/K), 𝔽) is infinite-dimensional (by Artin-Schreier theory; see [NSW08, § VI.1]) and it maps to H^{1}(K, ρ̄ ⊗ ρ̄^{∨}) with finite dimensional kernel, by the following long exact sequence

H^{0}(K, ρ̄ ⊗ ρ̄^{∨}/\overline{1}) ⟶ H^{1}(K, \overline{1}) ⟶ H^{1}(K, ρ̄ ⊗ ρ̄^{∨}) .

Definition 8.3. Let d be a non-negative integer. We define a subfunctor D□,⩽d ; ρ̄ of D□ ; ρ̄ so that for ρ_{B} ∈ D□ ; ρ̄(B), we have ρ_{B} ∈ D□,⩽d ; ρ̄(B) if and only if ρ_{B} admits an effective torsion local shtuka model with height ⩽ d as a torsion A_{ε}[Gal(K^{sep}/K)]-module; in other words, there exist a torsion local shtuka \underline{M̂} satisfying (z − ζ)^{d}M̂ ⊂ τM̂(σ̂^{∗}M̂) ⊂ M̂, equipped with an A_{ε}-linear Galois-equivariant isomorphism ρ_{B} ≅ Ť_{ε}\underline{M̂}. Since the condition defining the subfunctor D□,⩽d ; ρ̄ ⊂ D□ ; ρ̄ only depends on the equivalence classes of framed deformations (i.e., deformations), we may similarly define a subfunctor

D⩽d ; ρ̄ of Dρ̄.

Theorem 8.4 (cf. [Kim11, Theorem 4.2.1]). Let K be a finite extension of Q_{ε}. Then the framed deformation functor D□,⩽d ; ρ̄ is pro-representable by a complete local noetherian 𝔽[[z]]-algebra R□,⩽d ; ρ̄, and there exists a complete local noetherian 𝔽[[z]]-algebra R⩽d ; ρ̄ which is a pro-representable hull of the deformation functor D⩽d ; ρ̄ in the sense of Schlessinger [Sch68, Definition 2.7]. If we have End_{Gal(K^{sep}/K)}(ρ̄) = 𝔽, then D⩽d ; ρ̄ is pro-representable by R⩽d ; ρ̄.

Proof. The statement is an equi-characteristic analog of [Kim11, Theorem 1.3], and the proof can be easily adapted by working with torsion local shtukas instead of torsion φ-modules over 𝔖. Let us sketch the main ideas.

Let F be a set-valued functor on Art_{𝔽[[z]]} such that F (𝔽) is a singleton. By Schlessinger’s theorem [Sch68, Theorem 2.11], the following conditions are equivalent to pro-representability of F by a complete local noetherian 𝔽[[z]]-algebra:

(H1) For any surjective map B′ → → B with length-1 kernel and a map B′′ → B in Art_{𝔽[[z]]}, the natural map h : F (B′ ×_{B} B′′) → F (B′) ×_{F (B)} F (B′′) is surjective.

(H2) In the setting of (H1), h is bijective if B = 𝔽 and B′ = 𝔽[u]/(u^{2}). (This implies that F (𝔽[u]/(u^{2})) is an 𝔽-vector space; cf. [Sch68, Lemma 2.10].)

(H3) Under (H2), the 𝔽-vector space F (𝔽[u]/(u^{2})) is finite-dimensional.

(H4) For any surjective map B′ → → B with length-1 kernel, the natural map h : F (B′ ×_{B} B′) → F (B′) ×_{F (B)} F (B′) is bijective.

Furthermore, the existence of a pro-representable hull of F is equivalent to the conditions (H1)–(H3).

The proof of the pro-representability result in [Maz89, §1.2] actually shows that D□ ; ρ̄ and Dρ̄ satisfy (H1) and (H2), and (H4) is satisfied by D□ ; ρ̄ for any ρ̄ and by Dρ̄ if End_{Gal(K^{sep}/K)}(ρ̄) = 𝔽. Since the condition defining the sub-functors D□,⩽d ; ρ̄ and D⩽d ; ρ̄ is stable under finite direct products and subquotients (by Proposition 7.15), one can check without difficulty that D□,⩽d ; ρ̄ (respectively, D⩽d ; ρ̄) satisfies (H1), (H2), or (H4) if and only if the same holds for D□ ; ρ̄ (respectively, for Dρ̄).

It remains to verify (H3) for D□,⩽d ; ρ̄ and D⩽d ; ρ̄, which is the key step. Indeed, it suffices to show that D⩽d ; ρ̄(𝔽[u]/(u^{2})) is a finite set, since D⩽d ; ρ̄(𝔽[u]/(u^{2})) is the quotient of D□,⩽d ; ρ̄(𝔽[u]/(u^{2})) by the natural action of ĜL_{r}(𝔽[u]/(u^{2})) := ker[GL_{r}(𝔽[u]/(u^{2})) → GL_{r}(𝔽)]; cf. the definition of deformations as equivalence classes of framed deformations in Definition 8.1. The finiteness of D⩽d ; ρ̄(𝔽[u]/(u^{2})) can be obtained by repeating the argument in [Kim11, §1.6]. The main ideas behind the argument can be described roughly as follows:

(a) Any ρ ∈ D⩽d ; ρ̄(𝔽[u]/(u^{2})) admits an effective torsion local shtuka model \underline{M̂} with height ⩽ d with an action of 𝔽[u]/(u^{2}), which induces the correct scalar action on ρ; indeed, viewing ρ as a torsion A_{ε}[Gal(K^{sep}/K)]-module, the maximal torsion local shtuka model \underline{M̂}^{+} of ρ with height ⩽ d (cf. Lemma 7.19) inherits the action of 𝔽[u]/(u^{2}) by Lemma 7.19(b).

(b) Considering only the 𝔽-action on \underline{M̂}, it follows that \underline{M̂} is a torsion local 𝔽-shtuka by Lemma 7.24. Then by the 𝔽[u]/(u^{2}) action, \underline{M̂} fits in the following short exact sequence of torsion local 𝔽shtukas:

0 → \underline{M̂}_{1} → \underline{M̂} → \underline{M̂}_{0} → 0,

where \underline{M̂}_{0} and \underline{M̂}_{1} are effective torsion local 𝔽-shtuka models of ρ̄ with height ⩽ d, and there exists an 𝔽-linear map u : \underline{M̂}_{0} → \underline{M̂}_{1} of torsion local shtuka models of ρ̄ (induced by the action of u).

(c) By Lemma 7.19, there are only finitely many choices of \underline{M̂}_{0} and \underline{M̂}_{1}. (Recall that K is discretely valued.) Therefore, it suffices to show that for any fixed \underline{M̂}_{0} and \underline{M̂}_{1} as above there exist only finitely many extensions of \underline{M̂}_{0} by \underline{M̂}_{1} as effective torsion local 𝔽-shtukas with height ⩽ d. This finiteness assertion can be read off from [Kim11, §1.6.9ff ]. Note that the argument uses the finiteness of the residue field k of K as the extensions naturally form a k-vector space.

This concludes the proof.

Remark 8.5. One can define a notion of “ε-torsion A-motives” with good reduction at ε (over certain open subschemes U of a finite cover of the curve C from 1.1) in an analogous way to torsion local shtukas over R. It is possible to formulate a global deformation problem (with the height condition at ε analogous to D⩽d ; ρ̄) and prove a suitable global analog of Theorem 8.4.

For what follows, let us assume that End_{Gal(K^{sep}/K)}(ρ̄) = 𝔽 so that R⩽d ; ρ̄ pro-represents D⩽d ; ρ̄. Since R⩽d ; ρ̄ is a complete local noetherian 𝔽[[z]]-algebra, it is possible to associate to it a rigid analytic variety (Spf R⩽d ; ρ̄)^{rig} over 𝔽((z)), where ( .)^{rig} denotes the rigid analytic generic fiber. Keeping the analogy between p-adic crystalline representations and equi-characteristic crystalline representations, (Spf R⩽d ; ρ̄)^{rig} can be viewed as an equi-characteristic analog of a p-adic crystalline deformation space. Now we are going to prove some properties of equi-characteristic crystalline deformation spaces (such as smoothness and the dimension formula), which will strengthen the analogy with p-adic crystalline deformation spaces. If D⩽d ; ρ̄ is not pro-representable, then we can work with the framed deformation ring R□,⩽d ; ρ̄ instead of R⩽d ; ρ̄.

Let ρ^{⩽d} : Gal(K^{sep}/K) → GL_{r}(R⩽d ; ρ̄) denote (a representative of) the universal deformation (obtained by taking the limit of the deformations over Artinian quotients of R⩽d ; ρ̄). Then given a finitedimensional 𝔽((z))-algebra B and a continuous 𝔽[[z]]-homomorphism f_{B} : R⩽d ; ρ̄ → B, we obtain

f_{B}^{∗}(ρ^{⩽d}) : Gal(K^{sep}/K) ⟶^{ρ^{⩽d}} GL_{r}(R⩽d ; ρ̄) ⟶^{f_{B}} GL_{r}(B).

Let E be a finite extension of 𝔽((z)) with valuation ring O_{E}, and fix a continuous 𝔽[[z]]-morphism f_{E} : R⩽d ; ρ̄ → E. We set ρ_{E} := f_{E}^{∗}(ρ^{⩽d}). Since f_{E} factors through O_{E} by compactness, ρ_{E} comes equipped with a distinguished Galois-stable O_{E}-lattice.

Lemma 8.6. Let B be an Artinian local 𝔽((z))-algebra with residue field E, and let f_{B} : R⩽d ; ρ̄ → B be a continuous 𝔽[[z]]-homomorphism. Then there exists an effective local shtuka \underline{M̂} with (z − ζ)^{d}M̂ ⊂ τM̂(σ̂^{∗}M̂) ⊂ M̂, with a Q_{ε}[Gal(K^{sep}/K)]-isomorphism f_{B}^{∗}(ρ^{⩽d}) ≅ V̌_{ε}\underline{M̂}.

Conversely, let ρ_{B} : Gal(K^{sep}/K) → GL_{r}(B) be a lift of ρ_{E} such that for some effective local shtuka \underline{M̂} with (z − ζ)^{d}M̂ ⊂ τM̂(σ̂^{∗}M̂) ⊂ M̂, there exists a Q_{ε}[Gal(K^{sep}/K)]-isomorphism ρ_{B} ≅ V̌_{ε}\underline{M̂}. Then there exists a unique continuous 𝔽[[z]]-homomorphism f_{B} : R⩽d ; ρ̄ → B lifting f_{E}, such that there exists an isomorphism ρ_{B} ≅ f_{B}^{∗}(ρ^{⩽d}) which reduces to the identity map on ρ_{E}.

Proof. For the first claim, observe that by the usual compactness argument one can find an 𝔽[[z]]- subalgebra B_{0} ⊂ B, finitely generated as an 𝔽[[z]]-module, such that B_{0}[\frac{1}{z}] = B and f_{B} factors through B_{0}. By construction, f_{B}^{∗}(ρ^{⩽d}) has a distinguished Galois-stable B_{0}-lattice, whose Artinian quotients admit effective torsion local shtuka models with height ⩽ d. We now obtain the first claim by applying Lemma 7.20.

For the converse, let B^{∘} ⊂ B denote the preimage of O_{E}. Since ρ_{E} factors through GL_{r}(O_{E}), it follows that ρ_{B} factors through GL_{r}(B^{∘}). Since B^{∘} is a directed union of 𝔽[[z]]-sub-algebras of B which are finitely generated as 𝔽[[z]]-modules, the usual compactness argument shows that there exists an 𝔽[[z]]-subalgebra B_{0} ⊂ B, finitely generated as a 𝔽[[z]]-module, such that B_{0}[\frac{1}{z}] = B and ρ_{B} is factored by ρ_{B_{0}} : Gal(K^{sep}/K) → GL_{r}(B_{0}).

Now viewing ρ_{B} as a Galois representation over Q_{ε} isomorphic to V̌_{ε}\underline{M̂}, its B_{0}-lattice ρ_{B_{0}} yields an A_{ε}-lattice T′ ⊂ V̌_{ε}\underline{M̂}. By Proposition 4.17 and Lemma 2.5 there exists an effective local shtuka \underline{M̂}′ isogenous to \underline{M̂} with (z − ζ)^{d}M̂′ ⊂ τM̂′(σ̂^{∗}M̂′) ⊂ M̂′ with an isomorphism ρ_{B_{0}} ≅ Ť_{ε}\underline{M̂}′. By applying the universal property of R⩽d ; ρ̄ to the mod z^{n} reduction of ρ_{B_{0}} for each n, we obtain a continuous 𝔽[[z]]-morphism f_{B_{0}} : R⩽d ; ρ̄ → B_{0} giving rise to ρ_{B_{0}}. By composing f_{B_{0}} with the natural inclusion B_{0} ↪ B, we obtain the desired f_{B}.

Let us make a digression to the z-isocrystal with Hodge-Pink structure associated to an equicharacteristic crystalline representation “with coefficients in B”. Let B be a finite-dimensional Q_{ε}- algebra, and let ρ_{B} ∈ Rep_{B} Gal(K^{sep}/K) be a representation which is equi-characteristic crystalline if viewed as a Galois representation over Q_{ε}, that is ρ_{B} ≅ V̌_{ε}\underline{M̂} as Q_{ε}[Gal(K^{sep}/K)]-modules for a local shtuka \underline{M̂} over R. By full faithfulness of V̌_{ε} (Theorem 4.15), there exists a local shtuka \underline{M̂} := (M̂, τM̂), unique up to quasi-isogeny, such that B acts on \underline{M̂} by quasi-morphism and we have a B[Gal(K^{sep}/K)]- isomorphism ρ_{B} ≅ V̌_{ε}\underline{M̂}.

Lemma 8.7. In the above setting, M̂[\frac{1}{z}] is free over R[[z]]_{B} := R[[z]] ⊗_{A_{ε}} B with rank equal to rk_{B} ρ_{B}.

Proof. One can repeat the proof of [Kis08, Proposition 1.6.1] via Fitting ideals, with 𝔖 and E(u) replaced with R[[z]] and z − ζ.

Corollary 8.8. Let ρ_{B} and \underline{M̂} be as above. We define ℍ(ρ_{B}) := ℍ(\underline{M̂}) := (D, τ_{D}, 𝔮_{D}), where we give the unique rigidification to \underline{M̂}; cf. Lemma 5.8. Then the admissible z-isocrystal with Hodge-Pink structure ℍ(ρ_{B}) enjoys the following additional properties:

(a) D is free over k((z))_{B} := k((z)) ⊗_{Q_{ε}} B and τ_{D} is B-linear.

(b) 𝔮_{D} ⊂ D⊗_{k((z))} K((z −ζ)) is a finitely generated free submodule over K[[z −ζ]]_{B} := K[[z −ζ]]⊗_{Q_{ε}} B.

Proof. This corollary follows directly from Lemma 8.7 and the construction of D and 𝔮_{D}; cf. (5.6).

Assume that End_{Gal(K^{sep}/K)}(ρ̄) = 𝔽, and fix f_{E} : R⩽d ; ρ̄ → E for some finite extension E of 𝔽((z)). Set ρ_{E} := f_{E}^{∗}(ρ^{⩽d}) and ℍ(ρ_{E}) := (D_{E}, τ_{D_{E}}, 𝔮_{D_{E}}), and write

ℍ(ρ_{E}) ⊗ ℍ(ρ_{E})^{∨} = (Ad(D_{E}), τ_{Ad(D_{E})}, 𝔮_{Ad(D_{E})}) ,

with 𝔭_{Ad(D_{E})} denoting the tautological lattice; cf. Definition 5.2. Identifying Ad(D_{E})⊗_{K[[z]]_{E}} K((z−ζ))_{E} with End_{K((z−ζ))_{E}}(D_{E} ⊗_{K((z))_{E}} K((z − ζ))_{E}) we can describe 𝔭_{Ad(D_{E})} and 𝔮_{Ad(D_{E})} as follows:

𝔭_{Ad(D_{E})} = End_{K[[z−ζ]]_{E}}(𝔭_{D_{E}}) and 𝔮_{Ad(D_{E})} = End_{K[[z−ζ]]_{E}}(𝔮_{D_{E}});

in other words, as a K[[z − ζ]]_{E}-lattice of End_{K((z−ζ))_{E}}(D_{E} ⊗_{K((z))_{E}} K((z − ζ))_{E}), 𝔭_{Ad(D_{E})} (respectively, 𝔮_{Ad(D_{E})}) consists of endomorphisms preserving 𝔭_{D_{E}} (respectively, 𝔮_{D_{E}}).

Theorem 8.9. In the above setting, the completion of R⩽d ; ρ̄ ⊗_{𝔽((z))} E with respect to the kernel of f_{E} ⊗1 is isomorphic to a formal power series ring over E of dimension

1 + dim_{E} \frac{𝔭_{Ad(D_{E})}}{𝔭_{Ad(D_{E})} ∩ 𝔮_{Ad(D_{E})}}

(This is a finite number since K is a finite extension of Q_{ε}.) In particular, (Spf R⩽d ; ρ̄)^{rig} is a smooth rigid analytic variety over 𝔽((z)).

Remark 8.10. If we do not assume that End_{Gal(K^{sep}/K)}(ρ̄) = 𝔽, one can still show that (Spf R□,⩽d ; ρ̄)^{rig} is a smooth rigid analytic variety over 𝔽((z)), and its dimension at an E-point f_{E} : R□,⩽d ; ρ̄ → E is

r^{2} + dim_{E} \frac{𝔭_{Ad(D_{E})}}{𝔭_{Ad(D_{E})} ∩ 𝔮_{Ad(D_{E})}} ,

with the obvious notation, where r = dim_{𝔽} ρ̄ = dim_{E} ρ_{E}. This formula is compatible with Theorem 8.9 if End_{Gal(K^{sep}/K)}(ρ̄) = 𝔽, since the natural morphism Spf R□,⩽d ; ρ̄ → Spf R⩽d ; ρ̄ is formally smooth of relative dimension r^{2} − 1. (In fact, formal smoothness is clear, and to obtain the relative dimension it suffices to look at the tangent spaces. Since we have End_{Gal(K^{sep}/K)}(ρ̄) = 𝔽, the stabilizer of the natural ĜL_{r}(𝔽[u]/(u^{2}))-action on D□,⩽d ; ρ̄(𝔽[u]/(u^{2})) is the subgroup of scalar matrices 𝔾̂_{m}(𝔽[u]/(u^{2})), which shows that D□,⩽d ; ρ̄(𝔽[u]/(u^{2})) is a { \atop PGL}_{r}(𝔽[u]/(u^{2}))-torsor over D⩽d ; ρ̄(𝔽[u]/(u^{2})). Since { \atop PGL}_{r}(𝔽[u]/(u^{2})) ≅ 𝔭𝔤𝔩_{r}(𝔽), where the latter is an (r^{2} −1)-dimensional 𝔽-vector space, we obtain the desired numerology.)

This result should be thought of as an equi-characteristic analog of the formal smoothness and the dimension formula for the (rigid analytic) generic fibers of crystalline deformation rings; cf. the case with N = 0 of [Kis08, §3]. Note also that in the dimension formula, we have terms involving the Hodge-Pink structure associated to ρ_{E} instead of the Hodge-Pink filtration, which supports that the Hodge-Pink structure is the right equi-characteristic analog of the Hodge filtration associated to a p-adic crystalline representation. (Compare with [Kis08, Theorems 3.3.4, 3.3.8].)

Example 8.11. Assume that 𝔽 = 𝔽_{ε} so that R[[z]]_{𝔽} = R. Consider a local shtuka \underline{M̂} where M̂ = R[[z]]^{2} and τM̂ is given by the matrix ( \begin{matrix}1 & 1 \\ 0 & (z−ζ)^{d} \end{matrix} ). Then the isomorphism δM̂ from Lemma 5.8 equals ( \begin{matrix}1 & g \\ 0 & ℓ_{−}^{−d} \end{matrix} )

for some g ∈ R[[z, z^{−1}\}[ℓ−1_{−}] and so 𝔮_{D} = 𝔭_{D} + (z − ζ)^{−d} ( \begin{matrix}σ̂(g)−1 \\ σ̂(ℓ_{−})^{−d} \end{matrix} ) K[[z − ζ]]. One can check that

ρ̄:= Ť_{ε}\underline{M̂} ⊗_{A_{ε}} 𝔽_{ε} is a non-split extension of (χ_{ε})^{−d} mod z by the trivial character. Therefore, we have End_{Gal(K^{sep}/K)}(ρ̄) = 𝔽 if and only if q̂ − 1 does not divide d, which we assume; cf. Example 7.14.

Clearly, Ť_{ε}\underline{M̂} defines an A_{ε}-point f : R⩽d ; ρ̄ → A_{ε}. By Theorem 8.9, this implies that the dimension of (Spf R⩽d ; ρ̄)^{rig} at f is given by

1 + dim_{Q_{ε}} (K[[z − ζ]]/(z − ζ)^{d}) = 1 + d · [K : Q_{ε}],

because End_{K[[z−ζ]]}(𝔭_{D}) = K[[z − ζ]]^{2×2} and End_{K[[z−ζ]]}(𝔭_{D}) ∩ End_{K[[z−ζ]]}(𝔮_{D}) =

\{ ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ K[[z − ζ]]^{2×2} : b ≡ c(σ̂(g) − 1)^{2}σ̂(ℓ_{−})^{2d} + (d − a)(σ̂(g) − 1)σ̂(ℓ_{−})^{d} mod (z − ζ)^{d} \}.

Let us begin the proof of Theorem 8.9. Let B be an Artinian local E-algebra with residue field E, and let I ⊂ B be an ideal annihilated by the maximal ideal of B. We fix a lift f_{B/I} : R⩽d ; ρ̄ → B/I of f_{E}, and set ρ_{B/I} := f_{B/I}^{∗}(ρ^{⩽d}).

In order to study a lift of f_{B/I} to a B-point, we introduce the following complex concentrated in degrees 0 and 1:

C^{•}(ρ_{E}) := [Ad(D_{E}) ⟶^{(τ_{D_{E}} −1,incl)} Ad(D_{E}) ⊕ \frac{𝔭_{Ad(D_{E})}}{𝔭_{Ad(D_{E})} ∩ 𝔮_{Ad(D_{E})}} ] .  (8.1)

Let H^{0}(ρ_{E}) and H^{1}(ρ_{E}) respectively denote the 0th and the 1st cohomology of C^{•}(ρ_{E}). Clearly, H^{0}(ρ_{E}) is naturally isomorphic to the E-vector space of E-linear endomorphisms of ℍ(ρ_{E}), which is isomorphic to End_{Gal(K^{sep}/K)}(ρ_{E}) by full faithfulness results (Theorem 4.15, Proposition 5.12).

Lemma 8.12. The set of maps f_{B} : R⩽d ; ρ̄ → B lifting f_{B/I} is a principal homogeneous space under H^{1}(ρ_{E}) ⊗_{E} I; in particular, there exists a lift f_{B} of f_{B/I}.

Proof. By filtering I if necessary, it suffices to show the lemma when I is 1-dimensional as an E-vector space, which we assume from now on. Let u ∈ I denote a principal generator.

Let us first show the existence of a lift f_{B}. Write ℍ(ρ_{B/I}) = (\overline{D}, τ_{\overline{D}}, \overline{𝔮}). Since \overline{D} is free over k((z))_{B/I} by Corollary 8.8, we pick a free k((z))_{B}-module D that lifts \overline{D} and lift τ_{\overline{D}} arbitrarily to τ_{D}. Note that (D, τ_{D}) is a z-isocrystal. Since \overline{𝔮} is free over K[[z − ζ]]_{B/I} by Corollary 8.8, we choose a free K[[z − ζ]]_{B}-module 𝔮 lifting \overline{𝔮} and an injective map 𝔮 ↪ (z − ζ)^{−d}𝔭 lifting the natural inclusion \overline{𝔮} ↪ (z − ζ)^{−d}\overline{𝔭}. (Here, \overline{𝔭} and 𝔭 respectively denote the tautological lattices for \overline{D} and D.) We set \underline{D} := (D, τ_{D}, 𝔮).

Let us first show that \underline{D} is admissible. Indeed, choosing an E-basis u ∈ I, we have an isomorphism ℍ(ρ_{E}) ^{∼} ⟶ \underline{D} ⊗_{B} I (with the obvious notation), so we have the following short exact sequence of z-isocrystals with Hodge-Pink structure:

0 → ℍ(ρ_{E}) → \underline{D} → ℍ(ρ_{B/I}) → 0.

Now, one can show that any extension of admissible z-isocrystals with Hodge-Pink structure is again admissible, using the characterization of admissibility in terms of the associated σ̂-bundle F(\underline{D}) (Theorem 6.10(a)) and the classification of σ̂-bundles (Theorem 6.7(a)).

Let \underline{M̂} denote a local shtuka with ℍ(\underline{M̂}) ≅ \underline{D}, which exists by admissibility. By construction, all the Hodge-Pink weights of \underline{M̂} are between 0 and d, so \underline{M̂} is effective with (z −ζ)^{d}M̂ ⊂ τM̂(σ̂^{∗}M̂) ⊂ M̂. By full faithfulness (cf. Lemma 5.8 and Proposition 5.12), B acts on \underline{M̂} by quasi-morphism, so V̌_{ε}(\underline{M̂}) has a natural B-action commuting with the Galois action. On the other hand, V̌_{ε}(\underline{M̂}) is a free Bmodule lifting ρ_{B/I} by comparing the E-dimension and the minimal number of generators obtained by the Nakayama lemma. By Lemma 8.6 it follows that V̌_{ε}(\underline{M̂}) defines a B-point of R⩽d ; ρ̄ lifting f_{B/I}.

Let us now show that the set of f_{B} lifting f_{B/I} is a principal homogeneous space under H^{1}(ρ_{E}) ≅ H^{1}(ρ_{E}) ⊗_{E} I. Let \underline{D} be as above. Given γ ∈ Ad(D_{E}) ≅ End_{k((z))_{E}}(D_{E}) and δ ∈ 𝔭_{Ad(D_{E})} ≅ End_{K[[z−ζ]]_{E}}(𝔭_{D_{E}}), we obtain another z-isocrystal with Hodge-Pink structure as follows:

\underline{D}_{(γ,δ)} := (D, (1 + uγ)τ_{D}, (1 + uδ)𝔮).

By repeating the previous argument for \underline{D}_{(γ,δ)} instead of \underline{D}, it follows that \underline{D}_{(γ,δ)} defines a B-lift of f_{B/I}, thus we obtain an E-linear transitive action of Ad(D_{E}) ⊕ 𝔭_{Ad(D_{E})} on the set of B-lifts of f_{B/I}. (Transitivity follows from Corollary 8.8.) One can easily check that \underline{D}_{(γ,δ)} and \underline{D}_{(γ′,δ′)} define the same B-lift of f_{B/I} if and only if (γ, δ) and (γ′, δ′) define the same class in H^{1}(ρ_{E}). This shows the desired claim.

Proof of Theorem 8.9. Lemma 8.12 shows that the completion of R⩽d ; ρ̄ ⊗_{𝔽[[z]]} E at the maximal ideal corresponding to f_{E} is a formal power series ring of dimension equal to dim_{E} H^{1}(ρ_{E}). It remains to compute dim_{E} H^{1}(ρ_{E}). Note that

dim_{E} H^{0}(ρ_{E}) − dim_{E} H^{1}(ρ_{E}) = dim_{E} Ad(D_{E}) − (dim_{E} Ad(D_{E}) + dim_{E} \frac{𝔭_{Ad(D_{E})}}{𝔭_{Ad(D_{E})} ∩ 𝔮_{Ad(D_{E})}} ) ,

and H^{0}(ρ_{E}) ≅ End_{Gal(K^{sep}/K)}(ρ_{E}) is 1-dimensional since End_{Gal(K^{sep}/K)}(ρ_{𝔽}) ≅ 𝔽. This shows that dim_{E} H^{1}(ρ_{E}) has the expected dimension.

By construction of rigid analytic generic fibers (cf. [dJ95, §7.1]), (Spf R⩽d ; ρ̄)^{rig} is the direct union of affinoid rigid analytic spaces Sp B_{n}, where B_{n} is the completion of R⩽d ; ρ̄ ⊗_{𝔽[[z]]} 𝔽((z)) with respect to the norm such that R⩽d ; ρ̄[\frac{𝔪^{n}}{z}] is the set of norm ⩽ 1 elements in R⩽d ; ρ̄ ⊗_{𝔽[[z]]} 𝔽((z)). Since \{Sp B_{n}\} is an admissible covering of (Spf R⩽d ; ρ̄)^{rig}, it remains to show that each B_{n} is a topologically smooth affinoid algebra over 𝔽((z)). Since B_{n} is Jacobson [BGR84, §5.2.6, Theorem 3] and all ideals of B_{n} are closed [BGR84, §5.2.7, Corollary 2], it suffices to show that the completion of B_{n} at any closed maximal ideal (i.e., classical points of Sp B_{n}) is geometrically regular. (Note that the invertibility of Jacobian can be captured by the completions at a dense set of prime ideals.)

By [dJ95, Lemma 7.1.9], any (closed) maximal ideal of B_{n} restricts to a maximal ideal of R⩽d ; ρ̄ ⊗_{𝔽[[z]]} 𝔽((z)), and the completion of B_{n} at a maximal ideal is isomorphic to the completion of R⩽d ; ρ̄ ⊗_{𝔽[[z]]} 𝔽((z)) at the corresponding maximal ideal. Since Lemma 8.12 shows that the local ring of R⩽d ; ρ̄ ⊗_{𝔽[[z]]} 𝔽((z)) at any maximal ideal is geometrically regular, we obtain the desired smoothness claim.

We end the section with discussing the equi-characteristic analog of moduli of finite flat group schemes [Kis09a]. We first need the following definition:

Definition 8.13. Let B be an A_{ε}-algebra with #B < ∞ (not a finite Q_{ε}-algebra), and let T_{B} be a finitely generated free B-module equipped with a discrete action of Gal(K^{sep}/K). Then a torsion local B-shtuka model of T_{B} is a torsion local B-shtuka \underline{M̂}_{B} equipped with a B[Gal(K^{sep}/K)]-isomorphism T_{B} ≅ Ť_{ε}\underline{M̂}_{B}.

We take the obvious notion of equivalence for torsion local B-shtuka models of T_{B}. We can understand an equivalence class of torsion local B-shtuka models of T_{B} as a certain R[[z]]_{B}-lattice of T_{B} ⊗_{A_{ε}} K^{sep}[[z]] stable under 1 ⊗ σ̂ and invariant under the Galois action, using the isomorphism (4.1).

Theorem 8.14 ([Kim11, Proposition 4.3.1]). We fix ρ̄: Gal(K^{sep}/K) → GL_{d}(𝔽), and for simplicity we assume that End_{Gal(K^{sep}/K)}(ρ̄) ≅ 𝔽. Then there exists a projective scheme GR⩽d ; ρ̄ over Spec R⩽d ; ρ̄ with the following property. For any local R⩽d ; ρ̄-algebra B with #B < ∞, Hom_{R⩽d ; ρ̄}(Spec B, GR⩽d ; ρ̄) is

functorially bijective with the set of equivalence classes of torsion local B-shtuka models of ρ^{⩽d} ⊗_{R⩽d ; ρ̄} B which are effective and of height ⩽ d.

Proof. Indeed, GR⩽d ; ρ̄ can be represented by a closed subscheme of a certain affine Grassmannian for GL_{r}, following the same idea as Kisin’s construction of moduli of finite flat group schemes; cf. [Kis09a, Proposition 2.1.10].

Remark 8.15. In the p-adic setting, Kisin’s construction of moduli of finite flat group schemes over flat deformation rings was “globalized” by Pappas and Rapoport; cf. [PR09]. Namely, the mod p local Galois representation ρ̄ is allowed to vary in the moduli space constructed by Pappas and Rapoport, and we recover moduli of finite flat group schemes by “fixing ρ̄”. Such “coefficient spaces” (which can be thought of as fixing a base of p-divisible groups and varying coefficients) bear somewhat striking similarities with moduli spaces of p-divisible groups (with fixed coefficients ℤ_{p} and varying its base), commonly known as Rapoport-Zink spaces. For example, there exists a natural “period-morphisms”

[PR09, §5] on the rigid analytic generic fiber of a Pappas-Rapoport coefficient space whose image was computed by Hellmann and the first author; see [Hel13, Theorem 7.8] and [HH13, Corollary 6.12].

By repeating the construction in [PR09] or [HH13] by working with A_{ε} instead of ℤ_{p}, one can obtain an equi-characteristic analog of Pappas-Rapoport coefficient spaces, which can roughly be thought of as the moduli space of local shtukas with fixed base R and varying coefficients. Such equi-characteristic coefficient spaces could be interesting objects to study; for example, one can ask about a description of the image of the rigid analytic period morphism, analogous to [Hel13, HH13].

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Urs Hartl

Universität Münster

Mathematisches Institut

Einsteinstr. 62

D – 48149 Münster

Germany

www.math.uni-muenster.de/u/urs.hartl/

Wansu Kim

Department of Mathematics King’s College London Strand London, WC2R 2LS United Kingdom

www.mth.kcl.ac.uk/∼wansu/